HPSCHD-L discussions
This is a digest of my responses to various questions about tuning, in the online forum HPSCHD-L.
In each posting I try to quote back the point to which I am responding directly.
These are explanations and defenses of my reasoning, perspective, priorities, and methods
of testing the evidence.
Understandably, a reading of this digest may be like
listening to one side of a telephone conversation; but the other
respondents' comments are available in chronological sequence through the search function at
http://listserv.albany.edu:8080/cgi-bin/wa?S1=hpschd-l&D=1&H=0&O=D&T=0
An especially good posting (IMO) is the March 6th 2006 "What's the squiggle squabble about?",
summarizing the main thrust of discussions so far. That same date, 6 Mar 2006, also had
my presentation of Neidhardt's "fifth-circle #11" temperament from 1732, the one that features
an E-G# wider than Ab-C.
There are other relevant comments before the earliest date, too; I continue to work
chronologically backwards through the archive, toward the beginning of 2005 to coincide with the
release of the
Oxford article.
Bradley Lehman
[Feb 05]
[Mar 05]
[Apr 05]
[May 05]
[Jun 05]
[Jul 05]
[Aug 05]
[Sep 05]
[Oct 05]
[Nov 05]
[Dec 05]
[Jan 06]
[Feb 06]
[Mar 06]
[Apr 06]
[May 06]
[Jun 06]
[Jul 06]
[Aug 06]
[Sep 06]
[Oct 06]
[Nov 06]
[Dec 06]
[Back to the LaripS FAQ]
------------------------------
Date: Fri, 11 Feb 2005 11:47:08 -0500
From: Brad Lehman
Subject: JS Bach's tuning
The first half of my article "Bach's Extraordinary Temperament: Our Rosetta
Stone" is now available in the February issue of _Early Music_, and at
http://em.oupjournals.org/current.dtl
It describes Bach's specific keyboard tuning, its basic sound, and the
historical background. The second half of the article (May 2005) gets more
deeply into the musical and mathematical analysis.
I am also releasing today a new web site to accompany that article:
http://www.larips.com
That includes by-ear and electronic instructions to set up this tuning,
along with various other spin-off essays. For comparison with the Bach
sound I have included by-ear instructions to set up Vallotti, three of the
Neidhardts, and two of the Sorges.
The Oxford web site will soon include downloadable supplementary materials,
comparing Bach's with forty other temperaments.
I know that several HPSCHD-L members have already served as tuners during
the testing of this temperament, setting it for concerts by The English
Concert and Apollo's Fire. And others here have assisted me more directly
during the writing of the article. Thank you!
Enjoy!
Bradley Lehman
2/11/05
------------------------------
Date: Sat, 12 Feb 2005 09:09:17 -0500
From: Brad Lehman
Subject: Re: JS Bach's tuning CORRECTED POST
Excellent posting by Daniel Jencka, at
http://listserv.albany.edu:8080/cgi-bin/wa?A2=ind0502&L=hpschd-l&D=1&O=D&P=7861
The issue here is starting from C fork, or starting from A fork. And the
sub-issue is getting the regular 1/6 set of fifths going by listening to
some measured fifth/fourth, vs listening to some measured major third.
Indeed there are several ways to start all this, to get those first five
regular 1/6 fifths F-C-G-D-A-E.
And in practice I do use a C fork just as often as I use an A fork. In
that case I listen to the beat rate of the major third C-E instead of the
F-A (with a different rate, of course). That set of rates is stashed into
one of my footnotes.
The geometric exercise of tuning and retuning a couple of notes, to get the
fifths and fourths exactly into 2 to 3 relationship (on my "practical
instructions" web page to do this by ear), isn't due to starting from any
particular fork. Rather, it's a process of splitting things in half and
then half again to do an exact quarter "fold" of whatever major third is my
measurement, instead of three cumulative fifths/fourths. All the major
third start does for me is to set the limit exactly for the total of the
four fifths, so I'm not introducing any cumulative errors into those
regular fifths. That is, it's irrespective of which fork I've started
from. I set the endpoints first, whether one of those endpoints is a C or
an A.
All those resulting beat rates of every interval are at my "mathematical
analyses" web page there at www.larips.com, for anybody wishing to devise
some other method of getting the regular fifths (or indeed any other method
of doing the whole thing in some different sequence). For example, C down
to F as a fifth is -1.19 if we're in "A 440", or -1.12 if we're in "A
415". That's metronome 71 or 67. Might be even easier to go from middle C
up a fourth to the F, in which case it's 142 (at "A 440" from a C fork of
520whatever), or 134 (at "A 415" from a C fork that's really a B fork from
an old equal-tempered 440 set....).
Whatever. The broader point is simply to get cleanly regular 1/6 fifths
from F-C-G-D-A-E, somehow, as the ground plan. Daniel's way to do so works
well too. These C to F numbers here today supply his "X" value.
Then the whole Bach recipe (for reference), regardless of the baking steps
or sequence, is:
- F-C-G-D-A-E regular 1/6
- E-B-F#-C# pure
- C#-G#-D#-A# regular 1/12
- residual 1/12 wide A#-F
Any reasonable method to deliver that accurate result is fine! I suspect
that Bach himself just went straight across like that, at least some of the
time.
(As I noted inside the article, the only reason I do the F-Bb-Eb pure step
along my way is to stick the D# exactly where it belongs, early. That's
just my personal penchant to establish accurate endpoints wherever I can.)
Bradley Lehman
------------------------------
Date: Sun, 13 Feb 2005 15:11:21 -0500
From: Brad Lehman
Subject: cent charts, and Temperament Units etc
WRT: Ibo's chart of cent values for my proposed Bach temperament, in the
message
http://listserv.albany.edu:8080/cgi-bin/wa?A2=ind0502&L=hpschd-l&D=1&O=D&P=12264
Yes, that is a handy supplement laying out the fifths and the major/minor
thirds with cents. I deliberately did *not* use cents, for the reasons I
explained both in the article and at my web site www.larips.com (my several
pages about mathematical analysis). I presented the fifths with the
fractions of Pythagorean comma adjustments: -1/6, -1/12, +1/12, or 0. And
I presented the major and minor thirds as percentages of the *syntonic*
comma error, since that is the comma that matters in judgment of thirds.
But, it's good to see this complementary method of measurement (cents)
presented as well, for completeness. Thanks, Ibo!
There are also some 50 pages of additional mathematical analysis by me that
should be available from Oxford's web site, as soon as they get it put up
there. (I believe it will be in their free area where they normally post
recordings and facsimiles that support other articles of this journal.)
But again, I deliberately did *not* use cents in there either because I
believe they present a skewed way of thinking about the basic issues of the
comma fractions. The cents give us the inserted modern bias of equal
temperament as a norm, viewing history backwards...through the rose-colored
lens of Ellis, Barbour, et al. The whole Barbour book presents such a
skewed way of assessing temperaments, measuring their deviation from equal,
as if that's what really matters! Cents merely steamroll over the comma
relationships and turn the whole thing into a bunch of arbitrary numbers,
based on logarithmic metrics (based on 100) that do not really apply to the
18th century.
So, part of my decision here to disuse cents is obviously a
didactic/pedantic one!.... :) I myself could not see this proposed Bach
temperament clearly in my mind last spring, until I threw all the cents
stuff out the window first as too distracting, and did the ground-work
analysis with the "Temperament Units" measurement I describe along with the
percentages of commas. With *those* metrics and that geometric method of
paperwork, the whole thing became perfectly clear to me, and so I pass that
along as the standards I have employed in the paper. Mr Brombaugh is
pleased, too, that I have been able to bring his TU metrics to a wider
audience through this. Those show up especially in the Oxford
supplementary files, showing clear differentiation between syntonic comma
temperaments and Pythagorean comma temperaments.
Brad Lehman
------------------------------
Date: Sun, 13 Feb 2005 16:45:42 -0500
From: Brad Lehman
Subject: the WTC calligraphy
>When I was thinking about this ornament last year I was wondering, as
>Bach obviously marked the c, and connected it to the letter C in the
>word "Clavier", if not the D of the word "Das" could mean the position of
>the Dis (D-sharp).
Actually, the big C and the little C do not touch each other at all. They
appear to touch, in modern reproductions after the page has deteriorated
during the 20th century, but they clearly do not in the 1911 photograph for
the old Grove dictionary (see the reproduction from that on my site, in the
FAQ page). Direct address to that graphic,
http://www-personal.umich.edu/~bpl/larips/wtc-1722-from-1911-grove.gif
The article has some suggestive bits about the way there appears to be a
big "C D E" on the page among the capital letters of the title, and also a
big "C D Es" nearby....both the Ut Re Mi and the Re Mi Fa there. Note the
way that the "s" of "Es" gets slashed midway, and the equidistant dots
around that line. (This is all *in* the big putative capital D of
"Das"!) Sure enough, the temperament has some symmetries with mean
placements of the accidentals...but most of those symmetries happen
specifically around the note C#, i.e. "Cis".
That's all to suggest, the page as a whole probably has a bunch of other
calligraphic stuff in it that I don't know about yet. I have a hunch that
the bottom flourish might have something to do with the way of listening to
tempered fifths at all, on harpsichord: spiral of noise, followed by the
evening out of a regular series of beats. But, other interpretations would
also be possible. And why is that closing flourish so far off center,
unless it's merely balancing out the word funnel that has drifted too far
to the right for the page's gross symmetry?
Also, I have a remark in the article somewhere about copying the spiral as
a right-handed person (re David Pickett's question).
Here at home yesterday (with my wife and our 2-year-old) we were playing a
bit more with a printout of the page, trying things upside down and
backwards with mirrors. With or without the mirror, the whole page
upside-down looks a little bit like a Christmas tree with an elaborate
crowning ornament on it.... Then there's also the more obvious dismantled
treble clef of the big capital "P", looking at it upside-down and in a
mirror. My wife was asking yesterday if there's anything special about
JSB's signature down in the shaped portion of the words, with its
odd-looking capitals as well. I still believe that a good way to find
hidden shapes is to ask a 2-year-old, viewing it at various
distances. "Pismas pree!"
Brad Lehman
------------------------------
Date: Mon, 14 Feb 2005 08:59:25 -0500
From: Brad Lehman
Subject: Sorge 1758
Ibo asked:
>Btw, has someone a description of one of Sorge's temperaments in 1758
>(the other was equal temp.)
Yes, that Sorge 1758 is the big one featured in my article; it's also in
Lindley's "Stimmung und Temperatur" and his later Michaelstein article, and
in Dominique Devie's book _Le temperament musical_. Its recipe is on both
my page of practical by-ear instructions and my "comparisons" page, and in
the (forthcoming) Oxford web appendix to all this.
Sorge 1758 is the one that (I believe) Sorge copied off the Leipzig organs
himself, modified to make it a little closer to equal temperament than
Bach's had been (his changes to it are quite logical ones...), and then
published as his own. Again, that's explained in the first half of my article.
Brad Lehman
------------------------------
Date: Mon, 14 Feb 2005 17:38:02 -0500
From: Brad Lehman
Subject: Re: the WTC calligraphy plus Alternate Interpretation
At 12:38 PM 2/14/2005 -0800, Daniel Jencka wrote:
http://listserv.albany.edu:8080/cgi-bin/wa?A2=ind0502&L=hpschd-l&D=1&O=D&P=16795
Intriguing!
But, a few counter-questions so far:
- In your interpretation, what happens to the place where there are
supposed to be three "single" temperings consecutively? You've ended up
with only two. If you're starting the line (turned upside down) with C,
shouldn't you follow it all the way through to a tempered E# coming from
A#, at the right side? And what about the flourish out beyond that? [Or
perhaps I'm simply not understanding your presentation clearly enough,
yet.....]
- The idea of the notes being the loops rather than the valleys is
attractive; but then, why would Bach have notated the pure F and Bb (in
your interpretation) by drawing them so extraordinarily differently from
the other notes, hanging out the other side and so tiny? Wouldn't they
just be drawn as big empty loops like the others? As I suggested inside
the paper, figure 2 on page 7, I've been taking that little cauliflowery
thing with its three humps as a reminder that the F-A major 3rd is 3 beats
per second; it's the *only* constant rate that a tuner by ear really needs
to know as a memorized speed, and therefore part of Bach's diagram. But I
also like your suggestion a few days ago that maybe that part is the little
temporary step to go F-Bb-Eb to stick Eb exactly where it belongs, ahead of
time...as I described in the practical instructions. In any case, I have
not ignored that particular doodad. It means *something*.
- The important principle here, though, is twofold: (1) as you mentioned,
take *all* of the evidence, and (2) think about what Bach's other options
might have been, to draw an expression of whatever interpretation you wish
to test. That is, starting from your proposed reading today and thinking
backward into it, wouldn't Bach have found a simpler way to draw yours on
the page, representing your pure C-F-Bb fifths and the presence of only two
1/12 fifths?
- Also important here is the transposing situation, Chorton/Cammerton by a
whole step. See especially my graph at the bottom of p16 in the article,
doubly labeled as the organ would sound to the organist, vs how it would
sound to the singers and players reading their parts a step higher. In
that case, from the orchestra's perspective, F# major is the scharffest
key, and it ends up being pretty much what we'd expect from our familiarity
with the other circulating temps where the transposition Chorton/Cammerton
is not taken into account. My related discussion of that is on pp
17-18. Within that orientation, it becomes less surprising that *in solo
repertoire* the brightest key ends up being E.
- There's much more in part 2 of my article (May '05) supporting my current
interpretation. That's where we *really* get into the fun
stuff. Obviously I can't tip off all my points there ahead of time.
- Your proposed layout gets pretty close to the Neidhardts. I don't have
all of them on my desk at the moment, to see if it coincides exactly with
one of them. But, the biggest and most consistent practical flaw in the
Neidhardts that I've been noticing (playing through all of Bach's music) is
that the temperaments that start with C-F-Bb pure all end up sounding
rather raunchy in flat-key music. The accidentals end up too low to serve
really well as flats, in their enharmonic duties that are complex in tonal
music. This too is discussed in part 2.
- As for E major being an endpoint among normal tonic keys (outside the WTC
itself, I mean), like the top of a mountain or something, beyond which we
normally do not cross in the key signature, check out the Bach repertoire
for hpsi and organ. Among other things, I'm impressed by the "Big Brother
Chris" capriccio in E.
