Practical temperament instructions by ear

These temperaments, among others, appear at this summary comparison page.

Bonus 3: an easy Lehman temperament, emphasizing tasteful listening

I have formulated this temperament also from elements in the Bach drawing, but in a less obvious way. (Full discussion of it is in my review of O'Donnell's article.) I still like my first one better, overall, while this one works very well in the same repertoire. Both are equally easy to set up in practice.

• C from fork, and over to middle C.
• Set E from C, somewhere near the quality of regular 1/6 comma (i.e. with the familiar sound from Vallotti, or slightly sharper than in Werckmeister 3). If it happens to be a little high or low to taste, that's OK. We just want to stay out of the range where it's so high that C-E turn into a blur, or so low that there's no room left to work with in the following steps.
• Fit the G, D, and A into this so C-G-D-A-E all have the same quality as 5ths or 4ths. Whatever regular size we're setting here, this is our basic unit.
• From E, pure B, pure F#. G-B should sound slightly "harder" or brighter than C-E, and then D-F# even more so.
• From C, pure F. F-A has the same character as G-B. The C major, F major, and G major triads are our three best.
• From F, temper Bb as a narrow 5th (or wide 4th) of approximately the same quality as the others above. Listen also that Bb-D is similar quality to the F-A already available; and that F#-A# is high but acceptable-sounding.
• From Bb, pure Eb. Confirm that Eb-G and D-F# have the same quality as one another, both sounding much like they do in equal temperament. Confirm also that B-D# has approximately the same character as F#-A#, and the whole triad B-D#-F# is quite good, owing in part to the pure 5th.
• From Eb, make Ab a narrow 5th or wide 4th of approximately the same quality again, or perhaps a little bit gentler to taste. Test that E-G# is high and bright, but not quite as wide as Pythagorean. Ab-C should sound very slightly wider in character than Eb-G does, but still a good complete triad Ab-C-Eb.
• From F#, make C# a narrow 5th or wide 4th similarly. Check that A-C# makes a nice transitional character between D-F# and E-G#. Also check that Db-F has a character resembling that of B-D#. Finally, note that our leftover interval C# to G# is probably slightly wide as a 5th (or narrow as a 4th), but that the character of C#-G# sounds the same as Eb-G#. They simply happen to be tempered in opposite directions, but a similar amount, not that anyone would notice during the playing of music.

Play suitable music in a variety of keys, to test that everything works nicely.

Bonus 4: yet another Lehman circulating temperament based on taste (27 October 2007)

