# Comparing the key characters in three temperaments:

## 1/4 Comma Meantone, Kirnberger 3 "Well Temperament," and 12-note Equal Temperament

http://www-personal.umich.edu/~bpl/temper-examples.html
bpl@umich.edu

The charts presented below contain the beat rates from the intervals that are available in three selected keyboard temperaments. I have chosen these three as representative samples of common styles of temperament (while other more extreme temperaments are also possible). Working upward from the bottom of this article, these beat rates are then used to assign integers to intervals and triads. These integers describe the relative purity of intonation using a 0 to 10 rating scale. This exercise is an attempt to quantify key character in a manner that (I hope) is easy to understand. All the number-crunching is an attempt to simplify concepts that look more complex on paper than they are to the ear. To do this it was necessary to make several subjective choices as to weighting the qualities of various intervals. (The reader may notice that the only completely objective and scientifically verifiable charts are at the bottom of the article.)

Similar charts for other temperaments are easy to generate from the spreadsheet that is available at the main page. The charts below are simply clipped from that spreadsheet using three different selections of temperament.

To facilitate direct comparison, all these charts are calculated from a constant A=440 Hz. (This affects only the absolute beat rates and frequencies, not the ratios or the relative assessments of interval quality.) The spreadsheet can also calculate everything based on other standards, or working from C instead of A.

Finally, this article is intended mostly as a set of examples for the main page, not as a separate piece of work. (And there are certainly other ways to organize this material in a manner that would make better sense to different readers!)

### Temperament "Recipes"

These are brief introductions to the three chosen temperaments. Reading across the chart shows relationships of fifths. Reading up or down shows relationships of major thirds. The numbers between the note names show the fraction of the comma that is being split (Syntonic or Ditonic; see explanations at the main page). A 0 indicates that the interval is pure.

1/4 Syntonic Comma Meantone - a temperament common in the early 16th century and continuing. It is very easy to tune: set the tempered fifths C-G-D-A-E such that the C-E major third is pure. Then all other notes are derived as pure major thirds above or below these notes. Result: all usable fifths are tempered by the same amount. There is a "wolf" diminished sixth where the circle does not close. Diminished fourths are hardly usable as makeshift major thirds because they are very sharp; however, they can still be used in careful chord spacings or as special effects.
 Comma E B F# C# SC (0) (0) (0) (0) SC C (- 1/4 ) G (- 1/4 ) D (- 1/4 ) A (- 1/4 ) E SC (...) (0) (0) (0) G# Eb Bb F (0) (...) (...) (...) E (- 1/4 ) B (- 1/4 ) F# (- 1/4 ) C# (- 1/4 ) G# (0) (0) (0) (0) C G D A (...) (0) (0) (0) G# (...) Eb (- 1/4 ) Bb (- 1/4 ) F (- 1/4 ) C (0) (...) (...) (...) E B F# C#

Kirnberger 3: 1/4 Syntonic Comma - a typical "well temperament" of the mid-18th century. It too is easy to tune: set the tempered fifths C-G-D-A-E such that the C-E major third is pure. Then all other notes are derived as a circle of pure fifths around the flat side, and the fifth E-B is also pure. The last fifth (actually the diminished sixth B-Gb) is not tuned, but is nearly pure: off by a schisma. Result: all keys are usable. Sizes of major thirds vary: keys with more sharps or flats have "brighter" or sharper major thirds than the simpler keys. Every key has a unique character due to the different sizes of major thirds. Modulation around the circle of fifths smoothly increases or decreases the tension, depending if the new key has more or fewer accidentals than the original key.
 Comma E B Gb Db SC (0) ( 1/4 ) (...) (...) SC C (- 1/4 ) G (- 1/4 ) D (- 1/4 ) A (- 1/4 ) E SC (1) ( 3/4 ) ( 1/2 ) ( 1/4 ) Ab Eb Bb F (...) (...) (1) (1) E (0) B (...) Gb (0) Db (0) Ab (0) ( 1/4 ) (...) (...) C G D A (1) ( 3/4 ) ( 1/2 ) ( 1/4 ) Ab (0) Eb (0) Bb (0) F (0) C (...) (...) (1) (1) E B Gb Db

