# List of pitch intervals

Below is a list of intervals exprimable in terms of a prime limit (see Terminology), completed by a choice of intervals in various equal subdivisions of the octave or of other intervals.

For commonly encountered harmonic or melodic intervals between pairs of notes in contemporary Western music theory, without consideration of the way in which they are tuned, see Interval (music) § Main intervals.

Comparison between Pythagorean tuning (blue), equal-tempered (black), 1/4-comma meantone (red) and 1-3-comma meantone (green). For each, the common origin is arbitrarily chosen as C. The degrees are arranged in the order or the cycle of fiths; as in each of these tunings all fifths are of the same size, the tunings appear as straigth lines, the slope indicating the relative tempering with respect to Pythagorean, which has pure fifths (3:2, 702 cents). The Pythagorean Ab (at the left) is at 792 cents, G# (at the right) at 816 cents; the difference is the Pythagorean comma. Equal temperament by definition is such that Ab and G# are at the same level. 1/4 comma meantone produces the "just" major third (5:4, 386 cents, a syntonic comma lower than the Pythagorean one of 408 cents). 1/3 comma meantone produces the "just" minor third (6:5, 316 cents, a syntonic comma higher than the Pythagorean one of 294 cents). In both these meantone temperaments, the enharmony, here the difference between Ab and G#, is much larger than in Pythagorean, and with the flat degree higher than the sharp one.

## Terminology

• The prime limit, a concept introduced by Harry Partch,[1] henceforth referred to simply as the limit, is the largest prime number occurring in the factorizations of the numerator and denominator of the frequency ratio describing a rational interval. For instance, the limit of the just perfect fourth (4 : 3) is 3, but the just minor tone (10 : 9) has a limit of 5, because 10 can be factorized into 2·5 (and 9 in 3·3). There exists another type of limit, the odd limit (bigger of odd numbers obtained after dividing numerator and denominator by highest possible powers of 2), but it is not used here.
• By definition, every interval in a given limit can also be part of a limit of higher order. For instance, a 3-limit unit can also be part of a 5-limit tuning and so on. By sorting the limit columns in the table below, all intervals of a given limit can be brought together (sort backwards by clicking the button twice).
• Pythagorean tuning means 3-limit intonation—a ratio of numbers with prime factors no higher than three.
• Just intonation means 5-limit intonation—a ratio of numbers with prime factors no higher than five.
• Septimal, undecimal, tridecimal, and septendecimal mean, respectively, 7, 11, 13, and 17-limit intonation.
• Meantone refers to meantone temperament, where the whole tone is the mean of the major third. In general, a meantone is constructed in the same way as Pythagorean tuning, as a stack of fifths: the tone is reached after two fifths, the major third after four, so that as all fifths are the same, the tone is the mean of the third. In a meantone temperament, each fifth is narrowed ("tempered") by the same small amount. The most common of meantone temperaments is the quarter-comma meantone, in which each fifth is tempered by 1/4 of the syntonic comma, so that after four steps the major third (as C-G-D-A-E) is a full syntonic comma lower than the Pythagorean one. The extremes of the meantone systems encountered in historical practice are the Pythagorean tuning, where the whole tone corresponds to 9:8, i.e (3:2)2/2, the mean of the major third (3:2)4/4, and the fifth (3:2) is not tempered; and the 1/3-comma meantone, where the fifth is tempered to the extent that three ascending fifths produce a pure minor third.(See Meantone temperaments). The music program Logic Pro uses also 1/2-comma meantone temperament.
• Equal-tempered refers to X-tone equal temperament with intervals corresponding to X divisions per octave.
• Tempered intervals however cannot be expressed in terms of prime limits and, unless exceptions, are not found in the table below.
• The table can also be sorted by frequency ratio, by cents, or alphabetically.