I'll look into generating one of those Beat Rate Charts for your reading,
when I get the chance. Or, pick up my old (1997) spreadsheet from
http://www-personal.umich.edu/~bpl/temper.html and plug in your layout
yourself, to get as many of those charts as you want for that and any other
layouts you want to check.
Brad Lehman
------------------------------
Date: Wed, 16 Feb 2005 10:18:32 -0500
From: Brad Lehman
Subject: Re: More Bach WTC tuning scrip interpretations: 18ths and 24ths
solutions
At 09:19 PM 2/15/2005 -0800, Daniel Jencka wrote:
>(...) I am sure that Bradley must have considered this other
>tones-in-the-valleys interpretation, which is simply that the three 5ths
>from C# to G# to D# to Bb would divide the remaining 1/6th comma into
>1/18th each. Just very slightly narrower than perfect 5ths. The last 5th
>from Bb to F would then be perfect, corresponding to that single loop at
>the very end.
>
>So that makes five 5ths of 1/6th comma each, which is the same as 3/18ths,
>and then three 5ths at 1/18th each. 18/18ths total.
>
>This interpretation makes the most sense to me because that last Bb to F
>interval in actual practice does not get set as part of the listening and
>tuning process. It is just the result of the tuning process that starts at
>the beginning with F and C. It wouldn't have to bi given in the script at
>all, but showing it as a smaller single loop completes the scheme
>theoretically.
So, why expect it to be pure, then? Wouldn't that be an astounding piece
of luck to end up with a pure fifth residually after setting up eight
tempered fifths? And, wouldn't he have drawn the diagram differently if
expecting a pure F-Bb? (Occam's Razor again.)
In fact, as explained in the paper, if one reads the whole line as a 1/13
PC division (!) since there are 13 little jots in total, it comes out in
practice being a 1/12 SC division off by only a couple of microns, and
there's indeed a residual *pure* Bb-F fifth. This was my own preferred
reading for a couple of weeks, until I was convinced that it does not suit
either the diagram or the milieu as well as the 1/12 PC reading as
presented. I have a footnote about that. My 1/13 PC reading (i.e. 1/12
syntonic comma), for completeness, *is* presented in the supplementary web
files for Oxford's site.
>(...)An interpretation applying a strictly proportional approach would
>necessarily have you divide the comma into 24 parts, because that is how
>many loops there are in total when you add them all up.. One would then
>assign 1/24th of a comma to the three single loop, 2/24ths to each of the
>three double loops, and 3/24ths to each of the triple loops.
>
>Could that be? How would one actually tune such a temperament? Could this
>be some surprising interpretation? The 2/24ths solution? Charts anyone?
That's getting into the realm of Owen Jorgensen's "Handel" temperament,
theoretically working with 1/25th PC portions and spraying them around
according to a vague set of rules.
>(...)That's it for the moment, though still wondering about how to make a
>beat chart on a Mac.
Good exercise is to develop your own spreadsheet, which lots of people have
done.
Or, did you try picking up the Mac version of OpenOffice.org (free from
Sun) and opening mine that way? I haven't tried any of this on Mac myself,
but theoretically it should go fine.... I have a link over to
OpenOffice.org on my "system requirements" section at
http://www-personal.umich.edu/~bpl/temper.html
( or http://how.to/tune )
Brad Lehman
------------------------------
Date: Wed, 16 Feb 2005 11:40:39 -0500
From: Brad Lehman
Subject: The WTC Scrawl
>From: Gordon Collins
>Subject: The WTC Scrawl
>
>Here's this elegant title page (look at that "P" in "Praeludia"!), neatly
>drawn up, symmetrically arranged, with a fine calligraphic flourish at the
>end - and then this rather careless-looking Scrawl running across the
>top. To me it looks much like someone trying out a new pen 100 or so
>years later without realizing what that piece of paper is that is sticking
>out from the bottom of a stack of forgotten manuscripts..... The notion
>that the Scrawl is a deliberate attempt to convey *any* meaning certainly
>requires some justification. It's clearly an afterthought at best,
>crammed into and overflowing the margin. (...)
The notion that it's anywhere near "crammed into and overflowing the
margin" is from looking at modern facsimiles, after the 20th century
deterioration of the original. That's documented in the article with
footnote out to info about the source, plus a photographic reproduction
from 1911. That 1911 photo is also the one I used on the web site: click
on the little one from the first page, and it pops up to a second window
full-size. (Marginal space is still an open question, of course; but my
point is that the reproductions in the NBA and the _New Bach Reader_ with
crammed/cropped margins don't necessarily tell the story, in themselves.)
If Bach wanted to convey a hidden meaning and at the same time make it
*appear* careless and meaningless, with a sprezzatura-like disorder to it,
isn't this drawing a possible manifestation of it? The English word for
this is "steganography" - hiding meaning in an otherwise innocuous-looking
carrier. See especially the papers by Neil F Johnson, on that.
Meanwhile, my article presents both a motive for Bach to have done this,
and lots of other corroboration that he did: especially (in the second
half) in the music itself.
All these are interesting questions/speculations, but isn't it better to
read the article (BOTH halves, plus the 50+ pages of supplemental web
material that I submitted for Oxford's site) first *before* trying to rip
apart what might be in it, by hearsay? The first printed half of the
article, February, is less than 1/3 of the presented case. We're in the
unfortunate situation at the moment of having court in recess for another
three months; but at least these three months give everybody the
opportunity to get familiar with the proposed resulting sound, in a
practical way.
=====
Look at it from the opposite angle, the contrapositive. *If* my proposed
solution is indeed the sound that Bach wanted/expected, how would he convey
it through the means available to him, and why, and does it agree with a
close reading of all available materials in the historical record as to his
preferences and performances? I believe that, having looked at all that
record (to the best of my ability using the _Bach-Dokumente_ etc) *and* his
music, playing through it tuned this way, he's guilty (as I charge) of
writing down this specific temperament and making full use of it through
his career. So is CPE, although he was more vague in *his* writing-down of
same. So is the corroboration especially by Bach's Mizler-buddy Sorge. So
is the corroboration in the resulting sound of both CPE's and Friedemann's
music. So is the way it solves all outstanding intonation problems in all
the Leipzig music by JSB, plus the Musical Offering ricercars for king Big
Fred and his court keyboardist.
[Contrapositive construction: "if {A} implies {B}, then {not B} implies
{not A}."
- My {A} is: "My proposed solution is correct; and indeed there's something
there at all to be known/knowable, that Bach had *some* specific
preference/expectation that's merely been lost."
- My {B} is: "All of Bach's music sounds [in]credibly beautiful this way
and all outstanding intonation problems are solved by this; and it fits all
the extant historical record about Bach's abilities and
preferences/expectations, working in situations where he had precise
control over intonation, as in harpsichord tuning done by himself."
- Study all of {B}, and (I believe) you will see/hear that there are no
substantial contradictions.
- This doesn't prove absolutely that {A} is correct, of course, but it
demonstrates that {not A} is very unlikely.]
The broader problem to be solved here is that of plausible *musical*
practice in all of Bach's music: practicality, plus beauty (as far as we
can trust the slippery slope of 18th century aesthetics), plus a plausible
expressive range within the music (i.e. does _Affekt_ do what the 18th
century people said it did?). The WTC title page drawing is just the major
clue as to what that solution is. Obviously it's paramount to interpret
the clue correctly. That interpretation and testing is done by a LOT more
than just staring at the drawing and trying out a boatload of possibilities
at keyboards and in spreadsheets. That's why, as the article presents, I
believe that the broader solution is in fact this one: and it's
corroborated by the WTC title page drawing and the rest of the historical
record, as to the body of music that Bach actually wrote within his
preferences/expectations of intonation.
That's why this is huge. It's really a book, slimmed down to a two-part
article and 50+ pages of mathematical analysis, and nearly a year of trying
the music in practice. It's properly a lifetime of work beyond that,
checking out item {B} noted above to see how solid {A} is. The clincher,
at least for me, is that *all* the other known stand-ins for {A} over the
past 200 years *do not* fit the evidence {B} as well as my proposed {A}
does. Not even the Neidhardts and Sorges, and Lindley's otherwise
excellent averagings of them (in his Michaelstein article). That is: plug
in whatever other {A} you want to, run it all forward, and you'll run into
contradictions in {B}, some pieces by Bach that just don't work very well.
=====
p.s. Gordon, I've taken your off-list notes and tidied up that bit on my
web site. Thanks!
Brad Lehman
------------------------------
Date: Wed, 16 Feb 2005 16:59:17 -0500
From: Brad Lehman
Subject: Re: The WTC Scrawl
I believed for about 18 years that Bach probably switched around a bit,
changing his mind variously during his career, as suggested here in Ray's
remarks. But now I don't believe that anymore. This was a result I didn't
expect or see coming, that there should be any constant through all or most
of his career, or into the next generation. (Hence, part of the "Rosetta
Stone" allusion....)
Others' mileage may vary.....
Brad Lehman
At 04:31 PM 2/16/2005 -0500, Ray Lurie wrote:
>Quoting Brad Lehman :
>
> > The
> > clincher,
> > at least for me, is that *all* the other known stand-ins for {A} over
> > the
> > past 200 years *do not* fit the evidence {B} as well as my proposed
> > {A}
> > does.
>Brad Lehmann is certainly correct in asking for some forebearance until
>the whole of his article is published, and I'm sure that it must be
>frustrating for him to see questions raised about Pt. 1 that he feels
>he's already answered in Pt. 2 or on the forthcoming web supplement. I
>don't want to join in the free-for-all, especially since I have only
>glanced at the published portion of his article, but I do want to
>question the logic of the argument he presents in the snippet included
>above. If I've understood this passage and Brad Lehmann's posts
>correctly, he is arguing that one argument in favor of his thesis is
>that the proof is in the pudding, and that the pudding is not just the
>WTC but the whole of Bach's keyboard output (with the cherry and
>whipped cream being the whole of C.P.E.'s and W.F.'s keyboard output as
>well). Why, though, should a single temperament fit all of it?
>Wouldn't we be surprised if Bach hadn't experimented and changed his
>decisions about temperament (in much the same way as his contemporary,
>Rameau)? Whereas Brad Lehman finds a single solution to be
>the "clincher," to me it seems highly suspect.
>
>Ray Lurie
------------------------------
Date: Sat, 19 Feb 2005 08:47:03 -0500
From: Brad Lehman
Subject: the big/little C
Ibo Ortgies wrote:
> > http://www-personal.umich.edu/~bpl/larips/wtc-1722-from-1911-grove.gif
>
>There is no little C
>What has been mistakenly read as little C is a typical ornament which
>was applied frequently to some capital letters especially C, S, E, F -
>sometimes K you can find easily similar examples in Bach's handwriting.
>
>For example the C in "Concerto" in the heading of the first Brandenburg
>Concerto (facsimile of this page in Alfred Dürr: Johann Sebastian Bach.
>Seine Handschrift Abbild seines Schaffens. Wiesbaden: Breitkopf &
>Härtel, 1984. Blatt 16)
>
>or
>
>"K" in "Kirchbach" in the title page of BWV 198 ("Laß, Fürstin, laß noch
>einen Strahl") op.cit., Blatt 35
>
>or "C" in "Concertino"; heading of the "Violino Concertino"-part of the
>a-minor violin concerto (BWV 1041) op.cit., Blatt 40
>
>It is therefore an ornament (no little C) of the capital letter C in
>"Clavier", which does hardly touch one of the loops in the calligraphic
>ornament.
Also the title page of the double violin concerto 1043: see facsimile on
p235 of Wolff's _Essays_. Prominently overlapping the big C of "Concerto".
And (maybe) part of the big F in "Friedemann" on the 1720 title page of his
little book.
And part of the big K at the top of the _Entwurff_.
In all three of those, the little hook-C-ornamenty thing touches its
capital. On the WTC title page it doesn't touch the capital, but it
touches the spiral drawing *instead*. (Merely an observation....) That
bit of the detail was revealed to me when I got the photo reproduction from
Brombaugh and Leedy, scanned from the old 1911 Grove dictionary.
Brad Lehman
------------------------------
Date: Wed, 23 Feb 2005 09:29:09 -0500
From: Brad Lehman
Subject: BachScholarJokes
http://listserv.albany.edu:8080/cgi-bin/wa?A2=ind0502&L=hpschd-l&D=1&O=D&P=28912
>Ja ja. 'did you ever tune your claviharpsitoot in equal temperament?'
>Answer: 'I stopped after I read Early Music' - sorry, couldn't resist
:)
So, he stopped doing it after reading Barnes (1979) in EM, and then started
doing it again after Williams (1983) in EM, and now has stopped again in 2005?
Anyway, have a Spiel at the f# toccata 910, the g English suite's sarabande
808, the ouverture (all of it) 831 in both c and b, and both the fraternal
capriccii in E 993 and Bb 992, and the praeludium/toccata 566 in both C and E.
Brad Lehman
------------------------------
Date: Wed, 23 Feb 2005 12:36:43 -0500
From: Brad Lehman
Subject: paper organists playing pre-composed music
Ibo wrote:
http://listserv.albany.edu:8080/cgi-bin/wa?A2=ind0502&L=hpschd-l&D=1&O=D&P=29567
and
http://listserv.albany.edu:8080/cgi-bin/wa?A2=ind0502&L=hpschd-l&D=1&O=D&P=28732
But what about Sorge's report? I believe that--at least by implication,
tied to your remarks here about "paper organists"--quite a bit of playing
of published music (and manuscript music by masters) did go on, in
practice...by well-respected organists, not only beginners.