Equal Temperament - the 20th-century "standard." It is difficult to tune because there are no pure intervals. The fifths beat slowly and are difficult to judge accurately. Result: all keys are usable. The size of major thirds is constant. Every key has the same character (same interval relationships), and the only audible distinguishing feature of keys is an absolute difference in pitch level.
 Comma E B F# C# approx SC ( 67/100 ) ( 67/100 ) ( 67/100 ) ( 67/100 ) DC C (- 1/12 ) G (- 1/12 ) D (- 1/12 ) A (- 1/12 ) E approx SC (...) ( 67/100 ) ( 67/100 ) ( 67/100 ) G# Eb Bb F ( 67/100 ) (...) (...) (...) E (- 1/12 ) B (- 1/12 ) F# (- 1/12 ) C# (- 1/12 ) G# ( 67/100 ) ( 67/100 ) ( 67/100 ) ( 67/100 ) C G D A (...) ( 67/100 ) ( 67/100 ) ( 67/100 ) G# (...) Eb (- 1/12 ) Bb (- 1/12 ) F (- 1/12 ) C ( 67/100 ) (...) (...) (...) E B F# C#

### How nearly in-tune are the triads?

A grading scale of 0 to 10 is employed in the charts of this section. Low numbers indicate that the triads are nearly pure; 10 is at the limit of usability (usually due to the third being too sharp). Numbers over 10 indicate a "wolf" interval. The derivation of these numbers is explained more fully in the next section, but briefly this a weighted consideration of the fifth and the major or minor thirds above the root. Most of the perceived quality is due to the third unless the fifth is a wolf. The weighting formula is subjective and based on experience.

Capital letters as column headings indicate major triads; lowercase letters indicate minor or diminished triads. "Neap" indicates a "Neapolitan" chord: the fourth degree, flat sixth degree, and flat ninth degree. (In C major, for example, this chord is F-Ab-Db. In E major it is A-C-F.)

1/4 Comma Syntonic Meantone - notice that all the commonly-used keys are very nearly in tune and most of these are the same as one another; the unusable keys are far out of tune. Composers working with this temperament had to be careful to avoid the problematic areas. This temperament generally works very well in pieces whose key signatures have fewer than four sharps and fewer than three flats.
 Harmony Chart: subjective tension of common chords when playing in the given key Key I i Neap ii iio III iii IV iv V V7 v VI vi VII viio viio7 C 1 2 3 2 2 1 2 1 1 1 4 2 31 2 1 2 2 G 1 2 6 2 2 1 2 1 2 1 4 2 1 2 1 2 1 D 1 2 1 2 2 1 2 1 2 1 4 2 1 2 1 2 1 A 1 2 1 2 2 1 2 1 2 1 4 2 1 2 1 2 1 E 1 2 1 2 1 1 31 1 2 17 10 2 1 2 1 2 2 B 17 2 1 2 1 1 1 1 2 17 10 2 1 31 1 2 2 F# 17 2 1 31 1 1 1 17 2 17 10 2 1 1 1 2 2 C# 17 2 1 1 2 1 1 17 2 31 16 31 1 1 17 2 2 G# 31 31 1 1 2 17 2 17 2 1 1 1 1 1 17 2 2 Eb 1 1 1 1 2 17 2 31 31 1 1 1 17 2 17 2 2 Bb 1 1 5 2 2 17 2 1 1 1 4 1 17 2 31 2 2 F 1 1 3 2 2 31 2 1 1 1 4 2 17 2 1 2 2

Kirnberger 3 - notice that the keys with more sharps or flats are farther out of tune than the simpler keys, but all keys are now musically usable. Sharps are sharper than in meantone, and flats are flatter. The natural notes remain close to where they were in meantone. Looking at this in the other direction: half the major keys are better (more nearly in tune) than their counterparts in equal temperament, and the other half are more spicy. This temperament and other similar "well temperaments" give a particularly dramatic and dynamic profile to tonal music: modulation affects intonation from moment to moment.
 Harmony Chart: subjective tension of common chords when playing in the given key Key I i Neap ii iio III iii IV iv V V7 v VI vi VII viio viio7 C 1 9 8 3 5 7 1 1 9 2 4 7 9 2 2 4 7 G 2 7 8 2 6 2 3 1 9 3 4 3 7 1 1 4 7 D 3 3 5 1 7 1 3 2 7 4 5 2 2 3 1 5 7 A 4 2 3 3 7 1 8 3 3 8 7 1 1 3 2 6 8 E 8 1 1 3 7 2 8 4 2 8 6 3 1 8 3 7 8 B 8 3 1 8 7 3 9 8 1 9 5 3 2 8 4 8 7 Gb 9 3 3 8 8 4 9 8 3 9 5 8 3 9 8 8 6 Db 9 8 4 9 8 8 9 9 3 9 4 8 4 9 8 9 5 Ab 9 8 7 9 7 8 9 9 8 7 3 9 8 9 9 8 4 Eb 7 9 7 9 6 9 7 9 8 2 1 9 8 9 9 7 5 Bb 2 9 8 9 5 9 3 7 9 1 0 9 9 7 9 6 6 F 1 9 8 7 4 9 2 2 9 1 0 9 9 3 7 4 7

