# LucyTuning compared with 1/4 comma meantone and 12-note equal temperament

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

Results generated using temperament spreadsheet v1.3, which see for explanation of the charts.

This is a brief comparison among three varieties of meantone temperament: "LucyTuning," 1/4 syntonic comma meantone, and 12-note equal temperament. "Meantone" means that any major third (such as C-E) is built of two "mean" (average) tones of the same size; that is, the interval C-D and the interval D-E are the same size. (This would not be true in some just intonation systems, for example, where the frequencies of D and C are in a 9/8 ratio, and E and D are 10/9.)

Each of these three systems (among many others) arises from a different definition of the size of a "large interval" (whole tone, e.g. C-D or D-E), in constructing a seven-note diatonic scale of five large and two small intervals arranged L-L-S-L-L-L-S. (Look at the intervals between the notes C-D-E-F-G-A-B-C: E-F and B-C are small intervals or "half steps," and the others are large intervals or "whole steps.")

In 12-note equal temperament, where every fifth is narrow by 1/12 of a ditonic comma, the large interval ("whole step") is exactly the size of two small intervals ("half step"). The large interval is 2^(1/6), or 200 cents; the small interval is 2^(1/12), or 100 cents. (The definition of cents is of course based directly on this. The cent system is logarithmic, so addition of cents is a shortcut for multiplication of interval ratios.) The major third (composite of two large intervals) of 400 cents is wider than a pure 5/4 ratio, which is about 386.31 cents.

In 1/4 syntonic comma meantone, where every fifth is narrow by 1/4 of a syntonic comma, the size of the large interval ("whole step") is (5/4)^(1/2), or about 193.16 cents. This leaves the small interval ("semitone" or "half step") to be about 117.11 cents. A pure major third such as C-E is 5/4, so to put the D exactly between them ("a mean tone" where C-D and D-E are the same-sized intervals), the intervals C-D and D-E are each the square root of 5/4. Note: there is also a distinction between diatonic semitones (notes of different names, such as B and C) and chromatic semitones (with same basic note names, as B and Bb, chromatic="color change"): chromatic semitones are only about 76.05 cents in 1/4 comma meantone temperament. The only temperament in which these two types of semitones are equivalent is equal temperament.

In LucyTuning, developed by Charles Lucy, the fifth is even narrower, to generate a large interval defined as 2^(1/(2*pi)), or about 190.99 cents, leaving a small interval of about 122.53 cents. The major third is narrower than a pure 5/4 ratio, about 381.97 cents. LucyTuning (major and minor thirds all narrow) gives an effect between those of 1/4 syntonic comma meantone (major thirds pure, minor thirds narrow) and 1/3 syntonic comma meantone (minor thirds pure, major thirds narrow). That is, major and minor thirds all sound almost pure. Also, an interval of a fifth above a given note beats faster than the minor third, but slower than the major third.

The charts below show comparative beat rates of all intervals in these systems. These are followed by charts showing subjective interval and triad perception, as defined on the temperament spreadsheet page. I have chosen the notes Eb-Bb-F-C-G-D-A-E-B-F#-C#-G# in all cases. Note: these charts show how these temperaments sound on a twelve-note keyboard; "wolf" intervals and triads show up here. For instance, C#-F is treated as a major third in the charts, but is actually a diminished fourth.