Here's the preface of Sorge's 1st book of chorale preludes, 1750, as
explicated in Wolff's _Essays_ p113ff. (His broader discussion here in
this essay is the various chorale preludes by JSB, Sorge, and others
collected together in the Neumeister MS, having concordances
elsewhere.) Wolff presents a facsimile of the title page and one of the
compositions in Sorge's separate print, and gives the following translation
and commentary:
WOLFF]]Sorge's preface discusses the function of the three-part manualiter
chorale settings, compares them in general with the difficult and demanding
organ chorales in Johann Sebastian Bach's Clavier-Ubung III (1739),
suggests the possibility of performing them on both organ and clavier
(harpsichord, or another keyboard), and by analogy refers to the overall
purpose of the Neumeister collection:
"Next to the knowledge of figured bass, to which my 'Vorgemach der
musicalischen Compositionen' [Lobenstein, 1745-1747] gives sufficiently
comprehensive and detailed instructions, nothing is more important to the
organist than he be adroit in preluding to the various chorales, according
to their particular content, so that the congregation will be stimulated to
sing the subsequent chiorale with appropriate devotion. The preludes on
the Catechism Chorales by Herr Capellmeister Bach in Leipzig are examples
of this kind of keyboard piece that deserve the great renown they
enjoy. But because works such as these are so difficult as to be all but
unusable by young beginners and others who may lack the considerable
proficiency they require, I have prepared, at the suggestion of my good
friends as well as my own pupils, the following eight simple preludes, to
be played only on the manuals, and I herewith publicly present them to
those members of our musical youth who are eager to learn and to all
devotees of this type of playing."[[WOLFF
Now, why would Sorge bother to write any of that if organists really
weren't using pre-composed pieces in church, but *only* used pre-composed
pieces pedagogically to learn how to improvise? (Ibo, I apologize if I'm
over-simplifying or misunderstanding your thesis on this; I very much would
like to read your diss. to understand better what your position is, in this
regard.)
And, why couldn't there have been some change in common practice between
Niedt (first decade of 00's) and mid-century, as to the respectability of
playing from scores?
=====
In _Early Keyboard Journal_ 21 (2003 - MHKS and SEHKS joint publication)
there's an excellent review by David Yearsley of three books: Russell
Stinson's books about the _Orgelbuechlein_ and the Great 18, and the
facsimile edition of G F Kauffmann's _Harmonische Seelenlust_. On the
strength of this review I'm eager to go read Stinson's books directly.
And I like Yearsley's remark on p110 about the Bach/Kauffmann connection:
"Looking west out of the window of the second floor composing room in his
apartments in the St. Thomas School building in Leipzig, Bach could, on
clear days, see the distant spire of Merseburg Cathedral where Kauffmann
was organist and, later, director of church music to the Duke of
Saxe-Merseburg. Kauffmann is now chiefly remembered, to the extent he is
remembered at all, as one of Bach's competitors for the cantor's position
in Leipzig." Yearsley's point here is that Stinson probably should have
mentioned GFK as part of his discussion of playing pre-worked chorale
prelude settings.
Brad Lehman
------------------------------
Date: Wed, 23 Feb 2005 13:11:15 -0500
From: Brad Lehman
Subject: paper organists playing pre-composed music
An addendum to:
http://listserv.albany.edu:8080/cgi-bin/wa?A2=ind0502&L=hpschd-l&D=1&O=D&P=29846
And there are several Froberger toccatas that explicitly say they're for a
liturgical function: in communion at the elevation of the host. Why
shouldn't we take that title at its word, and assume that respected
organists (at least as early as the 17th century) DID really play from
written-out music like this, at least occasionally?
One of those Froberger elevation toccatas even visits harmonies all the way
from A-flat major to B major, inclusive. That piece is part of my
argument that Froberger's organ(s) probably had some manner of temperament
that allows such wide-ranging music to sound good.
Similarly, there are a Pachelbel fantasia in E-flat and ricercare in f#
minor that are harmonically adventurous (having lots of Dbs and Abs and a
Cb, and B#s and E#s, respectively); and why wouldn't these be liturgical
music? [And Pachelbel was JSB's grand-teacher, through the proxy of his
brother....]
Again, I'm trying to understand Ibo's assertions that such music as this
wasn't as practically liturgical as it claims/appears to be.
Brad Lehman
------------------------------
------------------------------
------------------------------
------------------------------
------------------------------
------------------------------
------------------------------
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------------------------------
Date: Wed, 11 Jan 2006 13:20:09 -0500
From: Brad Lehman
Subject: dentists who tune keyboards
> >http://groenewald-berlin.de/
>http://www.groenewald-berlin.de/Profile.htm
by Dr Grönewald, DDS.
On his page
http://www.groenewald-berlin.de/Mikro-Diff_bei_Bach-Stimmungen.html
he gives away a free PDF photocopy of pages 194-5 from Tessmer's
article, and compares only those four "Bach" proposals from the
1960s-70s. Tessmer's article (of which those pages are part of the
appendix) actually compares 18 temperaments. Furthermore, it's from
the "1996" dated volume 25 of _Acta organologica_ (published 1997),
not "1994" as claimed on that page.
http://www.groenewald-berlin.de/Profile.htm
Probably just a pseudo-3D representation of cent deviation from equal
temperament, nicht wahr? Everything is boiled down to cent
measurements, for example in this 1677 Trost temperament he gives:
http://groenewald-berlin.de/graphik_T051.pdf
http://groenewald-berlin.de/text/text_T051.html
Measure something with many significant digits of mathematical precision,
and make diagrams, in lieu of explaining why any of the numbers would
matter in the actual performance of music....... Vertical intervals
are measured with as much theoretical accuracy as anyone could reasonably
want. But what about melodic thinking, in the way scales are put
together with varied sequences of tones and semitones? (What
brush-strokes and sweeping lines in the painting make the resulting
piece look emotionally compelling, or sexy, or even merely *interesting*
to contemplate? No, rather, just analyze the atomic structure and
reflective characteristic in each individual pigment and hope that
sufficient meaning will emerge from a sufficient mass of measurements...?)
Here's the index of this online book:
http://www.groenewald-berlin.de/Inhaltsverzeichnis.htm
The more interesting book by another dentist, that I've seen, is the
forthcoming one by Thomas Donahue. He stresses a hands-on approach
to trying things out at the keyboard (both a musical keyboard and a
computer keyboard), and learning how to do any procedures *oneself*
rather than just reading them off somebody else's charts. His book
also gives a step-by-step procedure to construct one's own spreadsheets
from scratch. Likewise he suggests some procedures to convert any
on-paper temperament into a set of by-ear tuning instructions, by
looking for any similar beating coincidences and whatnot. There's
some good pedagogical value in that: learn how to build something
from scratch, and analyze it creatively hands-on, as a way to start
understanding how it works.
=====
Obligatory non-dentist material:
I have a chart ready for Lindley's 1994/7 Michaelstein proposal, the
one where he prepared a conference tape of various Bach repertoire
performed by Peter Sykes. That's the article where he suggested some
"ideal" shaping of a Bach organ temperament (according to his own,
Lindley's, premises!) and then provided some algebraic justification
of his various "lucubrations" (his $50 word). But I want to finish
formulating my commentary first, in response to his musical comments
within that article. Then it will be added to my roster at
http://www-personal.umich.edu/~bpl/larips/bachtemps.html
Bradley Lehman
------------------------------
Date: Tue, 17 Jan 2006 18:17:10 -0500
From: Brad Lehman
Subject: new organ and harpsichord recordings released
This is to announce two new CD sets released January 4th 2006,
recorded in March 2005. I am the performer and producer of
both of these sets. They use the specific keyboard tuning that
I believe was Bach's own, which evidence I have explained at
http://www.larips.com
and in various publications during 2005 (Early Music, The
Diapason, Clavichord International, BBC Radio 3 broadcast,
et al).
http://www-personal.umich.edu/~bpl/larips/articles.html
Press release about the two recordings:
http://www.goshen.edu/news/pressarchive/01-13-06-organ-cd.html
Ordering information:
http://www.gcmusiccenter.org/php/music.store/index.php
The organ set "A Joy Forever: Opus 41 at Goshen College" demonstrates
the new two-manual pipe organ at Goshen College (Indiana), built by
Taylor & Boody Organbuilders (Virginia). The music is by Bach,
Brahms, Walther, Fischer, Erbach, Zachow, and some others. To my
knowledge this is the first *complete* recording of Fischer's
_Ariadne musica_ including its five ricercars: a book that inspired
Bach's composition of the Well-Tempered Clavier. The details of this
set are at
http://www-personal.umich.edu/~bpl/larips/cd1002.html
3 CDs, $30.00 USD plus shipping. Total time slightly over 3 hours.
The single disc "Playing From Bach's Fancy" has nearly an hour of
harpsichord music by JS Bach and WF Bach, and 20 minutes on the
organ. Preludes, fugues, sinfonias, polonaises, duetti, chorale
preludes, excerpts from the Musical Offering and Art of Fugue, and
several other tidbits. The harpsichord is a Franco/Flemish style
double by Knight Vernon, and owned by Goshen College. The album
details are at
http://www-personal.umich.edu/~bpl/larips/cd1003.html
1 CD, $15.00 USD plus shipping. Total time 77 minutes.
The Taylor & Boody organ Opus 41 used in these recordings:
http://www.taylorandboody.com/opuses/opus_41.htm
http://www.gcmusiccenter.org/php/facility/special.features/organ.php
http://www-personal.umich.edu/~bpl/larips/tb41.html
=====
The trumpet + organ album "In Thee is Gladness" from January 2005
(recorded 1997) is also still available:
http://www-personal.umich.edu/~bpl/larips/cd1001.html
My colleague on that, Dr Martin Hodel, is a member of the Minnesota
Orchestra, and teaches trumpet and music theory at St Olaf College.
We recorded this album on two equal-tempered organs in northern
Germany. It includes a variety of compositions by Buxtehude,
Brahms, Bach, Viviani, Baldassare, Pachelbel, Cellier, Bernstein,
Starer, and Lehman. 1 CD, $15.00 USD plus shipping, ordered by
e-mail inquiry to hodel@stolaf.edu .
Enjoy the music,
Bradley Lehman
Dayton VA
http://www-personal.umich.edu/~bpl
17 January 2006
------------------------------
Date: Tue, 24 Jan 2006 10:34:45 -0500
From: Brad Lehman
Subject: Young #1
> Date: Mon, 23 Jan 2006 15:35:43 -0500
> From: David Jensen
>
> At least five days a week, I tune at least one harpsichord here at IU I tune
> in W-III, and two in K-III, and a couple in Vallotti, and one in (EGAD!!!)
> Vallotti&Young.
Why not use Young's first one, that he himself claimed to prefer ahead
of his second one? It's smooth in result, symmetrical in the same way
the physical layout of a keyboard is (all intervals reflected exactly
across D, or across G#), and very easy to prepare.
To set it up, do a complete Vallotti (F-C-G-D-A-E-B in 1/6 comma, and
all other 5ths pure); then tweak your F so it's averaged out between
Bb and C (1/12 each), and likewise the B averaged between E and F#.
That's all there is to it in practice: making those transitional
half-tempered 5ths at each boundary point with the pure 5ths.
He *described* #1 in the rather obscure way of 3/16 syntonic comma etc;
but as Barbour pointed out more than 50 years ago, the cent values do
not differ from using 1/6 PC on the natural 5ths.
And then Young's #2 is just his concession version for people who
couldn't figure out #1. That's the one that has the tempered 5ths
C-G-D-A-E-B-F#, i.e. six entirely different notes on F and all the
accidentals, as compared with Vallotti. EGAD!!! Because F and the
accidentals are all lower (as compared with Vallotti or with Young's
preferred #1), this temp doesn't work as well for flat-key music.
N.B. Watch out for references to Vallotti, because there is a bowdlerized
PC version that supplanted the real one. Vallotti himself used syntonic
comma, not Pythagorean comma; and in his the schisma gets absorbed
imperceptibly among the pure 5ths, giving each pseudo-pure 5th the
slightest bit of fringing. The biggest major 3rds then come out to be
slightly narrower than Pythagorean, instead of being a whole syntonic
comma sharp. Reference: Dominique Devie's book. I like Devie's quip
that the dumbing-down of syntonic comma temperaments to theoretical PC
modern temperaments is like getting a bad haircut. Big brutal chops
with the shears, instead of refined trimming.
My chart of real Vallotti is at page 15 in the "supplementary data" PDF
file #2, downloadable through
http://www-personal.umich.edu/~bpl/larips/outline.html
Bowdlerized Vallotti is at page 18 in that file.
My Young #1 chart is viewable through the page
http://www-personal.umich.edu/~bpl/larips/practical.html
where I put the distinction between the theoretical version and the
easier practical version.
Brad Lehman
http://www.larips.com
------------------------------
Date: Tue, 24 Jan 2006 10:55:47 -0500
From: Brad Lehman
Subject: geometric 5ths (any reasonable size) by ear
Paul Poletti wrote: (about Kirnberger's scheme)
>Surprisingly (or not),
>the irregular division is very easy to tune, which is probably exactly why
>he proposed it in the first place. This makes perfect sense when we remember
>than one of Kirnberger's basic arguments was that a good system must be easy
>to do by ear. In fact, his real alternative to the half comma fifths is much
>easier to do than the ubiquitous modern misinterpretation: 4 fifths all
>tempered by 1/4 Syntonic comma.
>
>The process is as follows (precise beat rates are for c1 = 249.6 Hz):
>
>(1) Tune c-e pure.
>(2) Tune the e1 above e.
>(3) Temper c-g so that it beats not quite once a second (0.835 to be exact).
>(4) Temper g-d1 so that it beats 1 1/2 times a second, or three times in 2
>second (the actual beat rate is 1.6/sec).
>(5) Tune d an octave below d1.
>(6) Temper d-a so that it beats exactly the same as g-d1.
>(7) Check that a-e1 beats ever so slightly faster than twice c-g (actual
>rate is 2.1/sec).
OK, but accurately geometric 5ths are even easier than that, by ear, and
without counting any specific beat rates for anything, at any specific
starting pitch. And all these steps together take less than a minute.
- c2 (octave above middle c) from the fork.
- c1 (middle c) pure to it; c pure to it.
- e1 pure to c1 (and check it with c as a 10th).
- g temporarily pure as a 4th under c1
- a temporarily pure as a 5th under e1
- d1 averaged out, comparing as a 5th above g and as a 4th above a, so
the beat rate of the 4th is exactly triplets against the beat rate of
the 5th, as duplets. (And what that beat rate is numerically, we don't
care at all!) Just listen for the 3-to-2 relationship. It will be fast
and easy to hear, as both of these (at this point) are 1/2 syntonic
comma...pretty big.
- g1 averaged out likewise, as a 5th above c1 and a 4th above d1. Triplets
of the 4th against duplets of the 5th.
- a1 averaged out likewise, as a 5th above d1 and a 4th above e1. Triplets
of the 4th against duplets of the 5th.
- Correct the g and a to be pure octaves from g1 and a1.