Equal Temperament - notice that all the keys are equally out of tune (or equally in tune). No triad has much repose, because the thirds are always beating rapidly. This is somewhat similar to the effect of using constant vibrato in string playing. Some people like it, while others do not.
 Harmony Chart: subjective tension of common chords when playing in the given key Key I i Neap ii iio III iii IV iv V V7 v VI vi VII viio viio7 C 3 6 5 6 6 3 6 3 6 3 3 6 3 6 3 6 6 G 3 6 5 6 6 3 6 3 6 3 3 6 3 6 3 6 6 D 3 6 5 6 6 3 6 3 6 3 3 6 3 6 3 6 6 A 3 6 5 6 6 3 6 3 6 3 3 6 3 6 3 6 6 E 3 6 5 6 6 3 6 3 6 3 3 6 3 6 3 6 6 B 3 6 5 6 6 3 6 3 6 3 3 6 3 6 3 6 6 F# 3 6 5 6 6 3 6 3 6 3 3 6 3 6 3 6 6 C# 3 6 5 6 6 3 6 3 6 3 3 6 3 6 3 6 6 G# 3 6 5 6 6 3 6 3 6 3 3 6 3 6 3 6 6 Eb 3 6 5 6 6 3 6 3 6 3 3 6 3 6 3 6 6 Bb 3 6 5 6 6 3 6 3 6 3 3 6 3 6 3 6 6 F 3 6 5 6 6 3 6 3 6 3 3 6 3 6 3 6 6

### Qualities of intervals

The left half of each chart shows the qualities of individual intervals (two notes). Again the scale is 0 (pure) to 10 (barely usable), with numbers greater than 10 indicating a wolf. The right half of each chart builds these into triads and seventh chords for use in the charts that were above. In the left half, negative numbers indicate intervals that are too narrow to be pure. In the right half, these have all been converted to positive numbers before calculation of the triad's values (i.e., a wide third and a narrow fifth do not cancel each other out, but each contributes additional beats to the triad).

The top half of each chart is arranged in "circle of fifths" order. The bottom half is in chromatic order.

The standard for assigning the numbers is as follows, explained also on the main page: For major thirds, 10 is a syntonic comma too wide. For fifths, 10 is 1/2 Ditonic comma (found in some other temperaments) too narrow.

For other intervals, the standard is more subjective and complex. A typical listener is more tolerant of out-of-tune minor thirds than out-of-tune major thirds. And there are two different minor thirds to be dealt with: the normal 6:5 and the rarer 7:6 (very nearly approximated in meantone as augmented seconds). A minor third's rating scale is assigned rather arbitrarily so that in equal temperament the major and minor thirds have the same rating (6): equally bland.

Similarly, intervals larger than fifths are harder to judge than fourths (4:3), fifths (3:2), and major thirds (5:4) are, and they too offer different options. For example, a musically usable minor seventh may be any of three intervals: two fourths (16:9), or a fifth plus a minor third (9:5), or the direct and rarely heard 7:4. A major sixth is usually heard as a fourth plus a major third (5:3), but a different major sixth may be heard directly as 12:7. These are all shown later in this article. Their rating scales are calculated as component intervals, but a pure or nearly pure direct interval may override this if present.

That is why these charts include "best" values for these intervals. The interval version with the lowest rating is chosen automatically, just as the ear organizes these to the nearest approximation of a pure interval.

Numbers between 0 and 10 are derived from ratios of beat rates (see below) divided into the frequency of the lower note. For example, Kirnberger 3's fifths are 1/4 comma narrow, and this is half as much as 1/2 comma (which would be assigned -10): therefore their value is -5.