• Get A from a tuning fork, and make a pure octave down to tenor A.
• Tune F as a major 3rd below tenor A, and make it 2 to 3 beats wide...to your taste. This will give us anything from about 1/5 comma to 1/6 on the naturals, automatically generated by the following set of instructions. If you pick 3 beats per second you get 1/6 comma.
• Build two temporary notes: tenor C below tenor F pure as a 4th, and tenor D below tenor A pure as a 5th. These are being done only so we can put the G exactly where it belongs between F and A, using these two reference points.
• Working from that C up to tenor G as a 5th, and from that D up to tenor G as a 4th, put G where it has exactly the same quality from both. It will be fairly rough from both. The 4th will be beating faster than the 5th, and if you care to count beats, you have reached the correct point when the 4th is beating as triplets against the 5th as duplets.
• Working from that tenor G and tenor A up to the D above middle C, similarly put the D into position so the G-D 5th and the A-D 4th have the same quality as one another. Again the 4th will be beating as triplets against the duplets of the 5th. This is always true when doing correctly equalized 5ths/4ths, playing the notes that are under the fingers 5-4-1 of the left hand.
• From this middle D, correct the lower temporary D to be a pure octave. Check it with both G and A for equal roughness.
• Working from tenor F and tenor G up to middle C, put C into place so the F-C 5th and G-C 4th are equally rough. Same triplets-against-duplets shtick.
• From middle C fix the lower C as pure octave, and check its 4th and 5th.
• Now F-C-G-D-A are all similarly tempered with one another, smoothly. Regular meantone!
• Copy tenor F up to middle F as a pure octave.
• From tenor A, tune middle C# as high as you can bear it, testing also as the Db to F major 3rd above it. I usually try to get these two as equal in quality as I can. Some others might prefer to put C# a bit lower, favoring it as A-C# instead of Db-F.
• From tenor A build a temporary pure E 4th under it. From that C# build a temporary tenor F# pure as a 5th. We are going to use both of these to build a correct B next to middle C.
• Make a B that is equally rough from both that E as a 5th, and F# as a 4th. As usual, the 4th makes triplets against the 5th's duplets.
• From B and C# build the F# above middle C. Both these intervals will be pure or nearly so. Copy it down to tenor F# erasing the temporary one we did a moment ago.
• From A and B build the E above middle C. Put most or all of the tempering into A-E, and little or none into B-E. Check E also with middle C; we don't want it to be worse in quality than our starting F-A was. It will be beating faster than F-A, merely because the notes are higher in pitch, but the quality should be the same...and B-E will work out to be pure or nearly so.
• Copy it down to tenor E erasing the temporary one.
• We're almost done. We already have F-C-G-D-A-E-B-F#-C#. We only lack G#, Bb, and Eb.
• Copy middle C# down to tenor C# as a pure octave.
• Build G# from C# as a 5th, either pure or slightly narrow. Check it as a major 3rd above E also, that it is plenty high...and that this resembles the A-C# major 3rd's quality.
• Tune Bb as a 5th under F, either pure or slightly wide (which will improve it as a major 3rd in F#-A#, without much cost to Bb-D).
• Working from G# and Bb put middle Eb where it sounds best.
• Copy middle Eb down to tenor Eb.
• Play through all twelve major triads listening for usability of all of them, but also an interesting variety. The sharp keys should be bright and penetrating, and the flat keys comparatively mellow. F major, C major, and G major should be calmest.
• Go back to C# and check it one more time, maybe nudging it slightly, until you like both A major and Db major as much as you want to.
• Finish the instrument by octaves.
• Play through something suitably intense and expressionistic to test the instrument. The slow movements in B minor and C minor by Wilhelm Friedemann Bach, pages 20 and 32 in the Henle edition #452, do very nicely with all the cross-relations and such.

Bonus 5: Lehman ordinary temperament for Buxtehude, Reincken, Froberger, Kuhnau, Böhm, et al (20 June 2008)

It comes from analysis of the actual keys, scales, and notes used by these composers, plus temperament ordinaire principles of sharpening sharps and flattening flats. Smoothness and resonance are emphasized in the keys most often used as tonics: C, F, G, D, A, and B-flat majors, and D, A, G, E, C, B, F, F# minors.

At the same time, individual notes are all carefully placed harmonically and melodically, so they can be used as needed within these compositions: F#, C#, G#, D#, A#, E#, B#, Bb, Eb, Ab, Db, Gb, Cb. The occasional triads such as B major, F# major, Ab major, and C# major are spicy with high 3rds, but nowhere near as raucous as their counterparts in regular meantone layouts (where some of those notes are normally misspelled). These triads are also gentler and smoother than they are in Werckmeister III.

Major 3rds better than (smaller than) their counterparts in equal temperament: C-E, G-B, F-A, D-F#, A-C#. For music in the major and minor scales, and in the various church modes, those are the major 3rds most frequently needed to be consonant.

Major 3rds slightly wider than in equal temperament: Bb-D, B-D#, F#-A#, Eb-G, E-G#, Ab-C, Db-F. This spiciness is an expressive asset whenever diminished 4ths occur in the music...and that is a remarkably frequent occurrence in this repertoire: C#-F (often within D minor), G#-C (in A minor), A#-D (in B minor), B-Eb (in C minor), F#-Bb (in G minor), D#-G (in E minor), E-Ab (in F minor).