- We now have a geometrically accurate C-G-D-A-E with identical tempering
in all of those 5ths/4ths.
This procedure also works for 1/4 comma meantone, or for any other regular
system (any size of "meantone" generated by a reasonably-sized starting
major 3rd somewhat bigger than pure) for the reasons I have described at
http://www-personal.umich.edu/~bpl/larips/tetrasect.html
That same trick of 4ths triplets vs 5ths duplets (under fingers 5-4-1 of
the left hand) can be used all the way up the treble of the keyboard, in
any *regular* temperament, to check the veracity of octaves. Eventually
it all gets too fast to hear, but the equal *quality* of the intervals
is easy to sense.
Starting from an A fork instead of a C fork? Same basic trick (assuming
our goal is to set up regular naturals). Establish the correctly-positioned
F under A; use the C and D below (temporarily pure to those) to erect
a correctly averaged G between them, and then correct the C and D as
averages from F/G and G/A.
Brad Lehman
------------------------------
Date: Tue, 24 Jan 2006 11:28:01 -0500
From: Brad Lehman
Subject: Werckmeister 3 by ear
And Werckmeister 3 from a C fork, geometrically, with no beat-counting? Easy.
[Assuming that one wants this non-harpsichord temperament on a harpsichord,
and horrible sounds when playing in all the flat keys beyond two flats in
the signature (sorry, I hate W3 but we might as well set it up accurately!)....]
- C from fork.
- C-F-Bb-Eb-Ab-Db-Gb-Cb-Fb all pure 5ths. (Yes, we'll move the last several
later.)
- This gives a C-E that is very slightly narrow; one schisma narrow.
- From this C-E do the 5ths construction trick as described below: temporary
G below the C and temporary A below the E, each pure. Use those to construct
the accurate mean D within that particular C-E (brutally 1/2 Pythagorean comma
from each of the G and A, a very rough sound). Use that D with the C and E
to construct the accurate G and A above them, and then correct our temporary
G and A as octaves.
- That gives us an accurate C-G-D-A-E cycle of 1/4 Pythagorean comma each.
- Retune E upward to be pure from A, and then B pure from this new E.
- Our last chunk of 1/4 comma has therefore been moved up to B-F# (Gb); same
quality as our other C-G-D-A. That accounts for all four of these
equally-sized chunks.
Brad Lehman
------------------------------
Date: Tue, 24 Jan 2006 11:40:52 -0500
From: Brad Lehman
Subject: arithmetically faked roots
>>Incidentally I ought to stick up for K's mathematics - an arithmetic
division of the comma is for all musical purposes the same as a
'correct' geometric one.<<
I'll second that. The general principle is that we can fake a fourth root
by using four neighboring superparticular fractions.
Viz: if we want to whack 81/80 (the syntonic comma) into four geometrically
almost-equal pieces (i.e. a fourth root) we multiply the whole thing top and
bottom by 4, and then deal it out thus:
81/80 = 321/320 x 322/321 x 323/322 x 324/323 = 324/320
Starting from a fixed middle C, compare the geometrically correct G above
it (a tempered 5th of 1/4 syntonic comma) against the arithmetically
approximated 5th where a pure 3/2 has been flattened by 321/320. The
difference is less than 0.005 Hz. That's closer than any acoustic instrument
is going to stay in tune for an hour.... Ergo, such fake roots are good
enough for practical music.
Likewise, the fake square roots of 81/80 are 161/160 and 162/161.
Brad Lehman
(personally not interested in monochord construction anyway)
------------------------------
Date: Tue, 24 Jan 2006 16:12:48 -0500
From: Brad Lehman
Subject: B major
Jensen:
>>I am reminded that on one occasion I was stuck for a temperament due to the
variety of keys (including B major) being used in an orchestral concert, and
I chickened out and used ET. Upon being asked, I told the leader, a
violinist, that I had indeed set equal temperament. He looked shocked and
exclaimed, "But I can't play in equal temperament!"<<
Recently (i.e. today over lunch) I had a good Baroque violinist in for a
reading session, and we played through a rack of mid-17th-C German stuff.
I of course had my Bach temp set up on the main hpsi, and I had put
Werckmeister 3 onto the second one just in case we'd like to try it; and
there was the remainder of meantone from a couple weeks ago on the virginal
nearby.
We got near the end of one piece in E minor and I found that I had a whole
page of pedal point, reiterating the dominant B major for at least a good
45 seconds. It all went fine on the Bach and I wouldn't want it to be any
spicier than that. At the end of the page I leapt up, sat back down at
the Werckmeister, and said, "Let's do that page again to hear how it sounds
in this." It sounded pretty much like garbage, with that excessively strong
B-D# that Werckmeister has. Then for nut's sake I played the B major harmony
a couple of times on the meantone, and it was even worse than that because
the D# was tuned as Eb.
Brad Lehman
http://www.larips.com
------------------------------
Date: Tue, 24 Jan 2006 16:46:27 -0500
From: Brad Lehman
Subject: Re: B major
> We got near the end of one piece in E minor and I found that I had a whole
page of pedal point, reiterating the dominant B major for at least a good 45
seconds. It all went fine on the Bach and I wouldn't want it to be any spicier
than that. At the end of the page I leapt up, sat back down at the Werckmeister,
and said, "Let's do that page again to hear how it sounds in this." It
sounded pretty much like garbage, with that excessively strong B-D# that
Werckmeister has.
I should add: the trouble wasn't so much from the B-D# interval *in isolation*.
(The size of B-D# is almost the same in both of these, and Werckmeister's is
actually a smidge smaller.) To my ear at least, it was the difference of
having a heavily tempered (1/4 PC) 5th there on the B-F# in the Werckmeister,
vs having a pure one. 45 seconds of a B major chord, it jangles incessantly
with those several beats in there vying for supremacy. If I leave the 5th
out of the B major chord it calms down quite a bit, but who wants to remember
to do that while playing?
That plus the way the leading A# in Werckmeister is so bloomin' high on the
way into B minor/major, either melodically or as part of its dominant. I
was playing through the B minor Ouverture/partita (BWV 831) last night, in
this Werckmeister to remind myself what used to seem normal and unavoidable,
and I found the notes A#, E#, and B# to be way too high when they come up
in their various melodic and harmonic contexts.
A few more solo pieces to try, and then I'm switching this hpsi back out of
W3 (ick!) for at least a good month. Probably putting back the Young #1
that I mentioned this morning.
Brad Lehman
------------------------------
Date: Wed, 25 Jan 2006 12:33:44 -0500
From: Brad Lehman
Subject: Re: 1 step forward and 2 steps back; was B major
Poletti wrote:
http://listserv.albany.edu:8080/cgi-bin/wa?A2=ind0601&L=hpschd-l&D=1&T=0&O=D&F=&S=&P=54361
>>[basically a diatribe Brad-is-a-moron-for-even-setting-up-W3-at-all, deleted...]<<
As I mentioned yesterday, and I thought I explained clearly enough (but maybe
not), the main reason I had W3 on my 2nd harpsichord at all this week was to
play back through old stuff, to remind myself what seemed "normal" some years
ago when I first learned those pieces. The key-colors of the harmonies and
some of the melodic intervals: features that seemed somewhat interestingly
expressive at the time, but that now sound to me like clunky nonsense.
I was also checking out somebody's comment elsewhere that he actually *likes*
playing Clavieruebung II (Italian Concerto and Ouverture 831) in Werckmeister
3 better than in the temperament I've proposed, because W3 is more
interestingly spicy. So, I wanted to remind myself how 831 actually sounds
in W3. The Italian Concerto is almost indestructible as it uses only a
small handful of Ab occurrences, while the entire rest of the piece sticks
to the twelve classic meantone notes. Almost any reasonable temperament
will sound OK in that Italian Concerto. But in 831, a temperament has to
handle D#, A#, E#, B#, and Fx in various contexts...and W3 is (in my opinion)
horrible at this. The other gentleman playing 831 with W3 still fancies it
anyway; OK, whatever.
I would never dream of using W3 willingly in any concert on harpsichord,
or even any serious practice session whether solo or ensemble. I dislike
it *that* much, musically, along with believing it fundamentally
inappropriate to harpsichords. I even said as much (or at least half that
much) yesterday, when providing instructions to set it up accurately:
http://listserv.albany.edu:8080/cgi-bin/wa?A2=ind0601&L=hpschd-l&D=1&T=0&O=D&F=&S=&P=52146
Brad Lehman
------------------------------
Date: Wed, 25 Jan 2006 15:53:23 -0500
From: Brad Lehman
Subject: steps forward
>> I did a survey of the 10 or so
harpsi students I will be teaching tuning to next semester to find out what
their level is and how much they know. Temperaments:
meantone, WIII, KIII, Young/Vallotti
One did know "French tuning", by which I assume he meant some kind of
ordinaire.
So I got me work cut out for me.<<
Cool.
Suggested lab exercise: have them set up an ax in the way I recommend,
and then play through Figure 179 in the CPE Bach _Versuch_ (English
translation by Mitchell, p164), along with carefully working out what
CPE's paragraph 29 thereon is about. Gist: bring out dissonances more
prominently than their consonant resolutions, and especially emphasize
any notes that are coming in from outside the prevailing scale.
Same figured bass exercise that is reproduced in the middle of the page at
http://www-personal.umich.edu/~bpl/larips/cpeb.html
. If you're working with the Breitkopf facsimile of the 1753/62 editions
(Lother Hoffmann-Erbrecht, 1958/92), it's Figure XIV at the bottom of the
"Tab VI" pullout page of musical examples, and it's pages 129-31 in the
text.
If they have ears to hear, playing through these harmonic progressions,
they will notice that the spots CPE Bach marked with "f" are already
being emphasized somewhat, as a naturally-occurring result of that particular
temperament. (Hypothetically,) CPE here has simply written into his dynamic
indications a reinforcement of the normal musical priorities that are
*already* in the harpsichord's sound. In every one of his harmonies marked
"f" there is at least one note that is especially strong or noticeable,
due to a slight frustration of intonational expectations...i.e. whatever
note is creeping in from outside the prevailing scale. And it draws
attention to itself, making the harmony as a whole seem louder or more
restless: the thing that a dissonance is supposed to accomplish.
And yes, I've checked this in Vallotti, Young, K3, W3, various meantones,
and various Neidhardts...where this special resultant effect does not
happen with such a startling clarity (or much of anything). The point
is seriously lost, unless one actually listens to it by playing through
the exercise in the temperament I suggest that CPE was probably using.
Brad Lehman
------------------------------
Date: Wed, 25 Jan 2006 15:14:50 -0500
From: Brad Lehman
Subject: E minor
>>As to what a harpsichord owner did in the 17th century when faced with
E minor, I should imagine it was obvious.<<
Right -- "do something intelligent" or else pass out suppositories for
the ears. If said hypothetical harpsichord owner is playing straightforward
stuff, like the extant Reincken or Buxtehude pieces in E minor, it's easy
enough to flick all the Eb keys on the keyboard down to a better D# for
at least the duration of the piece.
But if said protagonist is essaying the Froberger toccata in this key,
from 1656, that simply doesn't work. Eb and D# are both in there, both
prominently. It happens to work in this piece, and it's maybe a fluke,
that one can leave all the Eb intact except switching the one nearest
middle C down to a D#. Who needs anything resembling an octave there
if the octave is not actually played in the piece, right?
Or else--whether as general habit or exceptionally for such pieces--pick
some compromised positions *between* D# and Eb, and likewise for some or
all of the other raised keys...at least. And maybe also F and C, so
they can act like E# and B# some of the time.
Or else play different music.
Or else sit there wondering why the harpsichord just sounds suddenly
rotten and then fine again, throughout some compositions in some otherwise
innocuous-looking keys, without really bothering to figure out why...or
to do anything about it.
Or get some split-key accidentals built into the instrument, but then
still sit there wondering why E# and B# in otherwise normal-looking
music still sound rotten.
=====
I meanwhile have a hard time believing any historical conjectures that
Buxtehude had Werckmeister 3 on the organ...at least at the point when
he wrote the F# minor Praeludium. The piece sounds like wounded dog (@@)
in W3, whenever the A# or E# or B# come up. The Cx and the Fx (in the
G# minor section) aren't so much of a problem here, because we're flipped
so far around the other side it starts to sound decent again.
Whatever circumstantial evidence argues for W3 in Buxtehude
(names/dates/places/whatnot), the existence of *this* piece has to be
circumstantial evidence on the other side of that argument...assuming
of course that the piece was ever intended to be played at all, as written,
and assuming that any musicians within earshot didn't really relish a
serving of wounded dog. That's also assuming that extant musical
compositions are allowed to *be* evidence of practical matters: evidence
with some value approaching that of words and dates and monochord calculations.
@@ "Wounded dog" is of course both a technical term and a euphemism for
unprintables. I probably should have made some synaesthetic reference
to olfactory trauma, instead.
Brad Lehman
------------------------------
Date: Thu, 26 Jan 2006 11:01:16 -0500
From: Brad Lehman
Subject: African or European swallow? (really Vallotti/Young)
Poletti ribbed Jensen:
>>
>>Oh - reminding myself - I tune Vallotti&Young with an electronic meter.
>
>How do you decide which day to tune Vallotti and which day to tune Young? Or
>do you tune V. on one harpsichord and Young on the next? Or alternate
>between the two each succesive octave? Or . . .
>
> ;-)
>
>But seriously, I don't think we professionals should propogate the error
>created by the Korg company, which I think was the first to conflate the
>two. At least in my experience, the first time I ever saw them "joined at
>the HIP" was in the MT-1200 instruction manual.
Yeah, and what do they *mean* by the conflation "Vallotti&Young", anyway,
to make a preset on the electronic device? Which of Young's?
Young's first one is considerably closer to Vallotti than his second one
is, even though it looks horrendously more complicated on the page.
Starting from Vallotti, move two notes (F slightly lower and B slightly
higher: that's all there is to it in practice!). It alters four of the
major 3rds slightly by 1/11 syntonic comma each. Db-F and B-D# both get
a bit gentler than Pythagorean; F-A and G-B both get a little more vibrant
than they were. The other eight major 3rds stay where they were in Vallotti.