Meantone
 Best Best Best Best Maj Min Dim As Master Min3 Maj3 Per4 Per5 Min6 Maj6 Min7 Maj7 | Triad Triad Triad Mm7 MM7 mm7 dim7 Neap C -2 0 4 -5 5 2 -6 -7 | 1 2 2 4 4 4 2 6 C G -2 0 4 -5 0 2 -6 -7 | 1 2 2 4 4 4 2 1 G D -2 0 4 -5 0 2 -6 -7 | 1 2 2 4 4 4 2 1 D A -2 0 4 -5 0 2 -6 -7 | 1 2 2 4 4 4 2 1 A E -2 0 4 -5 0 2 -6 44 | 1 2 2 4 27 4 2 1 E B -2 17 4 -5 0 2 -6 44 | 17 2 2 10 33 4 2 1 B F# -2 17 4 -5 0 1 -6 44 | 17 2 2 10 33 4 1 1 F# C# -2 17 4 -5 0 1 -6 44 | 17 2 2 10 33 4 1 1 C# G# -2 17 4 31 0 1 -6 44 | 31 31 2 16 39 16 1 1 G# Eb 1 0 -29 -5 9 2 -1 -7 | 1 1 2 1 4 1 2 5 Eb Bb 1 0 4 -5 5 2 -1 -7 | 1 1 2 1 4 1 2 3 Bb F 1 0 4 -5 5 2 -6 -7 | 1 1 2 4 4 4 2 3 F C -2 0 4 -5 5 2 -6 -7 | 1 2 2 4 4 4 2 6 C C# -2 17 4 -5 0 1 -6 44 | 17 2 2 10 33 4 1 1 C# D -2 0 4 -5 0 2 -6 -7 | 1 2 2 4 4 4 2 1 D Eb 1 0 -29 -5 9 2 -1 -7 | 1 1 2 1 4 1 2 5 Eb E -2 0 4 -5 0 2 -6 44 | 1 2 2 4 27 4 2 1 E F 1 0 4 -5 5 2 -6 -7 | 1 1 2 4 4 4 2 3 F F# -2 17 4 -5 0 1 -6 44 | 17 2 2 10 33 4 1 1 F# G -2 0 4 -5 0 2 -6 -7 | 1 2 2 4 4 4 2 1 G G# -2 17 4 31 0 1 -6 44 | 31 31 2 16 39 16 1 1 G# A -2 0 4 -5 0 2 -6 -7 | 1 2 2 4 4 4 2 1 A Bb 1 0 4 -5 5 2 -1 -7 | 1 1 2 1 4 1 2 3 Bb B -2 17 4 -5 0 2 -6 44 | 17 2 2 10 33 4 2 1 B
Kirnberger
 Best Best Best Best Maj Min Dim As Master Min3 Maj3 Per4 Per5 Min6 Maj6 Min7 Maj7 | Triad Triad Triad Mm7 MM7 mm7 dim7 Neap C -9 0 0 -5 -9 2 0 0 | 1 9 9 0 0 4 5 8 C G -7 2 4 -5 -5 2 5 4 | 2 7 8 4 3 6 4 5 G D -4 4 4 -5 -2 4 -6 11 | 3 3 7 4 7 5 5 3 D A -2 6 4 -5 0 6 -6 17 | 4 2 6 5 12 4 6 1 A E -2 8 4 0 0 8 -6 24 | 8 1 4 7 17 4 7 1 E B -4 8 0 -2 -2 8 5 24 | 8 3 4 6 17 4 7 3 B Gb -6 9 2 0 -3 9 2 26 | 9 3 4 5 19 2 7 4 Gb Db -8 9 0 0 -6 9 2 26 | 9 8 5 5 19 4 7 7 Db Ab -8 9 0 0 -8 9 0 20 | 9 8 6 4 15 3 8 7 Ab Eb -9 7 0 0 -8 9 0 13 | 7 9 7 3 10 4 8 8 Eb Bb -9 4 0 0 -9 7 0 7 | 2 9 8 1 5 4 7 8 Bb F -9 2 0 0 -9 4 0 0 | 1 9 8 0 0 4 6 8 F C -9 0 0 -5 -9 2 0 0 | 1 9 9 0 0 4 5 8 C Db -8 9 0 0 -6 9 2 26 | 9 8 5 5 19 4 7 7 Db D -4 4 4 -5 -2 4 -6 11 | 3 3 7 4 7 5 5 3 D Eb -9 7 0 0 -8 9 0 13 | 7 9 7 3 10 4 8 8 Eb E -2 8 4 0 0 8 -6 24 | 8 1 4 7 17 4 7 1 E F -9 2 0 0 -9 4 0 0 | 1 9 8 0 0 4 6 8 F Gb -6 9 2 0 -3 9 2 26 | 9 3 4 5 19 2 7 4 Gb G -7 2 4 -5 -5 2 5 4 | 2 7 8 4 3 6 4 5 G Ab -8 9 0 0 -8 9 0 20 | 9 8 6 4 15 3 8 7 Ab A -2 6 4 -5 0 6 -6 17 | 4 2 6 5 12 4 6 1 A Bb -9 4 0 0 -9 7 0 7 | 2 9 8 1 5 4 7 8 Bb B -4 8 0 -2 -2 8 5 24 | 8 3 4 6 17 4 7 3 B