## List

Column Legend
TET X-tone equal temperament (12-tet, etc.).
Limit 3-limit intonation, or Pythagorean.
5-limit "just" intonation, or just.
7-limit intonation, or septimal.
11-limit intonation, or undecimal.
13-limit intonation, or tridecimal.
17-limit intonation, or septendecimal.
19-limit intonation, or novendecimal.
Higher limits.
M Meantone temperament or tuning.
S Superparticular ratio (no separate color code).
List of musical intervals
Cents Note (from C) Freq. ratio Prime Factors Interval name TET 3 5 7 11 13 17 19 H M S
0.00
C[2] 1 : 1 1 : 1 Unison,[3] monophony,[4] perfect prime,[3] tonic,[5] or fundamental 1, 12 3 5 7 11 13 17 19 H M
0.40
C- 4375 : 4374 54·7 : 2·37 7 11 13 17 19 H S
0.72
E+ 2401 : 2400 74 : 25·3·52 7 11 13 17 19 H S
1.00
21/1200 21/1200 Cent 1200
1.20
21/1000 21/1000 Millioctave 1000
1.95
B++ 32805 : 32768 38·5 : 215 5 7 11 13 17 19 H
3.99
101/1000 21/1000·51/1000 Savart or eptaméride 301.03
7.71
B 225 : 224 32·52 : 25·7 Septimal kleisma,[3][6] marvel comma 7 11 13 17 19 H S
8.11
B- 15625 : 15552 56 : 26·35 Kleisma or semicomma majeur[3][6] 5 7 11 13 17 19 H
10.06
A++ 2109375 : 2097152 33·57 : 221 Semicomma,[3][6] Fokker's comma[3] 5 7 11 13 17 19 H
11.98
C29 145 : 144 5·29 : 24·32 Difference between 29:16 & 9:5 H
12.50
21/96 21/96 Sixteenth-tone 96
13.07
B- 1728 : 1715 26·33 : 5·73 7 11 13 17 19 H
13.79
D 126 : 125 2·32·7 : 53 Small septimal semicomma,[6] small septimal comma,[3] starling comma 7 11 13 17 19 H S
14.37
C- 121 : 120 112 : 23·3·5 Undecimal seconds comma[3] 11 13 17 19 H S
16.67
21/72 21/72 1 step in 72 equal temperament 72
18.13
C 96 : 95 25·3 : 5·19 Difference between 19:16 & 6:5 19 H
19.55
D--[2] 2048 : 2025 211 : 34·52 Diaschisma,[3][6] minor comma 5 7 11 13 17 19 H
21.51
C+[2] 81 : 80 34 : 24·5 Syntonic comma,[3][5][6] major comma, komma, chromatic diesis, or comma of Didymus[3][6][8][9] 5 7 11 13 17 19 H S
22.64
21/53 21/53 Holdrian comma, Holder's comma, 1 step in 53 equal temperament 53
23.46
B+++ 531441 : 524288 312 : 219 Pythagorean comma,[3][5][6][8][9] ditonic comma[3][6] 3 5 7 11 13 17 19 H
25.00
21/48 21/48 Eighth-tone 48
26.84
C 65 : 64 5·13 : 26 Sixty-fifth harmonic,[5] 13th-partial chroma[3] 13 17 19 H S
27.26
C- 64 : 63 26 : 32·7 Septimal comma,[3][6][9] Archytas' comma[3] 7 11 13 17 19 H S
29.27
21/41 21/41 1 step in 41 equal temperament 41
31.19
D 56 : 55 23·7 : 5·11 Ptolemy's enharmonic:[5] difference between (11 : 8) and (7 : 5) tritone 11 13 17 19 H S
33.33
21/36 21/36 Sixth-tone 36
34.28
C 51 : 50 3·17 : 2·52 Difference between 17:16 & 25:24 17 19 H S
34.98
B- 50 : 49 2·52 : 72 Septimal sixth-tone or jubilisma, Erlich's decatonic comma or tritonic diesis[3][6] 7 11 13 17 19 H S
35.70