Young's second one ("nearly the same effect") transposes the whole
temperament of Vallotti up a 5th, and *none* of the sharps/flats or the
F are the same as they were in Vallotti. [If we keep a common C and A
to all of these comparisons.] Also, eight of the twelve major 3rds have
changed size, some of them by as much as 2/11 of a syntonic comma. I
guess that's "the same effect" for rather large values of "nearly". :)
Vallotti:
9 11 11 11 Ab-C, Db-F, F#-A#, B-D#
9 7 5 3 E-G#, A-C#, D-F#, G-B
3 3 5 7 C-E, F-A, Bb-D, Eb-G
Young #1 (i.e. Vallotti with slightly altered B and F):
9 10 11 10
9 7 5 4
3 4 5 7
Young #2 (i.e. Vallotti transposed up a 5th):
11 11 11 9
7 5 3 3
3 5 7 9
Further details about Young 1:
http://harpsichords.pbwiki.com/f/Young.html (thanks, Gordon)
Theoretical and practical versions:
http://www-personal.umich.edu/~bpl/larips/Young1syn.jpg
http://www-personal.umich.edu/~bpl/larips/Young1.jpg
Explanation (most of it posted in November; new link added):
http://www-personal.umich.edu/~bpl/larips/practical.html
Vallotti and Young 1 are both symmetrical on the keyboard, in the same
way the keyboard layout itself is: across D or across G#. Young 2 isn't.
It's symmetrical across G or Db.
Brad Lehman
------------------------------
Date: Thu, 26 Jan 2006 11:16:55 -0500
From: Brad Lehman
Subject: Re: African or European swallow? (really Vallotti/Young)
At 11:01 AM 1/26/2006, Brad Lehman wrote:
> Vallotti and Young 1 are both symmetrical on the keyboard, in the same
way the keyboard layout itself is: across D or across G#. Young 2 isn't.
It's symmetrical across G or Db.
Sorry, that last sentence about Young 2 should be: "It's symmetrical
across A or Eb."
Brad Lehman
------------------------------
Date: Fri, 27 Jan 2006 09:54:15 -0500
From: Brad Lehman
Subject: 'tis V
> C-F is definitely tempered, so normally I would say, yes it's V. But (even
> taking into account the variability inherent in the MT-1200, about which I
> readily agree), something's still not quite right.
>
> One of the list members sent me this "deviation from equal temperament" list
> for the Mt-1200 V/Y:
> MT1200, Vallotti & Young, ¢ deviation from =temp:
> C - +6
> # - 0
> D - +2
> # - +4
> E - -2
> F - +8
> # - -2
> G - +4
> # - +2
> A - 0
> # - +6
> B - -4
It's Vallotti with everything rounded off to the nearest cent, and lined
up with a start on A (if we were using a tuning fork).
For a version from a C standard, subtract 6 from all twelve numbers.
There's nothing compellingly "& Young" about it. Young's two are as I
described here yesterday, where four or six of the notes are different
from the above recipe, and four or eight of the major 3rds are different sizes:
http://listserv.albany.edu:8080/cgi-bin/wa?A2=ind0601&L=hpschd-l&D=1&T=0&O=D&F=&S=&P=58509
(But strike/correct the last sentence: Young's #2 is symmetric across A
and Eb, not G/Db.)
When can we get around to talking about temperament structures as something
other than cent "deviation from equal temperament" measurements? I blame
Barbour. :)
When I hear "Vallotti" I immediately think, "Ah yes, all seven naturals
are spot-on to their positions in regular 1/6 comma, and all five accidentals
are at compromised positions rising by pure 5ths." That temperament is
so simple and elegant, that way. There doesn't need to be a "Vallotti
[with or without Young]" setting on a device, if the device already has
regular 1/6 comma on it. Just lay down all the naturals from 1/6, turn
off the device, and then stick five pure 5ths on the accidentals, coming
off either end from F or B. It then meets a residual pure 5th on the
other one, as checkpoint.
Want Barnes? Tune "Vallotti" and then knock the B upward so it's pure
from E. That's all there is to it, that single note difference.
Brad Lehman
------------------------------
Date: Fri, 10 Feb 2006 12:14:21 -0500
From: Brad Lehman
Subject: mozart tuning for todays modern a'=442 Hz
>
> It would be more useful to describe this or any other temperament in
> terms independent of reference pitch. Best for some of us is simply
> to express the circle of fifths in terms of either purity or
> fractions of a comma (whichever kind you want) wide or narrow.
>
> Some will want beat-rates, some won't bother. Some will wants
> cents-deviations from ET, some won't bother.
>
> But tying the thing to absolute frequencies seems not only cumbersome
> and inflexible, but a good way to make it particularly hard for
> beginners to see the underlying logic of the pattern.
I agree. It makes no sense to me to tie any temperament to any specific
starting frequency, as a rigid prescription. Temperaments are shapes, not
strings of numbers! The whole thing slides up or down together--as for any
specific frequencies and beat rates--if we've started from some lower or
higher pitch.
If it's a regular temperament ("meantone"), all the notes are in _regular_
positions, i.e. consistently spaced from one another, by 5ths/4ths. All
eleven 5ths are the same constant size as one another. The leftover
diminished 6th, wherever it may be, is some different leftover size (unless
we're using equal temperament, where it comes out the same as the eleven 5ths).
If it's an irregular temperament (a "temperament ordinaire" or a "well
temperament" or what-have-you), some of the notes have been nudged off
their regular positions, more sharpward or flatward. And the general
strategy is to get the sharps and flats to work more tolerably as one
another, when the wrongly-spelled enharmonic note gets played. (For
example, playing a D# in the music while the keyboard was tuned originally
to favor Eb.)
Beat rates are dependent on starting frequency, but a temperament's
identity IS NOT. Therefore, beat rates in a book are not a particularly
useful way to learn a temperament, because they all have to be adjusted
together if we'd start from some different frequency. And beat rates
don't clarify a shape; they're just a bunch of numbers measuring *results*
instead of the concept.
Vallotti at 440, Vallotti at 442, Vallotti at 396, Vallotti at 420 --
they're all equally Vallotti, even though they all have absolutely
different beat rates from one another, on all the corresponding notes.
The whole Vallotti shape just moves up or down a little bit in pitch,
as a constant set of relationships among its notes. Likewise for any
other temperament.
Vallotti has all seven naturals in the same position they are for a
regular 1/6 comma temperament. F-C-G-D-A-E-B. And then the sharps are
nudged a little higher, and the flats a little lower, until all of these
happen to meet each other in a chain of pure 5ths B-F#-C#-G#-D#-A#-E#,
or Cb-Gb-Db-Ab-Eb-Bb-F. Name that chain whichever one you want: sharps
or flats. It just works out conveniently that we can make a chain of
pure 5ths here.
Other related irregular temperaments treat this similarly: over the
core of regular naturals, nudge the sharps upward a little bit (and
maybe also B or E), and/or nudge the flats downward a little bit (and
maybe also F), until it connects with itself somewhere, working from
one or both ends of the naturals.
That is, the sharps generated by 5ths (being nudged upward some each)
get some amount of tempering *less* than the regular naturals. The
flats generated by 5ths (being nudged downward some each) get some amount
of tempering *less* than the regular naturals. Some of these 5ths,
here and there, might end up pure or maybe slightly wide: but the
perspective is that they're being narrowed *less* than regularity would
say. They're some bit wider than the regular amount, on average.
That distance from B through the sharps/flats back around to F has to
get spanned somehow. And the average size in there is pure 5ths,
unless we've also let the B/E drift up a little bit or the F down a little
bit. The main point is that these average 5ths across this region are
bigger than our regular 5ths on the naturals. And, the simplest (least
sophisticated) way to do it is to make them all pure...yielding the
Vallotti layout.
A hazard is: if we use four or more pure 5ths in succession, the resulting
major 3rd across them gets wider/rougher than we might want. So, the
simple chain of six 5ths from B across the accidentals to F might not
be best done as all pure. That's why it's useful to diddle the endpoints
F and/or B a little bit outward, so the stuff in between doesn't have
to average as wide as a pure 5th...it lets those extreme major 3rds
be softened a bit.
Various Neidhardt, Sorge, Bach, and Young #1 layouts are differently
sophisticated ways of spanning that region, giving the thing more
interesting musical nuances (or smoother transitions) than the Vallotti
layout does, and less harsh major 3rds as well. Some of the 5ths get
tempered only half as much as the normal amount would be. It therefore
makes that note tastefully a little higher (if a sharp) or lower
(if a flat) as we go along, working through B-F#-C#-G#-D# etc, or
from F-Bb-Eb-Ab etc.
=====
Getting back round to the original question about tuning for Mozart's
music: we don't know how Mozart tuned keyboards. But, the extant
music by him makes it obvious that the sharps and flats have to work
very well as one another, regardless of their spelling; we can't stick
with strictly regular 1/6 comma all the way through 12 notes, or
we'd be retuning the instrument for every composition...and some go
beyond 12 notes anyway.
So--start from that basic Vallotti shape mentioned above, with 1/6 on
all the naturals, and then experiment a bit with the way you're moving
the sharps up (going upward by 5ths off the B end) or flats down (going
flatward by 5ths off the F end). Play with the ideas of taking F
itself down a bit, or B and/or E up a bit, before continuing that
tasteful chain in either direction.
On our church's piano I have the layout F-C-G-D-A-E 1/6, E-B-F#-C# pure,
C#-G#-D#-A# 1/12 each. (That is, the basic Bach layout I've been
writing about....) And it works marvelously for Mozart/Haydn/Beethoven
and on into the 19th century. My setup instructions:
http://www-personal.umich.edu/~bpl/larips/praxis.html
This Sunday as part of the service I'm planning to play the first movement
of Mozart's sonata K282 (E-flat major, with lots of Ab-Db-Gb-Cb in it),
and the Rondeau en Polonaise second movement of sonata K284 (A major,
with G#-D#-A#-E#-B# in it). Noticeably different colors, these two
pieces, with everything working smoothly.
Brad Lehman
------------------------------
Date: Tue, 14 Feb 2006 16:59:34 -0500
From: Brad Lehman
Subject: naked major 17ths in Bach
> Well we had a big Italian, and our friend JSB, as I've been noticing
> more and more and more in recent months, has an almost intentionally
> perverse habit of exposing raw two-note intervals of two-octaves and
> a third. So that the upper third is measuring its FUNDAMENTAL against
> the lower note's fifth partial. Which is just a complicated way of
> saying that NOTHING will make those moments of this exposed interval
> tolerable. It is bloody AWFUL.
>
> I've been thinking of getting disciplined and, just in the pieces
> I've been working on, tallying-up all the places where Bach plays
> this very mean trick. It's as though he's just rubbing our noses in
> it.
I'd like to see such a list, too.
By my quickish tally in the four Duetti BWV 802-5 -- where a main point
might have been to rub organbuilders' noses in it with a set of test
pieces for tuning -- the major 17th comes up 35, 19, 33, and 6 times
respectively. That first Duetto has the remarkable feature in bar 48
of hitting seven of these in succession, parallel major 17ths on 32nd
notes!
The downbeat of bar 74 in Duetto 2 really nails it on an accented
entrance of the inverted subject, with Ab in the bass and C in treble.
Yow. Elsewhere, this piece contents itself with leaping parallel
major 10ths....
In Duetto 3, most of the 33 occurrences hit in weak parts of the beat.
Intrigued by this, I did an experimental alternate take for the
recording, registering it with the tierce riding the bass line to hear
these guys duke it out. The bass's tierce therefore creates some tight
beats against the treble's fundamental. Both versions are in this set:
http://www-personal.umich.edu/~bpl/larips/cd1002.html
and the more sanely registered version is also on this disc:
http://www-personal.umich.edu/~bpl/larips/cd1003.html
What surprised me was that these look plenty awful on paper and sound
plenty awful on harpsichords that have strong upper partials...but
the tierce registration somehow worked anyway in the context of real
music, on organ.
The mother of these, C#-E# at the beginning of the C# prelude in WTC 1
-- I think it's interesting that Bach's earlier drafts of that piece
did not have it hitting on the downbeat, but the right hand went
G#-C#-E#-C#-E#-C#....
What I'm wondering is: did Bach perhaps use this bold sound deliberately
to keep pieces moving forward melodically, instead of stagnating
harmonically? There's just enough irritation to keep the listener
aroused and alert, which doesn't happen so much if all of these major
17ths are the same as one another (i.e. equal temperament), or if they're
too well in tune (in the small number of Bach pieces that happen to
work OK in meantone). It's irritation that gets an oyster to hork up
a pearl.
The effect of battling major 17ths also falls off rapidly,
exponentially(?), as distance increases from the instrument. The
harpsichordist hears these strongly while playing, but they sound
less gnarly to the audience or microphones. Is this related to a
string player's and singer's fondness for vibrato, both to warm up
the sound and to help it project presence at a distance?
Brad Lehman
------------------------------
Date: Thu, 16 Feb 2006 13:38:40 -0500
From: Brad Lehman
Subject: Re: "Far out keys"
>(1) Is the aesthetic of "the farther out you go, the more dissonant it gets"
>documented in the historical record for well temperament, or is this a
>modern-day interpretation we have unconsciously applied?
See Rita Steblin's dissertation, now in its 2nd edition as a regular book:
http://www.urpress.com/80460410.HTM
(Easy to get by Interlibrary Loan...)
Brad Lehman
------------------------------
Date: Thu, 16 Feb 2006 13:47:12 -0500
From: Brad Lehman
Subject: Tuned Bach really an ugly wolf A#-->F?
>> (...) the "upside-down" attempt with an wolf 5th: A#-->F broadley
claims that Bach had been so incompetent, that he would had
tuned such an ugly 704 cents wide sharp wolf (...) <<
Is that intended as a joke at my expense?
An "ugly 704 cents wide sharp wolf"?!? Ever heard one? In musical practice
on acoustic keyboard instruments, its sound is scarcely distinguishable
from a 700-cent 5th of equal temperament. Both are the same distance on
either side of the pure 5th, ~702. They both sound like a nearly pure 5th
with a very gentle vibrato. The beats on 704 are simply wobbling in the
opposite direction, subtly.
IMO, anyone who objects to the straightforward sound of a 704 cent "5th"
(or diminished 6th) in musical practice is just stirring up the proverbial
tempest in a teapot...stocked with red herring.
Bradley Lehman
------------------------------
Date: Thu, 16 Feb 2006 14:10:08 -0500
From: Brad Lehman
Subject: "Far out keys" - dominants of minor keys
[RE: Very strong dominants of simple minor keys, e.g. B major cadencing
into E minor, or F# major cadencing into B minor...]