Equal
 Best Best Best Best Maj Min Dim As Master Min3 Maj3 Per4 Per5 Min6 Maj6 Min7 Maj7 | Triad Triad Triad Mm7 MM7 mm7 dim7 Neap C -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 C G -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 G D -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 D A -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 A E -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 E B -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 B F# -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 F# C# -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 C# G# -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 G# Eb -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 Eb Bb -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 Bb F -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 F C -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 C C# -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 C# D -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 D Eb -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 Eb E -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 E F -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 F F# -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 F# G -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 G G# -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 G# A -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 A Bb -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 Bb B -6 6 2 -2 -5 6 4 14 | 3 6 6 3 10 5 6 5 B

### Beat Rate Charts

The column values indicate the audible "beats" in the given intervals above the row note. Beats occur when an interval is almost a simple ratio (almost pure) but not quite. This is a "wah-wah-wah" sound, a pulse of varying amplitude. A person tuning a keyboard instrument counts these beats per second to ensure that the interval is correctly tempered according to the recipe. The beat rates are given in cycles per second (Hz).

These beat rates are the source for the 0 to 10 tolerability ratings used above. The beat rate for an interval is divided into the frequency of the lower note. This result is then mapped onto a rating scale such that 10 is the subjective limit where the ear will still hear the interval as "not too far out of tune to be generally usable." (Values over 10 are still somewhat usable in certain musical contexts or spacings, but these intervals must be handled with special care by the composer.)