My postings about this on HPSCHD-L, including rosters of music by Buxtehude and Böhm analyzed for the notes needed: [6/20] [6/23] [6/23] [6/24] [6/26]

Try the following sequence:

• Get C from a tuning fork. Copy to middle C.
• Set E as a major 3rd above middle C beating about 3 times per second. (This is the same as it does in Werckmeister III, or in 1/5 syntonic comma meantone. Regular 1/5 was popular in the 17th century; see Mark Lindley's article "Temperaments" in New Grove.)
• Build two temporary notes: tenor G below middle C pure as a 4th, and tenor A below middle E pure as a 5th. These are being done only so we can put the D exactly where it belongs between C and E, using these two reference points.
• Working from that G up to middle D as a 5th, and from that A up to middle D as a 4th, put D where it has exactly the same quality from both. It will be fairly rough from both, but we will fix this in the next two steps. The 4th will be beating faster than the 5th, and if you care to count beats, you have reached the correct point when the 4th is beating as triplets against the 5th as duplets.
• Move the temporary G down slightly until the G-C 4th and G-D 5th have the same average quality as one another.
• Move the temporary A up slightly until the D-A 4th and A-E 5th have the same average quality as one another.
• Check C-G-D-A-E to be sure all these 4ths or 5ths have similar tempering.
• Tune F pure from C.
• Tune B pure from E.
• Tune Bb pure from F, and then Eb pure from Bb.
• Tune F# pure as a 5th from B, and then nudge it slightly flat so it barely begins to beat.
• Tune C# pure from F#, and nudge it slightly flat until it barely begins to beat.
• Tune G# pure from C#, and check that it is beating slightly with Eb.
• Go back to Bb and nudge it downward slightly, so the 5th from Bb-F begins to beat very slowly, wide.
• Finish the instrument by octaves.
• Play through mid-to-late 17th century German repertoire: Buxtehude, Pachelbel, Böhm, Kuhnau, et al.

Demonstrations of this temperament: [Part 1] [Part 2]

Bonus 6: Interpretations of Rameau's 1726 temperaments: C version in 1/4 syntonic comma, and his preferred Bb version in 1/6 (19 November 2008)

(Preliminary presentation; further details being worked out for a more substantial article about this.)

In Jean-Philippe Rameau's Nouveau systeme de Musique (1726), chapter 24, [Transcription] he described several temperaments that are modifications of regular meantone. One of them has the series of regularly-tempered intervals starting on C, and the other includes F and Bb as well. He preferred the latter layout, as it "conserve toute la justesse possible dans les modulations les plus usitees" (keeps the best purity in the most-used keys; bottom of page 110). He brought up this Bb temperament as a contrast against the 1/4 comma C temperament he had just described.

Rameau later (1737) changed his preference to equal temperament.

Rameau's 1726 presentation of a C-based temperament is clearly about the use of 1/4 syntonic comma division. His section about his preferred Bb-based temperament, however, appears to be about the use of 1/6 comma division instead of 1/4. It begins at the bottom of page 110; I have been studying it from two facsimile editions: one edited by Kremer (Zurfluh 1996), and the other by Jacobi (American Institute of Musicology 1967). He says that this temperament unites the differences between the two commas, i.e. the syntonic and the Pythagorean, into a single solution. It is a practical and musical solution, not a theoretical/mathematical abstraction that fits neatly into the exact marks hit by overtones.

Furthermore, simple experimentation reveals that a 1/4 comma division starting on Bb does not work well in practice; it has to be a gentler division such as 1/6 for the notes around A# (D#, E#, and B#) to sound sufficiently smooth in Rameau's music.