>> PPS. Brad's ... temperament addresses this, doesn't it?
>
>ANY circulating temperament will, even those with Pythagorian thirds.
Not if we at least halfway accept the retorts of Marpurg to Kirnberger
during their protracted debate. (Bach-Dokumente III, #815; or page 449
in the old _Bach Reader_ (1966), or Steblin's chapter about these guys.)
Marpurg asserted that "The late Capellmeister Joh Seb Bach, who did not
have an ear spoiled by bad calculation, must have felt that a major
third enlarged by 81:80 is an execrable interval."
Granted, Marpurg's reasoning is flawed elsewhere in the passage, where
he asserts that it is not even possible to have such Pythagorean major
3rds in existence, as long as _all_ major 3rds are at least somewhat
sharp of pure. He's taking what looks like a hot-headed poke at
Kirnberger here, the pure-interval-boy, and he's crossed the edge of
reason. But still, at least Marpurg is an 18th century writer with
general tuning expertise (proven elsewhere) making a remark about
Bach's taste.
Why indeed should an 18th century temperament connoisseur sit still
for a performance stuffed with Pythagorean major 3rds, in this common
situation of dominant-tonic? (Obviously, some such connoisseurs did,
and some did not!) People wrote and improvised B-minor music, and
presumably the listeners liked it.
Want to see a simple temperament that has all 12 major 3rds somewhat
sharp of pure, but *three* of those sharpened by 81:80? Famous old
Werckmeister 3.
http://www-personal.umich.edu/~bpl/larips/33Werck3.jpg
And its B-D#-F# is very lively, too, due to having a 1/4 Pythagorean
comma narrowing of the 5th....
Brad Lehman
------------------------------
Date: Thu, 16 Feb 2006 18:38:20 -0500
From: Brad Lehman
Subject: Re: Tuned Bach really an ugly wolf A#-->F?
>>Which brings me to an unanswered question of mine regarding said
A#/Bb-F disputed 5th: Why would one refer to it as a diminished 6th in
the context of a circulating temperament? I mean, the lower tone is an
A# or a Bb depending on musical context, is it not? In setting the WTC
Bach temperament (whether your interpretation or my variation of your
interpretation) it is not a leftover, remainder, or any other kind of
different-than-any-other such interval. That's the theoretical and
practical point of a circulating system: Every tone can be either a
sharp or a flat depending on the musical context. The fact that one
may, while tuning, arrive at the A#/Bb tone from a "D#" below doesn't
make it an A#, because of course that "D#" is also an Eb in a
circulating temperament. The use of sharps and flats going off in
opposite directions from C is a convention that has historical and
theoretical and musical import in non-circulating temperaments, but to
carry over the key-specific identity of certain tones from those
non-circulating systems makes no sense.<<
The main point about calling that note "A#", and about having the
whole diagram start on F, is (I believe) an elegant and straightforward
concept. One does all seven naturals, and then one does all five sharps,
and the work is done. A start on F is the only way to do all seven
naturals in succession as ascending 5ths, without crossing into some
of the accidentals first! (And one may "start on F" by getting *any*
of the nearby naturals from a fork; I use either A or C, set my F from
there with appropriate quality, and away we go, simply being careful
not to move the one that came from the fork!) By "5ths" here I mean
as pitch-classes; one can break it back by an octave here or there,
or do some of them as 4ths, keeping the same sequence of all the note
names.
One does not tune the A#-F interval directly; it really *is* a residual
coming off both ends of the diagram. So, why not call it the diminished
6th? Every temperament has to have one, if we want to stay really
technical about things. So does Pythagorean. Run 11 5ths in one
direction or the other, either ascending or descending, and whatever
is left over -- not tuned directly -- *is* the diminished 6th of
whatever size.
I've had an FAQ section about this very point of the A#-F for at least
half a year:
http://www-personal.umich.edu/~bpl/larips/faq3.html
with a couple of additional remarks at
http://www-personal.umich.edu/~bpl/larips/errata.html
Perhaps I did not explain that point well enough in the paper, but
I have endeavored to do so on the web site to help people over that
particular point of confusion! It also addresses the people who
would bomb the "little C" hook-stroke (above the big C of "Clavier")
all the way out of the diagram as meaningful. I believe the diagram
*still* starts on F, with or without that stroke, for the same reason
as explained above. Do all the naturals, and then do all the accidentals.
And the thingy still looks like a very nicely rounded little C to
me, whether any bean-counters vote it in or out of the diagram!
The "little C" business has also been fully addressed on the web site,
and in the paper (a footnote within the second half). Anyone is of
course free to disagree with this, and claim variable mileage.
Myself, I'd rather go play music than niggle about nomenclature too
much. If we want to call the same key-lever "E#" or "F" on two
different occasions, in different musical contexts, who *really*
cares that much as long as the sound coming out sounds good in both
contexts? Likewise, whatever "B" is set up on the keyboard, it
had better sound decent played as a Cb wherever those come up in
the music. That is a point of my "enharmonics" analyses.
>>By the way Brad, I am using my slightly altered version of your WTC
"Bach" temperament in my upcoming concert this weekend. I find that it
just works so well (no pun intended) on much 18th century literature.
Thanks.<<
Best wishes on the concert! What are you playing?
Brad Lehman
------------------------------
Date: Thu, 16 Feb 2006 18:50:23 -0500
From: Brad Lehman
Subject: circulating temp with a wide 5th in it...
>>I say this as someone
who also is politely troubled by a wide 5th in your interpretation on
the grounds that it seems to be unusual for circulating "well"
temperaments of the 18th century (...) <<
Is the wide interval in Neidhardt's third-circle #4 (1732) disturbing?
http://www-personal.umich.edu/~bpl/larips/00NeidhardtThirdCircle4.jpg
Eb to Bb (or D# to Bb, or call it whatever) is 1/12 comma wide of a
pure 5th; 704 cents.
Here's my quick and easy way to set that temp up on a harpsichord:
NEIDHARDT 1732 "Third-Circle #4"
Ab -60 Eb +60 Bb -120 F 0 C -120 G -120 D -120 A 0 E -120 B 0 F# -60 C# -60 G#
1. F-C-G-D-A-E in regular 1/6 Pyth.
2. C-F-Bb-Eb-Ab pure.
3. E-B-F# pure.
4. C# equally tempered from F# and G#.
5. Retune F down so it's now pure from C.
6. Retune E up so it's now pure from A.
7. Nick D# downward the slightest bit (1/12) which improves it with B,
and makes it slightly wide from Bb.
from:
http://www-personal.umich.edu/~bpl/larips/practical.html
Likewise, some of the lesser-known Werckmeisters also have wide 5ths,
considerably wider than 1/12, back in the daddy era of this topic....
Granted, not the 18th century on those.
Brad Lehman
------------------------------
Date: Fri, 17 Feb 2006 14:13:27 -0500
From: Brad Lehman
Subject: Re: circulating temp with a wide 5th in it...
> Is the wide interval in Neidhardt's third-circle #4 (1732) disturbing?
> http://www-personal.umich.edu/~bpl/larips/00NeidhardtThirdCircle4.jpg
> Eb to Bb (or D# to Bb, or call it whatever) is 1/12 comma wide of a
> pure 5th; 704 cents.
>
> NEIDHARDT 1732 "Third-Circle #4"
> Ab -60 Eb +60 Bb -120 F 0 C -120 G -120 D -120 A 0 E -120 B 0 F# -60
> C# -60 G#
Das ist:
Ab -1/12 Eb +1/12 Bb -1/6 F 0 C -1/6 G -1/6 D -1/6 A 0 E -1/6 B 0 F# -1/12 C# -1/12 G#
oder...
Ab -1 Eb +1 Bb -2 F 0 C -2 G -2 D -2 A 0 E -2 B 0 F# -1 C# -1 G#.
=====
And his third-circle #3, right next to it on his chart, has *two* wide
5ths of 1/6 comma each! Also remarkably, this temperament has zero pure
5ths in it.
Ab +2 Eb -2 Bb -1 F -1 C -2 G -2 D -1 A -2 E -2 B +2 F# -2 C# -1 G#.
Note that feature with the note F#. It's first tuned pure with both B
and C# on either side, and then it's knocked upward 1/6 comma...making
it serve as a brighter sharp, smoother flat, and coincidentally a wide
5th (706 cents) on one side. Likewise with Eb: originally pure between
both Ab and Bb, and then knocked upward 1/6 comma: it too becoming a
smoother flat and brighter sharp.
Neidhardt *could have* had four pure 5ths in this temperament (B-F#-C#
and Ab-Eb-Bb), but he chose not to have any, focusing instead on the
qualities of his major 3rds by nudging some of the notes off the
expected position.
Knocking the F# upward takes plenty of the edge off an excessively
bright F#-A#, as a trade-off of a less calm D-F# (which incidentally
becomes wider than A-C#!). Knocking the Eb upward softens a very wide
Eb-G, and widens B-D# so the latter becomes the same size as F#-A#,
and so Eb-G isn't wider than Eb-G.
So again, not such an odd thing to do. One just has to stare at these
temperaments for a while, and better yet try them out directly at the
harpsichord, to figure out what the strategy is in them. Nudge a
single note up or down by some amount, and two 5ths change accordingly,
and two major 3rds change accordingly. If the temperament ends up
with one or more slightly wide 5ths, so be it...as long as the
temperament as a whole sounds good for the music to be played!
And isn't this the sort of thing referred to by the Sorge teacher/student
quote, a dozen years later? One *can* make up interesting and useful
temperaments that have some wide 5ths in them [via just this sort of
dinking around with individual notes, thinking at the keyboard], but
"it's unnecessary"....
Barbour's chart of this Third-Circle #3, "table 158" at the top of page
171, has both the notes Bb and B transcribed incorrectly, along with the
# missing from C#. Yet more of so many typos in this otherwise fine book.
Barbour has mis-transcribed all three of the Neidhardt 1724 temperaments
(tables 151/155/156), and this one from 1732 (table 158), among others;
I haven't found the patience yet to go through everything in the book
looking specifically for corrections. His MGG article is riddled with
a similar batch of typos.
(Obviously we should cut him some slack, though: imagine performing
this incredibly detailed book on a tracker typewriter, through more
than one draft!)
Reference on these two Neidhardt third-circles: section VII of
http://harpsichords.pbwiki.com/f/Neidhardt_1732_ascii.html
...thx again Gordon for making this widely available....
Brad Lehman
------------------------------
Date: Fri, 17 Feb 2006 23:57:51 -0500
From: Brad Lehman
Subject: irritating vs active
Tom Dent wrote:
> ... one man's 'irritating and unusable' is very close to another man's
> 'active'. Just to be clear, Brad's tuning makes the F# major triad
> *better* than A major, E major and B major, and also B major better
> than A major, so you have to consider what does make musical sense to
> you. Would you extend the bathroom to be larger than the kitchen?
That is an excellent point, Tom. And, I spent plenty of time in that
skepticism/incredulity myself. Then, I gradually realized that this
temperament fosters a primarily melodic and contrapuntal manner of
playing, ahead of any triad-thwacking that would over-emphasize ANY
of the simple harmonies or their relative qualities.
It has transformed my approach to the instruments, as a player
(harpsichord/organ/clavichord). Music is full of all manner of
interesting dissonant/consonant contrasts and horizontal tensions,
which I sort of knew before, but which has been hammered home to me
by playing in this temperament for a long time now.
And, I feel that I have learned more about excellent practical keyboard
musicianship in the past 22 months than in the preceding ten years
(I stopped formal study 12 years ago...), and my independent fingerwork
has improved quite a bit as well. The thing makes me *want* to play
in the manner that Bach said was desirable, on the title page of the
Inventions/Sinfonias.
If I were an autoharp player, I would do it in meantone -- and have
done, experimentally. That instrument fosters chord-thwacking as its
basic sound. Strong and sweet harmony are paramount, and no complex
harmonies are even possible. Therefore, meantone sounds tremendous
on autoharp. But, the classic keyboards playing Bach's complex music
are not anything resembling an autoharp.
Brad Lehman
------------------------------
Date: Thu, 2 Mar 2006 17:24:28 -0500
From: Brad Lehman
Subject: discoveries: some relevant squiggles ....
> > As many here know I don't favor Vallotti or Young, of course, but as
> > scholar/scientist I have to accept the authority of the autograph, which
> > clearly shows Bach's intention to everyone.
>
> In case you plan to publish your discovery, might I suggest inverting the
> diagram to secure popular appeal?
Enough of such caustic smirking (the latter sentence) that both
misrepresents and belittles my work. More than enough.
What our resident cynics and rhetoric-spinners fail to grasp, or even
perhaps to know, is that I had my formula installed and in use on my
harpsichord for at least TWO MONTHS, before the day I realized that
*another* convenient way of explaining it in the paper would be to rotate
the page 180 degrees on a tabletop.
I simply read the thing from right to left, before that, during the
course of exploring various possibilities in both directions (March
into April 2004), and being especially drawn to the way the right-to-left
sounds on real harpsichords in my manner of interpreting it.
My argument doesn't stand *or* fall on flipping the book. It never did.
Nor does it stand or fall on accepting or rejecting the "little C",
which I have also explained in many ways on my web site. Those who
have been picking and plucking at my work, for their own (dis)satisfactions
on these and other little points of argumentation, are apparently unable
or unwilling to grasp the larger musical picture...at least as I see
it. Let me try to explain.
=====
"Popular appeal" is, I believe in largest part, because my resulting
layout sounds musically attractive and useful on the types of acoustic
keyboard instruments that Bach knew and worked with.
Whatever anybody thinks of my argumentation or my writing style is
really neither here nor there. The thing works so well for Bach's
keyboard music, and indeed for all tonal music, ...that's the appeal.
The gist of my spring 2004 discovery is this, boiled down to a single
sentence about the musical phenomenon:
>>>Keyboard instruments playing tonal music sound extraordinarily
rich when the interval C-E is 3/11 syntonic comma (or environs) sharp
in size, E-G# is noticeably wider than Ab-C, and all the other major
3rds are arranged in a smooth progression through these three fixed
checkpoints.<<<
Through some complex confluence of acoustics, human musical perception,
and the normally rule-based behaviors of tonal music: layouts with
those specific parameters make tonal music sound strong, colorful,
emotionally involving, interesting, intellectually satisfying, and
in some way seeming perfectly natural.