D 49 : 48 72 : 24·3 Septimal diesis, slendro diesis or septimal 1/6-tone[3] 7 11 13 17 19 H S
38.05
C23 46 : 45 2·23 : 32·5 Difference between 23:16 & 45:32 H
38.71
21/31 21/31 1 step in 31 equal temperament 31
40.00
21/30 21/30 Fifth-tone 30
41.06
D- 128 : 125 27 : 53 Enharmonic diesis or 5-limit limma, minor diesis,[6] diminished second,[5][6] minor diesis or diesis[3] 5 7 11 13 17 19 H
48.77
C 36 : 35 22·32 : 5·7 Septimal quarter tone, septimal diesis,[3][6] septimal comma,[2] superior quarter-tone[5] 7 11 13 17 19 H S
50.00
C/D 21/24 21/24 Equal-tempered quarter tone 24
53.27
C 33 : 32 3·11 : 25 Thirty-third harmonic,[5] undecimal comma, undecimal quarter-tone 11 13 17 19 H S
56.77
C31 31 : 30 31 : 2·3·5 Difference between 31:16 & 15:8 H
62.96
C- 28 : 27 22·7 : 33 Septimal minor second, small minor second, inferior quarter-tone[5] 7 11 13 17 19 H S
63.81
(3 : 2)1/11 31/11 : 21/11 Beta scale step 18.75
65.34
C+ 27 : 26 33 : 2·13 Chromatic diesis,[10] tridecimal comma[3] 13 17 19 H S
66.67
21/18 21/18 Third-tone 18
70.67
C[2] 25 : 24 52 : 23·3 Just chromatic semitone or minor chroma,[3] lesser chromatic semitone, small (just) semitone[9] or minor second,[4] minor chromatic semitone,[11] or minor semitone,[5] 2/7-comma meantone chromatic semitone 5 7 11 13 17 19 H S
78.00
(3 : 2)1/9 31/9 : 21/9 Alpha scale step 15.39
79.31
67 : 64 67 : 26 Sixty-seventh harmonic[5] H
84.47
D 21 : 20 3·7 : 22·5 Septimal chromatic semitone, minor semitone[3] 7 11 13 17 19 H S
90.22
D--[2] 256 : 243 28 : 35 Pythagorean minor second or limma,[3][6][9] Pythagorean diatonic semitone, Low Semitone[12] 3 5 7 11 13 17 19 H
92.18
C+[2] 135 : 128 33·5 : 27 Greater chromatic semitone, chromatic semitone, semitone medius, major chroma or major limma,[3] small limma,[9] major chromatic semitone,[11] limma ascendant[5] 5 7 11 13 17 19 H
98.95
D 18 : 17 2·32 : 17 Just minor semitone, Arabic lute index finger[3] 17 19 H S
100.00
C/D 21/12 21/12 Equal-tempered minor second or semitone 12 M
104.96
C[2] 17 : 16 17 : 24 Minor diatonic semitone, just major semitone, overtone semitone,[5] 17th harmonic,[3] limma[citation needed] 17 19 H S
111.73
D-[2] 16 : 15 24 : 3·5 Just minor second,[13] just diatonic semitone, large just semitone or major second,[4] major semitone,[5] limma, minor diatonic semitone,[3] diatonic second[14] semitone,[12] diatonic semitone,[9] 1/6-comma meantone minor second 5 7 11 13 17 19 H S
113.69
C++ 2187 : 2048 37 : 211 apotome[3][9] or Pythagorean major semitone,[6] Pythagorean augmented prime, Pythagorean chromatic semitone, or Pythagorean apotome 3 5 7 11 13 17 19 H
116.72
(18 : 5)1/19 21/19·32/19 : 51/19 Secor 10.28
119.44
C 15 : 14 3·5 : 2·7 Septimal diatonic semitone, major diatonic semitone[3] 7 11 13 17 19 H S
130.23