### Beat Rate Charts

LucyTuning, set from A=440
 14/9 12/7 7/4 16/9 Master Freq Cents Sm Min3 Min3 Maj3 Per4 Per5 Min6 Maj6 Min7 Maj7 Min6 Maj6 Min7 Min7 C 131.838 313.52 25.21 -0.97 -1.65 1.97 -1.47 -29.87 0.81 -5.87 -12.30 -1.03 -42.06 21.67 15.81 C# 137.156 381.97 26.22 -1.01 19.99 2.05 -1.53 2.75 22.63 -6.10 52.08 59.82 7.08 22.55 16.44 D 147.215 504.51 28.15 -1.08 -1.84 2.20 -1.65 2.96 0.90 -6.55 -13.74 64.21 -46.97 24.20 17.65 Eb 158.012 627.04 -4.74 -30.28 -1.98 -17.15 -1.77 -35.80 0.97 -50.55 -14.74 -1.24 -50.41 -8.84 -59.40 E 164.385 695.49 31.43 -1.21 -2.06 2.46 -1.84 3.30 1.01 -7.32 62.42 71.70 -52.44 27.02 19.71 F 176.442 818.03 -5.29 -33.82 -2.21 2.64 -1.97 -39.98 1.08 -7.85 -16.46 -1.38 -56.29 29.01 21.15 F# 183.558 886.48 35.09 -1.35 26.75 2.75 -2.05 3.69 30.28 -8.17 69.70 80.06 9.48 30.18 22.01 G 197.021 1009.01 37.67 -1.45 -2.47 2.95 -2.20 3.96 1.21 -8.77 -18.38 85.93 -62.86 32.39 23.62 G# 204.967 1077.47 39.19 -1.51 29.87 3.07 17.15 4.12 33.82 -9.12 77.83 89.40 10.58 33.70 24.57 A 220.000 0.00 42.06 -1.62 -2.75 3.29 -2.46 4.42 1.35 -9.79 -20.53 95.95 -70.19 36.17 26.38 Bb 236.136 122.53 -7.08 -45.26 -2.96 3.53 -2.64 -53.50 1.45 -75.55 -22.03 -1.85 -75.33 -13.21 -88.76 B 245.659 190.99 46.97 -1.80 35.80 3.67 -2.75 4.93 1.51 -10.93 93.28 107.15 -78.37 40.38 29.45 Middle C 263.677 (13.52) 50.41 -1.94 -3.30 3.94 -2.95 -59.74 1.62 -11.74 -24.60 -2.07 -84.12 43.35 31.61 C# 274.311 -(18.03) 52.44 -2.01 39.98 4.10 -3.07 5.51 45.26 -12.21 104.16 119.64 14.16 45.10 32.89 D 294.430 (4.51) 56.29 -2.16 -3.69 4.40 -3.29 5.91 1.80 -13.10 -27.47 128.42 -93.93 48.40 35.30 Eb 316.025 (27.04) -9.48 -60.57 -3.96 -34.30 -3.53 -71.61 1.94 -101.11 -29.49 -2.48 -100.82 -17.68 -118.79 E 328.771 -(4.51) 62.86 -2.42 -4.12 4.92 -3.67 6.60 2.01 -14.63 124.84 143.39 -104.89 54.05 39.42 F 352.884 (18.03) -10.58 -67.63 -4.42 5.28 -3.94 -79.96 2.16 -15.71 -32.93 -2.77 -112.58 58.01 42.31 F# 367.116 -(13.52) 70.19 -2.70 53.50 5.49 -4.10 7.37 60.57 -16.34 139.40 160.12 18.95 60.35 44.01 G 394.042 (9.01) 75.33 -2.89 -4.93 5.89 -4.40 7.91 2.42 -17.54 -36.77 171.86 -125.71 64.78 47.24 G# 409.934 -(22.53) 78.37 -3.01 59.74 6.13 34.30 8.23 67.63 -18.25 155.66 178.79 21.16 67.39 49.15 A 440.000 (0.00) 84.12 -3.23 -5.51 6.58 -4.92 8.84 2.70 -19.58 -41.06 191.91 -140.37 72.33 52.75 Bb 472.271 (22.53) -14.16 -90.51 -5.91 7.06 -5.28 -107.01 2.89 -151.10 -44.07 -3.71 -150.67 -26.43 -177.53 B 491.319 -(9.01) 93.93 -3.61 71.61 7.35 -5.49 9.87 3.01 -21.87 186.56 214.29 -156.75 80.77 58.90 C 527.354 (13.52) 100.82 -3.87 -6.60 7.89 -5.89 -119.49 3.23 -23.47 -49.21 -4.14 -168.24 86.69 63.22

1/4 Syntonic Comma Meantone, set from A=440
 14/9 12/7 7/4 16/9 Master 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

Equal Temperament, set from A=440
 14/9 12/7 7/4 16/9 Master 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