Meantone
 7/6 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 14/9 12/7 7/4 16/9 Root Freq Cents Sm Min3 Min3 Maj3 Per4 Per5 Min6 Maj6 Min7 Maj7 Min6 Maj6 Min7 Min7 C 131.591 310.26 23.38 -2.45 0.00 1.64 -1.22 -24.67 2.05 -7.33 -6.12 8.22 -39.09 20.45 13.12 C# 137.500 386.31 24.43 -2.56 16.50 1.71 -1.28 0.00 18.69 -7.66 42.95 55.00 -2.22 21.37 13.71 D 147.123 503.42 26.14 -2.74 0.00 1.83 -1.37 0.00 2.29 -8.20 -6.84 58.85 -43.70 22.87 14.67 Eb 157.419 620.53 1.49 -25.00 0.00 -12.85 -1.46 -29.52 2.45 -41.77 -7.32 9.84 -46.76 -1.93 -43.71 E 164.488 696.58 29.23 -3.06 0.00 2.05 -1.53 0.00 2.56 -9.17 51.38 65.80 -48.86 25.56 16.40 F 176.000 813.69 1.66 -27.95 0.00 2.19 -1.64 -33.00 2.74 -9.81 -8.19 11.00 -52.28 27.35 17.55 F# 183.904 889.74 32.67 -3.42 22.07 2.29 -1.71 0.00 25.00 -10.25 57.45 73.56 -2.97 28.58 18.33 G 196.774 1006.84 34.96 -3.66 0.00 2.45 -1.83 0.00 3.06 -10.97 -9.15 78.71 -58.45 30.58 19.62 G# 205.610 1082.89 36.53 -3.83 24.67 2.56 12.85 0.00 27.95 -11.46 64.23 82.24 -3.33 31.96 20.50 A 220.000 0.00 39.09 -4.09 0.00 2.74 -2.05 0.00 3.42 -12.26 -10.23 88.00 -65.35 34.19 21.93 Bb 235.397 117.11 2.22 -37.38 0.00 2.93 -2.19 -44.14 3.66 -62.46 -10.95 14.71 -69.92 -2.89 -65.36 B 245.967 193.16 43.70 -4.58 29.52 3.06 -2.29 0.00 3.83 -13.71 76.83 98.39 -73.06 38.23 24.52 Middle C 263.181 (10.26) 46.76 -4.90 0.00 3.27 -2.45 -49.35 4.09 -14.67 -12.24 16.45 -78.18 40.90 26.24 C# 275.000 -(13.69) 48.86 -5.12 33.00 3.42 -2.56 0.00 37.38 -15.33 85.90 110.00 -4.45 42.74 27.41 D 294.246 (3.42) 52.28 -5.47 0.00 3.66 -2.74 0.00 4.58 -16.40 -13.69 117.70 -87.40 45.73 29.33 Eb 314.838 (20.53) 2.97 -49.99 0.00 -25.69 -2.93 -59.03 4.90 -83.55 -14.64 19.68 -93.52 -3.87 -87.41 E 328.977 -(3.42) 58.45 -6.12 0.00 4.09 -3.06 0.00 5.12 -18.33 102.76 131.59 -97.72 51.13 32.80 F 352.000 (13.69) 3.33 -55.90 0.00 4.38 -3.27 -66.00 5.47 -19.62 -16.37 22.00 -104.56 54.71 35.09 F# 367.807 -(10.26) 65.35 -6.84 44.14 4.58 -3.42 0.00 49.99 -20.50 114.89 147.12 -5.95 57.16 36.67 G 393.548 (6.84) 69.92 -7.32 0.00 4.90 -3.66 0.00 6.12 -21.93 -18.30 157.42 -116.90 61.16 39.23 G# 411.221 -(17.11) 73.06 -7.65 49.35 5.12 25.69 0.00 55.90 -22.92 128.45 164.49 -6.65 63.91 40.99 A 440.000 (0.00) 78.18 -8.19 0.00 5.47 -4.09 0.00 6.84 -24.52 -20.47 176.00 -130.70 68.38 43.86 Bb 470.793 (17.11) 4.45 -74.76 0.00 5.86 -4.38 -88.27 7.32 -124.93 -21.90 29.42 -139.85 -5.78 -130.71 B 491.935 -(6.84) 87.40 -9.15 59.03 6.12 -4.58 0.00 7.65 -27.41 153.67 196.77 -146.13 76.46 49.04 C 526.363 (10.26) 93.52 -9.79 0.00 6.55 -4.90 -98.69 8.19 -29.33 -24.48 32.90 -156.35 81.81 52.47
(numbers in red indicate differences in cents from equal temperament)