The Bach diagram can be read in at least one plausible and simple
way that delivers exactly that result, with these C-E-G# checkpoints
and a smooth enough arrangement around them. Actually, I know of two,
and I am rather fond of both of them in my actual practice playing
the WTC on harpsichord: the right-to-left layout that I have published,
and a remarkably different left-to-right layout that I have not
(because the first one on balance makes what I feel is a better and
more elegant argument in public). I have put most of my energy into
that first one, because it to me looks the more intuitively obvious
reading. The two sound scarcely distinguishable from one another,
anyway, having the same pattern and relative strengths of the same
twelve major 3rds, and comparably smooth 5ths all around. Up one side
steeply from C major (calmest) to E major (brightest), and then
gradually transforming back around the other side until we reach C
again. All the scales have their own distinctive "signatures" in
their melodic/harmonic sounds.
Here's a broader point, though:
Any competent harpsichord tuner-by-ear should be able to set up a whole
manual with no fuss at all, in 10 minutes or considerably less, by this
simple expedient and counting *no* beats anywhere:
- Establish C-E at the right size (at or near 3/11 syntonic comma) by taste
and experience.
- Crank the G# up until it's just at the breaking point of being tolerable:
just barely inside where it goes as Pythagorean result if we'd do
E-B-F#-C#-G# all pure. The sharp keys around E major have a noticeable
brightness and hardness ("dur"...) to them.
- Ab-C is then, by residue, noticeably calmer and "warmer" than E-G#.
This gives a rich and mellow sound to the flat-side keys around Ab major,
as plenty of contrast against sharps-music or naturals-music. There are
these three distinctive characters, all gradually blending into one another.
- Average out all four of the 5ths C-G-D-A-E the same as one another
within the fixed endpoints C and E. (An aside: this evenness is also
important if we're going to have any string players tuning all their
open strings to the result...and we shouldn't forget that Bach himself
was at least a highly competent string player.)
- Average out all four of the 5ths E-B-F#-C#-G# the same as one another
within the fixed endpoints E and G#.
- Average out all four of the 5ths Ab-Eb-Bb-F-C the same as one another
within the fixed endpoints Ab and C.
- Either leave it at this point as finished; or do some tasteful
micro-adjustments of the same type common to "ordinaire" practice going
back into the 17th century: most notably, making each of Bb and Eb a
wee bit lower so the F#-A# and B-D# will have a slightly more friendly
quality in their musical contexts.
That's the theoretically optimal(?) object that I believe Bach was
describing by example: very closely approximating this particular shape
of the 12 major 3rds in their relative qualities. Its fundamental shape
has nothing to do, in particular, with exact measurements of 1/6 comma or
beats or cents or any other modern scientific voodoo; it's harmonic and
melodic relationships. 1/6 comma is merely a convenient way to measure
and describe the workable range of "good taste" where the shape works
itself out with internal consistency, and exhibits the right balance.
It has nothing to do with any electronic devices of any kind, which
Bach (obviously) didn't have anyway. The main point of
memory/taste/experience is to know the workable quality of C-E as a
direct interval, at whatever pitch they would happen to be in modern
Hz measurement.
Sit down and just *do* the thing, instead of arguing about any numbers
or any scientifically measurable refinements, or anybody's written
rhetoric. Just do the music, and the result makes its own case for
existence. Those who have ears to hear the music, let them hear.
On some days, if I feel like it, I just do that procedure above, straight
up and not caring about any of my current derivations from the Bach
diagram. The musical result is essentially the same, whether we're
dealing with the diagram or not, as long as we've got that particular
set of parameters on the resulting keyboard layout.
And then beyond that: who gives a &%#& if one 5th turns out to be some
micro-smidgeon wide; or if some people who don't tune harpsichords at
all (regularly or ever?) can't get their speculative minds around it;
or if the most thorough and overly cautious historians (which is generally
a good thing!) fail to accept anything as even *possibly* true, until
it's been demonstrated beyond any shadow of their *own* doubt (and
preferably in written documentation that's seemingly unambiguous words
and/or numbers)? No, the broader point is to play practical music,
both composed and improvised, to hear what works well.
My practical/theoretical connection then has been: I believe the Bach
diagram demonstrates (in at least one compelling way) evidence of that
strategic thought-process, setting up the C/E/G#/C to appropriate
spacing. One works directly at a harpsichord with the tuning lever
in hand, making such tiny adjustments until the music sounds optimally
good. And that is a major point of the published paper, and my later
web site clarifying it. I believe that Bach found a way to write down
that particular sound that happens to give outstanding results in music.
I'm trying to be a responsible historian about that, to the best of
my ability: bringing in as much plausible corroborating evidence as
I can find. Some might say that my attempt shows some decent ability,
or some might say it's near zero. Whatever anybody thinks of me
personally is less than relevant; the music still comes first. My
construction of any theoretical/historical/musical argument takes a
back seat to letting the extant music speak for itself, and the
diagram itself as evidence of same.
I personally wouldn't have seen or seriously tried this point about
that specific C-E tasteful size, plus the E-G# > Ab-C, because it
seems so crazy and far off the normally beaten paths; most of the
unquestionably documented temperaments have Ab-C > E-G#, or at a
limit they're equally sized. For many years until 2004 I had accepted
that situation as simply the only (or normal) way things work for
unequal temperaments, case pretty much closed, even though some of
the music sounds raunchy that way.... But, some things in the Bach
diagram sparked me to experiment with this other set of relationships
directly at the instrument, E-G# > Ab-C, to hear how it works in
his real music with the sharps turned up so high and the flats
correspondingly so gentle as tonics.
Everything else has been merely a writing exercise, along with a huge
amount of (continuing) research and experimentation. Anomalies grab
my attention, and they don't let go, and I try to puzzle them out to
the best of my ability. They wake me up at night, fairly often.
The anomaly of an irregular and asymmetric diagram has done so, and
continues to do: why would Bach write out such an irregular doodad
unless it is meaningful in some important way?
And *if* Bach had some excellent-sounding method in hand, as this
one happens to be, why would he care one way or another if everybody's
*published* temperaments around him couldn't see the efficacy of
E-G# > Ab-C? Excellent musicians, after all, are allowed to think
and do things that other people have not published or documented to
their own satisfaction; and part of the job of modern historians is
to find extant clues into practices that might not have been
documented to the satisfaction of positivists.
Historiography involves leaps of faith all over the place, and it's
just a question of where. I believe that expert musicians of the past
presumably had ears that were at least as sensitive to nuances, as
modern sensitive musicians claim to have--allowing that every individual
is somewhat different from every other, and that the cultures have
shifted a bit as well. But that's why they were experts in *their*
milieus: their ability to hear closely, and to control their materials!
Any positivistic argument has to cut both ways, to be
fair/meaningful/useful: one can't prove that the dead experts *did not*
do things that happened to work beautifully, but didn't trouble to write
down in some way sufficient to modern satisfaction. I personally am
quick to grant that the dead experts really did know what they were
doing, perhaps more than they're usually given credit for in modern
skeptical historiography. Those dead guys (the Bachs and their best
colleagues and contemporaries) are after all the witnesses to their
own, and one another's, expert practices. Any clues to those practices
are bound to show up, somewhere/somehow, in their extant music plus
any other written documentation they might have left...not necessarily
in words.
Documentation doesn't always have to look exactly like anyone later
might expect it to do. It only needs to have been sufficient for any
who used it originally. Would anybody seriously make a case, anymore,
that the existence of only unmarked orchestral parts to some composition
constitutes some kind of proof that any of those musicians played
it without any nuances, since those nuances aren't notated (or restricted)
to the satisfaction of the most cautious positivism? Nay, good musicians
living or dead will always try to use their best
instincts/training/sensitivity to make a compelling musical product
out of whatever sketchy piece of paper they happen to be looking at.
That's a difference between musical expertise, and mechanistic
sight-reading by some restrictive set of rules.
Bach knew how to make music sound good, and especially his own music
that he composed or improvised. Has that premise ever been in question
by anybody? Bach was a genius and a musical expert. Likewise, does
anybody seriously dispute that? Well then, does this particular
pattern about C/E/G#/C present some plausible view into his historically
"knowable" practice(s) on at least one occasion? That's what the
debate--on its good days--is about. The reports said that Bach was
brilliant in melody/harmony, and in thorough mastery of his materials,
and that he really didn't care about any numbers, and that he disliked
any unnecessary roughness within his musical practices. Those points,
to me, are some awfully important ones. Bach's task, as with any
excellent musician, was/is to do what all of one's instincts and
training and experience show to be simultaneously practical, effective,
and intense enough for the musical content.
And, I personally put exactly zero stock in any notion that any of the
flourishes in Werckmeister's book are in any musical way meaningful.
Sorry!
Bach's WTC title page scrawl still appears to me to be a one-off
notation of his own devising, as far as I have seen to date. But I
do still suspect that others in his family before him knew how to tune
similarly or identically, at least in the most important specifics,
to get their own music to work. They too were known as harmonic
innovators, especially uncle Christoph the organist in Eisenach around
1685. And obviously JSB's capriccio in E major (BWV 993) dedicated
to the other Christoph, big brother in Ohrdruf, had *some* way of
sounding musically compelling...which it happens to do in my layout
very nicely, all the way through its double-sharps.
=====
On another topic: what the author of a July 2005 Internet paper fails
to point out to anybody is: his own readings of the drawing right-to-left
amount to EXACTLY THE SAME RESULT he'd obtain if he'd flip the book
around 180 degrees himself. It's nothing more than rhetorical spin
by him as a red herring: one can read it from right-to-left, or flip
the book. The author just uses that point against flipping the book,
in more than a dozen occasions I've seen, to try to make Bradley
Lehman look less than credible in public. And, that's nothing more
than cynically-barbed verbal abuse by him. It says nothing about Bach,
or music, or scientific inquiry, or historical responsibility.
It's not even any manner of argument about harpsichord tuning, one
way or another.
I guess we're just supposed to overlook the fact that this same author's
June 2004 Internet paper proposed four seriously bizarre 1/3 comma
temperaments, claiming to stem directly from an esoteric interpretation
of a wholly different Bach document. That is, while he was making up
such things and trying to grab some limelight as some manner of
researcher, I and several other musicians had already been using my
now-published layout for several months already in real musical practice.
Likewise, I object to his incessant prattle/scoffing against any layout
where "the tuning scheme starts on F", while his own allegedly preferred
"Cornet-Ton" temperament does EXACTLY THE SAME THING putting F-C-G-etc
into exactly the same position, reading "right to left" as opposed to
flipping the whole page around. What is the *&@% difference, other
than his semantics to make his thing look somehow superior, in some
allegedly measurable way? Some public anti-Lehman joke or abuse by
him that he has now sustained for more than a year, for his own amusement?
Or is he simply unable to comprehend his own work? Has the man ever
actually tuned a harpsichord, by ear or otherwise? I asked him once,
some time ago, and he said that his work was done entirely by numerical
analysis....
Enough about that person in particular. His usual red herrings continue
to be dead fish argumentation, no matter how many times he presents them.
A strong and convincing musical sound always comes first...at least for me.
Bradley Lehman
http://www.larips.com
------------------------------
Date: Fri, 3 Mar 2006 12:19:27 -0500
From: Brad Lehman
Subject: meantone 101
>>>I think what is needed is some way of making tuning easier for the
beginner and amateur. I try to get my students to learn how tuning
works by doing it, but most are happier with a machine. I suppose it's
an issue of confidence. I don't use a machine and find the needle
movement more distracting than my ear, which I trust much more.
When I read much of our list's postings, the tuning 'recipes' do make
sense, but to the novice they may appear difficult. If anyone has found
a way to 'crack the code' in making this easy to amateurs and students I
would love to know. It's a bit like baking bread or making pasta- to
the unitiated I seems mysterious, yet others say how easy and simple it
is. Any ideas?<<<
Here's a rather quick and informal go at explaining "meantone" (i.e.
the class of all _regular_ temperaments where 11 of the 5ths are all
the same size as one another, and then there's one leftover diminished
6th that is not tuned directly). A No-Fear general method of regular
tempering, without using any specific beat rates anywhere.
- 1. It is important to understand that the concept of "same size" means
interval *quality*, and has nothing to do with beat rates or any other
type of numeration. It's the ability to hear a consistent amount of
wooziness in an interval, no matter where that interval may be located
on the keyboard. The interval is typically tuned pure first, and then
one of the notes is moved slightly so a little bit of deliberate dirt
is introduced into that clean sound. Our goal is to introduce the *same*
amount of dirt into all the places where it is appropriate, and to
recognize by ear when we have done so correctly.
- 2. To start, get one note (typically C) from a suitably reliable pitch
source. Establish middle C.
- 3. Set the major 3rd above it to be pure. C to E.
- 4. If our goal is to set something that's not exactly 1/4 comma (and
that *is* our goal for this exercise): move that upper note (E) upward
a little bit to taste. Introduce some vibration in that major 3rd, so
that it still sounds good to you without becoming obtrusive. That
vibrato in there is called "beats". The C to E major 3rd is widened.
- 5. The exact measurement of that amount only matters if we're trying
to be too scientific about it. But, for this exercise, we're focusing
on taste and the ability to hear. Just crank it up some little amount,
so it still sounds resonant but gets a little bit of activity to it.
The beats should be fast enough that they give the sound some nice
character, but not so fast that they turn into a machine-gun effect or
a blur, at middle C to E.
- 6. We really don't care for now if we're hitting "1/5 comma" or
"1/6 comma" exactly, or which of the two "commas" we're trying to split,
to please any theorist who might care about numbers. We're just
focusing on the ability to hear some tasteful amount by experience.
Anywhere in that continuum of "1/4 comma" up to about "1/7 comma" is
OK for this. (The nomenclature such as "1/6 comma" refers to the *specific*
amount of dirt we are introducing into each 5th/4th; it's not necessary
for this exercise, beyond that brief mention. And, "1/4 comma" is a
shorthand way of saying that our major 3rd is pure.) Importantly, we
want to stay shy of the place around 1/7 or 1/8 comma where the interval
of middle C up to E merely turns into an undifferentiated and ugly blur.