That harpsichord music is:

• Premier livre (1706)
• Pieces de Clavecin (1724, reprinted 1731)
• Nouvelles Suites de Pieces de Clavecin (c1728-9)
• Pieces de Clavecin en Concerts (1741, ensemble, including five rearranged as harpsichord solos)
• La Dauphine (manuscript 1747)

Rameau's published harpsichord music before 1737 uses these notes: Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#, C#, G#, D#, A#, E#, and B#. Because individual pieces go beyond 12 notes, and especially if we assume the harpsichord is not to be retuned during a suite, we must seek solutions that have good compromises for all these notes. We need good-sounding D#, A#, E#, B# in all their contexts without too much damage to Eb, Bb, F, C; we need placements of Db/C# and Ab/G# that work well as either note.

The harpsichord parts in the 1741 book similarly use all those notes from Db up to E# (no B#). In the second Menuet of the second concert, the chromatic phrases at bars 6-7 and 18-19 include enharmonic shifts from C# to Db. La Dauphine of 1747 requires all the notes from Ab to G#; see especially the enharmonic shift from G# in bar 35 to Ab in bar 36. That Ab in the bass (replacing the G#) is the only note that changes within those otherwise identically repeated phrases. It makes the music shift immediately from the D minor of the preceding bars (and a decorated A major dominant) into C minor.

In the 1/6 comma layouts, having put all of Bb-F-C-G-D-A-E-B-F# into position first, I placed the Ab/G# so the major 3rds Ab-C and E-G# are both the same size. Finally, I placed the C# and Eb into easy-to-find positions within F#-C#-G# and Ab-Eb-Bb, testing various triads and passages from Rameau's music. I present here several possibilities, all proceeding from a regular cycle of Bb-F-C-G-D-A-E-B. I worked these out at the harpsichord, only later transferring them into my spreadsheets for this report. The first four all sound good to me, especially the one with the lowered D#, and they are easy to do quickly by ear.

The fifth one alerted me that the history books are mistaken, with regard to Rameau's 1726 preference being a 1/4 syntonic comma layout beginning on Bb. It does not work, in Rameau's own published harpsichord music of the 1720s! The A#, D#, and E# are too high, given that the music uses them so frequently and in such direct harmonies as B major, F# major, and C# major triads. There is similarly a problem in Rameau's G minor and G major music of the 1720s, wherever there is a trilled Bb-A over D and F#. The premise of regular 1/4 comma from Bb-F-C-G-D-A-E-B leads inevitably to absurd musical conclusions in Rameau's music; therefore, the premise itself is wrong and must be discarded.

20th century writers (Barbour, Lindley, Devie, Jorgensen...) missed this switch away from 1/4 comma, apparently because they have assumed that Rameau's introductory phrase ("Pour que les Intervales conservent toute la justesse possible dans les Modulations les plus usitées...") on p110 takes "justesse" as referring to pure major 3rds. However, Rameau was clearly referring to purity of playability and regularity through all the most commonly used harmonies, not the short-sighted goal of beatless major 3rds. [The relevant sections: Barbour p135; Lindley Stimmung und Temperatur pp232-247; Devie pp95-105; Jorgensen Tuning pp189-213; Lindley New Grove pp252-3 and 256-7.] Lindley's New Grove article does not bother to mention that Rameau had two different temperaments in the 1726 book, although Lindley had presented hypothetical layouts of both on page 235 of Stimmung und Temperatur. Devie's three tables (figures 32-34 on p98) are all 1/4 comma layouts based on Gallimard (1754), a math book rather than a music book, instead of deriving them directly from Rameau's publication.

To test these temperaments in action, I played through all of Rameau's harpsichord music, and Francois Couperin's fourth book (published 1730, but it says he had completed all the music by 1727). Couperin's Ordres 21 and 23-27 in that fourth book make especially wide demands on a temperament. I played also the D major and C minor suites by Antoine Forqueray, since it makes similar demands of the notes D#, A#, Ab, Db, and extends as far as Gb (within the same C minor music that needs C# and F#). [Forqueray (c1671-1745). This music was published in 1747 by Forqueray's son, both as compositions for viola da gamba and as harpsichord-solo arrangements. Obviously, they had some suitable harpsichord temperament that handled all these notes well: to play not only the harpsichord version, but also for accompaniment in the ensemble version.]