Kirnberger
 7/6 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 14/9 12/7 7/4 16/9 Root Freq Cents Sm Min3 Min3 Maj3 Per4 Per5 Min6 Maj6 Min7 Maj7 Min6 Maj6 Min7 Min7 C 131.591 310.26 14.62 -9.75 0.00 0.00 -1.22 -13.00 2.05 -14.62 0.00 29.24 -39.09 14.62 0.00 Db 138.631 400.49 16.52 -9.34 8.66 0.00 0.00 -9.04 8.66 -14.01 25.99 39.17 -25.99 16.52 2.50 D 147.123 503.42 22.87 -5.47 3.75 1.83 -1.37 -7.29 4.58 -8.20 11.25 45.73 -38.35 22.87 14.67 Eb 155.959 604.40 17.33 -11.55 7.30 0.00 0.00 -14.01 9.75 -17.33 14.58 37.16 -29.24 17.33 0.00 E 164.488 696.58 29.23 -3.06 9.34 2.05 0.00 0.00 9.34 -9.17 28.02 65.80 -33.03 25.56 16.40 F 175.454 808.31 19.49 -13.00 2.73 0.00 0.00 -17.33 5.47 -19.49 0.00 38.99 -45.73 19.49 0.00 Gb 184.841 898.53 26.12 -9.04 11.55 0.83 0.00 -7.50 11.55 -18.68 34.66 60.44 -34.66 22.02 3.34 G 196.774 1006.84 26.22 -10.95 3.06 2.45 -1.83 -14.60 3.06 -16.42 5.84 52.43 -58.45 26.22 9.79 Ab 207.946 1102.44 24.77 -14.01 13.00 0.00 0.00 -18.68 13.00 -23.11 29.20 49.55 -38.99 23.11 0.00 A 220.000 0.00 39.09 -4.09 9.04 2.74 -2.05 -5.46 9.04 -12.26 27.13 78.18 -52.23 34.19 21.93 Bb 233.939 106.35 25.99 -17.33 7.29 0.00 0.00 -23.11 10.95 -25.99 10.91 51.99 -52.43 25.99 0.00 B 246.733 198.53 38.35 -9.17 14.01 0.00 -0.83 -6.12 14.01 -20.59 42.04 87.68 -49.55 32.87 12.28 Middle C 263.181 (10.26) 29.24 -19.49 0.00 0.00 -2.45 -25.99 4.09 -29.24 0.00 58.48 -78.18 29.24 0.00 Db 277.261 (0.49) 33.03 -18.68 17.33 0.00 0.00 -18.09 17.33 -28.02 51.99 78.35 -51.99 33.03 5.01 D 294.246 (3.42) 45.73 -10.93 7.50 3.66 -2.74 -14.58 9.17 -16.40 22.49 91.46 -76.69 45.73 29.33 Eb 311.919 (4.40) 34.66 -23.11 14.60 0.00 0.00 -28.02 19.49 -34.66 29.15 74.32 -58.48 34.66 0.00 E 328.977 -(3.42) 58.45 -6.12 18.68 4.09 0.00 0.00 18.68 -18.33 56.05 131.59 -66.07 51.13 32.80 F 350.909 (8.31) 38.99 -25.99 5.46 0.00 0.00 -34.66 10.93 -38.99 0.00 77.98 -91.46 38.99 0.00 Gb 369.681 -(1.47) 52.23 -18.09 23.11 1.67 0.00 -14.99 23.11 -37.37 69.32 120.88 -69.32 44.04 6.68 G 393.548 (6.84) 52.43 -21.90 6.12 4.90 -3.66 -29.20 6.12 -32.85 11.68 104.86 -116.90 52.43 19.59 Ab 415.892 (2.44) 49.55 -28.02 25.99 0.00 0.00 -37.37 25.99 -46.21 58.39 99.10 -77.98 46.21 0.00 A 440.000 (0.00) 78.18 -8.19 18.09 5.47 -4.09 -10.91 18.09 -24.52 54.27 156.35 -104.46 68.38 43.86 Bb 467.878 (6.35) 51.99 -34.66 14.58 0.00 0.00 -46.21 21.90 -51.99 21.83 103.97 -104.86 51.99 0.00 B 493.465 -(1.47) 76.69 -18.33 28.02 0.00 -1.67 -12.24 28.02 -41.19 84.07 175.35 -99.10 65.74 24.56 C 526.363 (10.26) 58.48 -38.99 0.00 0.00 -4.90 -51.99 8.19 -58.48 0.00 116.97 -156.35 58.48 0.00