That would negate a main goal of trying to do "meantone" in the first
place: which is to have as many excellent *and consistent* major 3rds
available as possible, and each one considerably smaller than they
would be in equal temperament.
- 7. OK, we've got our nice major 3rd as our boundary. Now we have to
fit four equal 5ths inside that boundary: C-G-D-A-E. This is most easily
done by alternating 4ths and 5ths. This is where we get our most basic
techniques of listening to the 5ths/4ths for consistency. A 4th is
merely a 5th in which we have jumped one of the notes over to the next
octave, on the other side. We are dealing with slightly *narrow* 5ths,
and correspondingly we need to have slightly *wide* 4ths. Our octaves
must always be pure. (Conceptually: start with a pure octave somewhere,
and put a pure 5th into it, up from the bottom note of the octave.
Now knock that middle note slightly lower, making the 5th narrow.
The 4th above it automatically becomes the same amount wide.)
- 8. Tune downward a 4th from C to G, making it pure, and then make the
G slightly flatter from that so a gentle wobble develops. The general
principle is to make 4ths wider than pure, by a tastefully small amount.
Since we are tuning the lower of the two notes here in C-G, the G,
make it flatter so the interval becomes wide.
- 9. Tune downward a 5th from E to A, making it pure, and then make the
A slightly sharper so a gentle wobble develops. The general principle
here with 5ths is to make them narrower than pure, by a tastefully small
amount. Since we are tuning the lower of the two notes here in E-A,
the A, make it sharper so the interval becomes narrow. Confirm that
the E-A 5th has a similar amount of wobbliness that we gave to the G-C
4th: not by counting their beats numerically, but by quality.
- 10. Tune upward a 5th from G to D, making it slightly narrow. In
this case, since we are tuning the upper of the two notes, it needs
to be *flattened* from the pure point. Check that this D makes a
similarly slightly wobbly sound as a 4th from the A already tuned: wider
than a pure 4th, i.e. sharper. An important concept is here: we are
putting our D at the geometric midpoint between D and A; we are also
putting it at the melodic midpoint between our original C and E.
This is the reason such temperaments are called "meantone": the "tone"
(whole step) within the major third is exactly "mean" or average, as
the midpoint.
- 11. Check all these 4ths and 5ths C-G-D-A-E again, listening closely
to make sure all of these have the same quality: the same amount of
wooziness or dirt away from purity. C-G, G-D, D-A, A-E. If some of
them are too nearly pure, at the expense of others wobbling too fast,
make the adjustment until they sound the same. The general principle,
again, is that all the 5ths have to be narrow and all the 4ths have to
be wide.
- 12. From this point forward, the rest of the temperament is easier.
All we have to do is to reproduce this same amount of dirt everywhere,
consistently, by testing both the 5ths/4ths and any major 3rds that
become available as we go along.
- 13. Tune a 5th downward from middle C to F, and make it slightly
narrower than pure, i.e. the F a bit raised. Test that this F-C has
the same amount of wooze/dirt to it as the G-D 5th next to it, already
completed. Also test that the major 3rd F-A has the same amount of
brightness to it as our original C-E major 3rd. We are trying to get
all of the major 3rds to have this same consistent sound, all equally
sharp of the point where they would have been pure.
- 14. Tune a 4th up from F to Bb, making it slightly wide. Test that
the F-Bb 4th has the same quality as the G-C 4th nearby; and that the
Bb-D major 3rd has the same quality as our original C-E, and as the F-A
that we just finished setting up.
- 15. Tune a 5th down from Bb to Eb, making it slightly narrow; test
its quality to be like F-C, and test the Eb-G quality to be like F-A.
- 16. Tune a pure octave Eb up to Eb, just so we don't forget to do it
later. Likewise, tune a pure octave from tenor F up to the F above
middle C.
- 17. That is as far as the flats go: Eb is the end of the road. Now
we must finish the B and all of the sharps. The notes such as Ab, Db,
Gb, etc *do not exist* in this temperament! But, don't worry about
that for now.
- 18. Tune a 4th down from E to B, wide; test its quality to be like
the A-D 4th, and test the G-B quality to be like both F-A and C-E.
- 19. Tune a 5th up from B to F#, narrow; test it to be like A-E, and
test D-F# to be like C-E.
- 20. Tune a pure octave F# down to F#. Also tune a pure octave down
from our original E to the tenor E. Test that the E-B 5th is still like
F-A and G-D, in quality. We are just doing a consistency check here.
- 21. Tune a 5th from tenor F# up to middle C#, narrow: same quality 5th
as G-D and F-C around it, and same quality major 3rd A-C# as the G-B and
the Bb-D already done.
- 22. Tune a 4th from that C# down to G#, wide: same quality 4th as G-C
and the A-D around it; same quality E-G# major 3rd as the F-A.
- 23. We now have all 12 notes: Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#. All 11 of
those 5ths or 4ths have the same quality as one another, the same geometric
size. All eight of the major 3rds Eb-G, E-G#, F-A, G-B, A-C#, Bb-D, C-E,
and D-F# have the same quality as one another. You might notice that
the vibrato gently increases in speed all the way up that test play,
but that the interval quality still sounds the same. That is the
confirmation that we have done it correctly. Beat speeds (the vibrato)
increase as we move up the keyboard, because all the strings are vibrating
more rapidly themselves to make the higher pitches.
- 24. Confirm that the following intervals are *not* particularly good!
G#-Eb will be quite a bit wide, and maybe unlistenably so, to the point
of raw ugliness. That is the "wolf" 5th, really a diminished 6th by its
spelling of the note-names! F#-Bb, G#-C, B-Eb, and C#-F are our four
diminished 4ths: they look like major 3rds on the keyboard, but the
spelling makes them diminished 4ths instead. They will all be much
wider than the eight major 3rds we set and tested earlier. The notes
named D#, A#, E#, B# *do not exist*, but are only simulated (roughly) by
a flat or a natural that was tuned from the other side. Likewise, Ab,
Db, Gb, Cb, Fb are not available but are only simulated (roughly) by a
sharp or a natural. Our strategy here is to set up the
sharps/flats/naturals that get used most often in music, making them
sound especially good by the major 3rds around them, and to let all the
other incorrectly-spelled notes be relatively rotten.
- 25. You have just met the five "wolf" intervals that are common to the
"meantone" or "regular" style of temperament. Most of the music that
would be appropriate to "meantone" style does not use these five wolf
intervals, except very carefully as special effects occasionally. The
emphasis, instead, is on the sonorous quality of the eight correctly
spelled major 3rds, all having the same gently tempered quality as one
another.
- 26. Play the chromatic scale all the way up from Eb, E, F, F#, etc.
listening closely to the sizes of these semitones. There are two different
sizes. Whenever we go between two notes of the same *name*, such as Eb
to E, or G to G#, that is a small-sized semitone and is called "chromatic".
Chroma: color change. When we go between two notes of *different* name,
such as E to F or from B to C, that is the large size and is called
"diatonic". All of the regular meantone layouts have two and only two
different sizes of semitones.
- 27. Test also the effects of playing the minor 3rd D-F immediately
followed by the major 3rd D-F#; the minor 3rd C-Eb followed by major 3rd C-E;
and the minor 3rd G-Bb followed by G-B. Try it also with the entire
triads G-Bb-D and G-B-D. This is another aspect of the color-change,
the chromatic phenomenon: these alternations give an effect that we are
staying somehow on the same harmony but merely changing its tone color.
That is one of the beauties of regular (meantone) temperament.
- 28. Look all the way back to steps 3 and 4, where we originally had
our C to E pure. If we had left it that way, instead of sharpening the
E a little bit, we could also go through all the rest of the instructions
using only pure major 3rds. (Try this on a different occasion.) With
the same principle of having all eight of the "good" major 3rds the same
as one another, all of our test points would give us pure major 3rds on
those eight: Eb-G, E-G#, F-A, G-B, A-C#, Bb-D, C-E, and D-F#. There are
several trade-offs to this otherwise lovely situation. All of our
5ths/4ths have to be more aggressively wobbly; the wolf intervals F#-Bb,
G#-C, B-Eb, C#-F, and G#-Eb end up sounding even more caustic; and our
two sizes of semitones become even more different from one another.
When we are listening closely to melodies played in meantone, these
different sizes of steps can start to become obtrusive...all being a
matter of taste, of course.
- 29. A general point is to be able to *control* all this, and to realize
that it all follows as a consequence of the original major 3rd size we
had chosen, way back in step 4. That single decision determines all
the sizes of all the 5ths, 4ths, semitones, major 3rds, and the wolves:
and the overall melodic character of the results, too!
- 30. Having a chromatic scale that we're happy with and understand,
all the way up from tenor Eb to the F# above middle C: it is now time
to finish the rest of the octaves. Everything from here forward is
simply pure octaves. From the tenor D, get a pure octave from the D
above it, and test that our new D-A 5th still has our consistent
quality. Go next to the C#, setting and testing it similarly...and
continue all the way down to the bottom of the keyboard, chromatically.
The only interval we cannot test as a 5th, remember, is G# to Eb.
And, you might notice that the vibrato speed continues to decrease,
smoothly, all the way down until it's imperceptible: but all our
5ths still have the same *quality* as one another.
- 31. Similarly, to do all the treble, chromatically upward from
the G above middle C. Get the pure octave from below. Going up
the treble range, it is a useful exercise and checkpoint to test
each note not only as a pure octave, but also with *both* the 5th
and 4th intervening. Those two need to have the same quality as
one another, and it is an excellent way to prove the purity of the
octave. Once again, remember that G#-Eb has to be skipped over: but
all the rest of the 5ths/4ths need to have the same quality as one
another. Also notice that they have a steadily increasing beat
speed, all the way up, until they turn into a blur each: and all
we can here then is the consistently woozy *quality*, not able to
count the beats in any useful way.
- 32. That crossover point from distinguishable beats into blur
happens somewhat differently on each individual instrument; and it
also depends on the size of major 3rd chosen all the way back at step #4.
- 33. Reviewing the basic concepts we have learned here: we have
learned how to install a basic temperament *shape* all the way up
and down the keyboard. That regular shape varies only by intensity,
all dependent on the size of major 3rd we chose at the beginning.
The narrower (or closer to pure, or maybe entirely pure) we have made
our original major 3rd, the more intense all the contrasts are
everywhere else as a result. Consistently we end up with five "wolf"
intervals in the same places; and they vary only in nastiness,
determined by the size of our original major 3rd.
- 34. We have also learned and heard how we may verify pure octaves
anywhere on the keyboard: by testing the 5th and the 4th within that
octave to be sure those two intervals have the same quality as one another.
This is useful during the setup of the whole temperament, or if we are
simply correcting a few weather-drifted notes without re-doing the whole
thing. Our 5ths and 4ths can quickly tell us which of the two notes
in our octave is the one that needs to be fixed. (Well, it is usually
the one that's farther away from middle C...but not always! The middle
register of a harpsichord tends to stay better in tune than the extreme
treble or bass do, through weather shifts. That is why we do it first:
plus the ability to hear the qualities most clearly in that region.)
- 35. This exercise, in part, is to train our ability to hear consistent
major 3rds; and to recognize the way their size depends on the consistency
of the four intervening 5ths/4ths. All major 3rds are generated by
a cycle of four 5ths/4ths. This knowledge can be used as checkpoints,
with the major 3rds proving the 5ths/4ths, and vice versa.
=====
All (or let's say "virtually all") of the other "modified meantone",
"ordinaire", and "well temperaments" are little more, conceptually,
than starting from a framework of some _regular_ layout...and then
making some of those 5ths *less* tempered than we would do if trying
to set all 11 consistently. By "less tempered" we mean that some of
the 5ths are less wobbly, i.e. wider than they would be in their regular
positions: and maybe even so wide that they become pure, or go slightly
beyond pure to become a little bit wider yet, and start wobbling again
in the opposite direction!
The larger goal there is to make the cycles of 5ths and major 3rds meet
somewhere around the back alley, the musical keys with plenty of sharps
or flats in the signature: so more of these major and minor scales/harmonies
sound reasonably playable, and so the wolf leftovers either go away
entirely or at least become less offensive to the ear. In those layouts,
it does become more important to get the starting major 3rd to some
known and controllably specific size, within reasonable tolerance.
But again, the first thing to learn is the ability to do a complete
_regular_ layout of all 11 of the 5ths/4ths--of various tasteful yet
consistent sizes--before starting to mess around with modifications.
Some of my other remarks about various examples of regular systems:
http://www-personal.umich.edu/~bpl/larips/meantone.html
There exist techniques -- easy ones by ear -- to make sure that all the
intervening 5ths/4ths within the major 3rd are really the same size as
one another, and not merely a guess. But, that's not the purpose of
this initial exercise. For completeness, I've described my favorite
one here with an analogy to paper-folding:
http://www-personal.umich.edu/~bpl/larips/tetrasect.html
Bradley Lehman
------------------------------
Date: Fri, 3 Mar 2006 16:04:41 -0500
From: Brad Lehman
Subject: Re: squiggles on the web
>>
http://www.rzuser.uni-heidelberg.de/~tdent/
For those that prefer to take their squiggles without hypotheses.<<
Heidelberg? I thought you were a researcher/professor of theoretical physics
in Greece, and then recently moved to England?
I think that at least the title page and several other pages of Fischer's
_Ariadne musica_ should also deserve to be there at your collection. We
know for sure that Bach was acquainted with that published book, while we
don't know one way or the other if he ever encountered Suppig's manuscript.
Several of the spirals in Fischer's book look virtually identical to Bach's
individual loops: especially the one at Fischer's name. (Designed by Fischer?
Or someone on his publisher's staff? Variously through the book some of them
look pre-fab, and some are especially ornate....)
As for Suppig, and the dilettantism displayed by both his handwritten
document and his interminable piece of music: what appeal would there be
for expert Bach to parody a dilettante? It's not as if Suppig's work had
anything to offer Bach's interest, but mainly some dull pages of numerical
calculations, accompanied by a thoroughly formulaic composition.
That's why I have suggested that *if* anyone was parodying anyone else
with their drawings (whether meaningfully or not) and/or their compositions,
probably Suppig and Bach were *both* imitating the Fischer work, separately.
Fischer's music offers an inspirational musical labyrinth, years before
either Suppig or Bach. See also Craig Wright's book for remarks about this:
http://www.amazon.com/gp/product/0674013638
...and his art |