### Subjective Tolerability

LucyTuning
 Best Best Best Best Maj Min Dim Master Min3 Maj3 Per4 Per5 Min6 Maj6 Min7 Maj7 | Triad Triad Triad Mm7 MM7 mm7 dim7 As Neap C -1 -2 5 -6 -1 1 -4 -13 | 2 1 2 3 9 3 1 5 C C# -1 20 5 -6 3 -3 -4 53 | 20 1 1 11 40 3 2 2 C# D -1 -2 5 -6 3 1 -4 -13 | 2 1 2 3 9 3 1 2 D Eb -3 -2 -38 -6 -1 1 -6 -13 | 2 2 2 4 9 4 1 2 Eb E -1 -2 5 -6 3 1 -4 53 | 2 1 1 3 33 3 1 2 E F -3 -2 5 -6 -1 1 -4 -13 | 2 2 2 3 9 4 1 2 F F# -1 20 5 -6 3 -3 -4 53 | 20 1 1 11 40 3 2 2 F# G -1 -2 5 -6 3 1 -4 -13 | 2 1 2 3 9 3 1 2 G G# -1 20 5 42 3 -3 -4 53 | 42 42 1 19 49 19 2 2 G# A -1 -2 5 -6 3 1 -4 -13 | 2 1 1 3 9 3 1 2 A Bb -3 -2 5 -6 -1 1 -6 -13 | 2 2 2 4 9 4 1 2 Bb B -1 20 5 -6 3 1 -4 53 | 20 1 1 11 40 3 1 2 B Arranged as circle of fifths: | C -1 -2 5 -6 -1 1 -4 -13 | 2 1 2 3 9 3 1 5 C G -1 -2 5 -6 3 1 -4 -13 | 2 1 2 3 9 3 1 2 G D -1 -2 5 -6 3 1 -4 -13 | 2 1 2 3 9 3 1 2 D A -1 -2 5 -6 3 1 -4 -13 | 2 1 1 3 9 3 1 2 A E -1 -2 5 -6 3 1 -4 53 | 2 1 1 3 33 3 1 2 E B -1 20 5 -6 3 1 -4 53 | 20 1 1 11 40 3 1 2 B F# -1 20 5 -6 3 -3 -4 53 | 20 1 1 11 40 3 2 2 F# C# -1 20 5 -6 3 -3 -4 53 | 20 1 1 11 40 3 2 2 C# G# -1 20 5 42 3 -3 -4 53 | 42 42 1 19 49 19 2 2 G# Eb -3 -2 -38 -6 -1 1 -6 -13 | 2 2 2 4 9 4 1 2 Eb Bb -3 -2 5 -6 -1 1 -6 -13 | 2 2 2 4 9 4 1 2 Bb F -3 -2 5 -6 -1 1 -4 -13 | 2 2 2 3 9 4 1 2 F

1/4 Syntonic Comma Meantone
 Best Best Best Best Maj Min Dim Master Min3 Maj3 Per4 Per5 Min6 Maj6 Min7 Maj7 | Triad Triad Triad Mm7 MM7 mm7 dim7 As Neap 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 Arranged as circle of fifths: | 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

Equal Temperament
 Best Best Best Best Maj Min Dim Master Min3 Maj3 Per4 Per5 Min6 Maj6 Min7 Maj7 | Triad Triad Triad Mm7 MM7 mm7 dim7 As Neap 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 Arranged as circle of fifths: | 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

### Harmony Chart: tension of common chords when playing in the given key

LucyTuning
 Key I i Neap ii iio III iii IV iv V V7 v VI vi VII viio viio7 C 2 1 2 1 1 2 1 2 2 2 3 1 42 1 2 1 1 G 2 1 5 1 1 2 1 2 1 2 3 1 2 1 2 1 2 D 2 1 2 1 1 2 1 2 1 2 3 1 2 1 2 1 2 A 2 1 2 1 1 2 1 2 1 2 3 1 2 1 2 1 2 E 2 1 2 1 2 2 42 2 1 20 11 1 2 1 2 2 1 B 20 1 2 1 2 2 2 2 1 20 11 1 2 42 2 2 1 F# 20 1 2 42 2 2 2 20 1 20 11 1 2 2 2 2 1 C# 20 1 2 2 1 2 2 20 1 42 19 42 2 2 20 2 1 G# 42 42 2 2 1 20 1 20 1 2 4 2 2 2 20 2 1 Eb 2 2 2 2 1 20 1 42 42 2 4 2 20 1 20 2 1 Bb 2 2 2 1 1 20 1 2 2 2 3 2 20 1 42 1 1 F 2 2 2 1 1 42 1 2 2 2 3 1 20 1 2 1 1

1/4 Syntonic Comma Meantone
 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

Equal Temperament
 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