Equal
 7/6 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 14/9 12/7 7/4 16/9 Root Freq Cents Sm Min3 Min3 Maj3 Per4 Per5 Min6 Maj6 Min7 Maj7 Min6 Maj6 Min7 Min7 C 130.813 300.00 17.69 -7.06 5.19 0.59 -0.44 -8.24 5.94 -11.91 13.34 37.49 -29.75 16.64 4.73 C# 138.591 400.00 18.74 -7.48 5.50 0.63 -0.47 -8.73 6.29 -12.61 14.13 39.72 -31.52 17.63 5.01 D 146.832 500.00 19.86 -7.92 5.83 0.66 -0.50 -9.25 6.66 -13.36 14.98 42.08 -33.40 18.68 5.31 Eb 155.563 600.00 21.04 -8.39 6.17 0.70 -0.53 -9.80 7.06 -14.16 15.87 44.59 -35.38 19.79 5.63 E 164.814 700.00 22.29 -8.89 6.54 0.74 -0.56 -10.38 7.48 -15.00 16.81 47.24 -37.49 20.96 5.96 F 174.614 800.00 23.62 -9.42 6.93 0.79 -0.59 -11.00 7.92 -15.89 17.81 50.05 -39.72 22.21 6.32 F# 184.997 900.00 25.02 -9.98 7.34 0.84 -0.63 -11.65 8.39 -16.84 18.87 53.02 -42.08 23.53 6.69 G 195.998 1000.00 26.51 -10.58 7.78 0.89 -0.66 -12.35 8.89 -17.84 19.99 56.17 -44.58 24.93 7.09 G# 207.652 1100.00 28.08 -11.21 8.24 0.94 -0.70 -13.08 9.42 -18.90 21.18 59.52 -47.23 26.41 7.51 A 220.000 0.00 29.75 -11.87 8.73 0.99 -0.74 -13.86 9.98 -20.02 22.44 63.05 -50.04 27.98 7.96 Bb 233.082 100.00 31.52 -12.58 9.25 1.05 -0.79 -14.68 10.58 -21.21 23.77 66.80 -53.01 29.65 8.43 B 246.942 200.00 33.40 -13.33 9.80 1.12 -0.84 -15.56 11.21 -22.47 25.19 70.78 -56.17 31.41 8.93 Middle C 261.626 (0.00) 35.38 -14.12 10.38 1.18 -0.89 -16.48 11.87 -23.81 26.68 74.98 -59.51 33.28 9.46 C# 277.183 (0.00) 37.49 -14.96 11.00 1.25 -0.94 -17.46 12.58 -25.23 28.27 79.44 -63.05 35.25 10.03 D 293.665 (0.00) 39.72 -15.85 11.65 1.33 -0.99 -18.50 13.33 -26.73 29.95 84.17 -66.79 37.35 10.62 Eb 311.127 (0.00) 42.08 -16.79 12.35 1.41 -1.05 -19.60 14.12 -28.32 31.73 89.17 -70.77 39.57 11.26 E 329.628 (0.00) 44.58 -17.79 13.08 1.49 -1.12 -20.76 14.96 -30.00 33.62 94.47 -74.97 41.93 11.92 F 349.228 (0.00) 47.23 -18.85 13.86 1.58 -1.18 -22.00 15.85 -31.78 35.62 100.09 -79.43 44.42 12.63 F# 369.994 (0.00) 50.04 -19.97 14.68 1.67 -1.25 -23.31 16.79 -33.67 37.74 106.04 -84.16 47.06 13.39 G 391.995 (0.00) 53.01 -21.15 15.56 1.77 -1.33 -24.69 17.79 -35.68 39.98 112.35 -89.16 49.86 14.18 G# 415.305 (0.00) 56.17 -22.41 16.48 1.88 -1.41 -26.16 18.85 -37.80 42.36 119.03 -94.46 52.82 15.02 A 440.000 (0.00) 59.51 -23.74 17.46 1.99 -1.49 -27.72 19.97 -40.05 44.88 126.11 -100.08 55.96 15.92 Bb 466.164 (0.00) 63.05 -25.16 18.50 2.11 -1.58 -29.37 21.15 -42.43 47.54 133.61 -106.03 59.29 16.86 B 493.883 (0.00) 66.79 -26.65 19.60 2.23 -1.67 -31.11 22.41 -44.95 50.37 141.55 -112.33 62.82 17.87 C 523.251 (0.00) 70.77 -28.24 20.76 2.36 -1.77 -32.96 23.74 -47.62 53.37 149.97 -119.01 66.55 18.93

### Rational Approximations

The following tables show the rational values of each note relative to C. In the cases where the numerator and denominator are both small integers, the interval is in tune as it would be in a Just Intonation scale. If there are three or more digits, it is merely an approximation as the spreadsheet rounds off the decimals.

Meantone
 C 1/1 C# 6329/6057 D 6119/5473 Eb 1286/1075 E 5/4 F 860/643 F# 3867/2767 G 643/430 G# 25/16 A 1075/643 Bb 10946/6119 B 643/344 C 2/1
Kirnberger 3
 C 1/1 Db 256/243 D 6119/5473 Eb 32/27 E 5/4 F 4/3 Gb 1024/729 G 643/430 Ab 128/81 A 1075/643 Bb 16/9 B 15/8 C 2/1
Equal
 C 1/1 C# 7893/7450 D 5252/4679 Eb 10754/9043 E 6064/4813 F 6793/5089 F# 8119/5741 G 10178/6793 G# 4813/3032 A 9043/5377 Bb 4679/2626 B 17843/9452 C 2/1