In all this music, particularly with the way it gets into the highest sharps (D#, A#, E#, B#) while also needing Ab and Db, anything tighter than a 1/6 comma base sounds too harsh to me, ruining the gracefulness of the music.

• 1 based on Bb: Bb-F-C-G-D-A-E-B 1/6 Pythagorean comma; B-F# 1/12; F#-C# pure; Ab/G# equalized within E-x-C as major 3rds; Eb tempered the same from both Ab below and Bb above. Result: the three wide 5ths are C#-G#-D#-A#, each only by a scarcely noticeable 1/12 comma. [Table] Four of the major 3rds have size 3. Db-F and B-D# are 12, and F#-A# is 14.
• 2 based on Bb (and lowering the D#): Same as above, but lowering the note D# by an additional 1/12, so the B-D# sounds better. Ab-Eb becomes pure, and Eb-Bb becomes 1/6 comma wide instead of 1/12. B-D# becomes the same size as in Pythagorean, rather than slightly wider than that. [Table] Four of the major 3rds have size 3. Db-F is 12, F#-A# is 14, and B-D# is 11.
• 3 based on Bb: Bb-F-C-G-D-A-E-B 1/6 syntonic comma; B-F#-C# each 1/12 comma; Ab/G# equalized within E-x-C as major 3rds; D# pure from G#; the two wide 5ths C#-G# and Eb-Bb are both the same size as each other. The crucial note to get right, tastefully, is the D#: beating slightly wide under the Bb. Then, tune the G# pure under D#, and check that it is only slightly wide from C#. [Table] Four of the major 3rds have size 3.7. Db-F is 11.8, F#-A# is 12.8, and B-D# is 10.5.
• 4 based on C: C-G-D-A-E-B-F#-C# 1/4 syntonic comma; C#-G# pure; C-F pure; Bb and Eb each made progressively wide as 5ths descending from F. [Table] Four of the major 3rds (C-E, G-B, D-F#, A-C#) have size 0. Eb-G is 12.5, F#-A# is 13.75, and Ab-C and Db-F are each 18.25. The notes D# and A# work well, at some expense to Eb and Bb.
• 5 (not as good) based on Bb: Bb-F-C-G-D-A-E-B 1/4 syntonic comma; B-F#-C#-G# each pure; Eb fit into an intermediate position, slightly better with Bb than with G#. [Table] Four of the major 3rds (Bb-D, F-A, C-E, and G-B) have size 0. Ab-C is 12.75, B-D# is 15, Db-F is 15.5, and F#-A# is 18.25. The A# seems to me to be too high for the music, occurring so frequently within the common keys of D major and E minor. D# is similarly too high for music in A minor or E minor.

My own preference as a practicing harpsichordist is for the 1/6 comma solutions 2 and (especially) 3: both treating the important note D# kindly, especially because so much of Rameau's harpsichord music of the 1720s is in A minor, A major, E minor, and E major. The sonority of D-F#-Bb (or D-F#-A#) also occurs frequently in the 1728 book, within the G minor and G major music.

Further evidence (admittedly circumstantial) away from 1/4 comma is in the "Remarques sur les Pieces de ce Livre, & sur les differens genres de Musique" of Rameau's c1728-9 volume of harpsichord music. [Scans from Bärenreiter 3800, 1972, ed. Erwin Jacobi: page 1, page 2; print it at a reduction to approximately 50%] Rameau devoted seven paragraphs to the theoretical explanation of enharmonic shifts. He called special attention to "L'enharmonique", and to a spot in "La Triomphante" that has B# in the left hand and C natural in the right hand at the same time. He carefully described the theoretical "difference of one Quarter-tone" between such pairs, theoretically, but emphasized that they are exactly the same key on the keyboard. He also explained the theoretical difference between Diatonic and Chromatic semitones.

Most importantly, in this section, his point was that the oddities come from musical usage by the composer: not from the physical properties of a strongly nuanced keyboard temperament. In his view, apparently, it does not come from a situation where one note of an enharmonic pair is in tune on the keyboard while the other is grossly out of tune (by as much as a "Quarter tone"), which is given by a 1/4 comma basis (where C# and Db are a diesis apart, about 42 cents). If this enharmonic shifting were directly perceptible, immediately, within his style of tuning (using the conjecture of 1/4 comma, and with sharps and flats compromised somewhere within a diesis span of difference), I believe he would have said so here. He would have explained why the G# in bars 15-16 and the Ab in bar 17 of "L'enharmonique" (or the C# in 52-3 vs the Db in 54) do not both sound reasonably in tune, simultaneously. He at least would have said something about getting the C# up high enough (obviously impure enough from A) to sound reasonable as a Db under the treble F.

Therefore, I must conclude that by 1728-9 he wasn't promoting as first preference a temperament as extreme as 1/4 comma among the naturals and Bb. It had to be more moderate than that, toward equal: at least as lightly tempered as 1/6 comma in the naturals and Bb, and with all the other sharps and flats at compromised/intermediate positions for their contexts, such that the enharmonic shifts in the music were not obvious. In the way he described his compositional theory here, the startling nature of enharmonic shifts came not from the physical phenomenon of one note (or the other) being far out of tune, but rather from subtle and clever features of the composition.

Coming to this from hands-on tuning practice, which makes the point most directly (in sound instead of on paper, or a computer screen): if it were based on 1/4 comma instead of 1/6, all for the questionable goal of getting the major 3rds Bb-D, F-A, C-E, and G-B all to be as small as pure 5:4 (or G-B slightly wider by raising the B), the losses elsewhere have to be greater. If we base it on 1/4, the notes D#, A#, E#, and B# are all garishly high, and the Ab and Db (and Gb) are disturbingly low. Did Rameau really want such an extreme approach, before changing his mind and going with equal? Or, was he writing about a more moderate middle ground?

Restating that practical point: if the Bb-F-C-G-D-A-E are all established first in regular 1/4 syntonic comma, it is not possible to fit the remaining notes into any places such that all of C#, G#, D#, A#, Eb, Ab, and Db will sound reasonable...let alone E# and B#. Try it. Among other problems in the vicinity, F# cannot be pulled up high enough to make F#-A# in any way consonant; even if B is raised aggressively from E, and B-F# is made as much as 1/3 comma wide (going against Rameau's instructions to make B-F# "less tempered"), it still does not work. F# does not get high enough to make a decent major 3rd under the given A#. Set up a temperament with regular Bb-F-C-G-D-A-E on a harpsichord. Raise the B-F#-C# with less than 1/4 comma tempering, or even (for the sake of demonstration) make them wider than pure, if you wish. Fit Eb and Ab/G# into place as wide 5ths, descending from Bb. Play all of the following spots: bars 47 and 86 of "La Poule", bar 30 of "Les Triolets", and bars 34 and 38 of "Les Sauvages", all from the G major and G minor pieces of c1728. Then, turn the page and play all of "L'enharmonique" as well. Turn back to "La Triomphante" and play through it, especially through the last couplet where Rameau's preface emphasized the use of B# and C simultaneously, and where he composed the C# major triad in the bar before that. 1/4 comma is too extreme for this music, especially when it runs into F# and Bb together, and when we encounter C# major on a downbeat.

Electronic device instructions for the Bach

Jon-O Addleman has generated a set of electronic pitches to match for this temperament, at A=415. I have not tried them myself, lacking a portable player to put next to the harpsichord (and preferring to work by ear anyway...). He has created recorded pitches for other temperaments also.

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