C23+ 69 : 64 3·23 : 26 Sixty-ninth harmonic[5] H
133.24
D 27 : 25 33 : 52 Semitone maximus, minor second, large limma or Bohlen-Pierce small semitone,[3] high semitone,[12] alternate Renaissance half-step,[5] large limma, acute minor second[citation needed] 5 7 11 13 17 19 H
150.00
C/D 23/24 21/8 Equal-tempered neutral second 8, 24
150.64
D↓[2] 12 : 11 22·3 : 11 3/4-tone or Undecimal neutral second,[3][5] trumpet three-quarter tone[9] 11 13 17 19 H S
155.14
D 35 : 32 5·7 : 25 Thirty-fifth harmonic[5] 7 11 13 17 19 H
160.90
D-- 800 : 729 25·52 : 36 Grave whole tone,[3] neutral second, grave major second[citation needed] 5 7 11 13 17 19 H
165.00
D-[2] 11 : 10 11 : 2·5 Greater undecimal minor/major/neutral second, 4/5-tone or Ptolemy's second[3] 11 13 17 19 H S
171.43
21/7 21/7 1 step in 7 equal temperament 7
179.70
71 : 64 71 : 26 Seventy-first harmonic[5] H
180.45
E--- 65536 : 59049 216 : 310 Pythagorean diminished third,[3][6] Pythagorean minor tone 3 5 7 11 13 17 19 H
182.40
D-[2] 10 : 9 2·5 : 32 Small just whole tone or major second,[4] minor whole tone,[3][5] lesser whole tone,[14] minor tone,[12] minor second,[9] half-comma meantone major second 5 7 11 13 17 19 H S
200.00
D 22/12 21/6 Equal-tempered major second 6, 12 M
203.91
D[2] 9 : 8 32 : 23 Pythagorean major second, Large just whole tone or major second[9] (sesquioctavan),[4] tonus, major whole tone,[3][5] greater whole tone,[14] major tone[12] 3 5 7 11 13 17 19 H S
223.46
E-[2] 256 : 225 28 : 32·52 5 7 11 13 17 19 H
227.79
73 : 64 73 : 26 Seventy-third harmonic[5] H
231.17
D-[2] 8 : 7 23 : 7 Septimal major second,[4] septimal whole tone[3][5] 7 11 13 17 19 H S
240.00
21/5 21/5 1 step in 5 equal temperament 5
251.34
37 : 32 37 : 25 Thirty-seventh harmonic[5] H
253.08
D- 125 : 108 53 : 22·33 Semi-augmented whole tone,[3] semi-augmented second[citation needed] 5 7 11 13 17 19 H
266.87
E[2] 7 : 6 7 : 2·3 Septimal minor third[3][4][9] or Sub minor Third[12] 7 11 13 17 19 H S
274.58
D[2] 75 : 64 3·52 : 26 Just augmented second,[14] Augmented Tone,[12] augmented second[5][11] 5 7 11 13 17 19 H
294.13
E-[2] 32 : 27 25 : 33 3 5 7 11 13 17 19 H
297.51
E[2] 19 : 16 19 : 24 19th harmonic,[3] 19-limit minor third, overtone minor third,[5] Pythagorean minor third[citation needed] 19 H
300.00
D/E 23/12 21/4 Equal-tempered minor third 4, 12 M
310.26
6:5÷(81:80)1/4 22 : 53/4 Quarter-comma meantone minor third M
311.98
(3 : 2)4/9 34/9 : 24/9 3.85
315.64
E[2] 6 : 5 2·3 : 5 Just minor third,[3][4][5][9][14] minor third,[12] 1/3-comma meantone minor third 5 7 11 13 17 19 H M S
317.60
D++ 19683 : 16384 39 : 214 Pythagorean augmented second[3][6] 3 5 7 11 13 17 19 H
320.14
77 : 64 7·11 : 26 Seventy-seventh harmonic[5] 11 13 17 19 H
337.15
E+ 243 : 200 35 : 23·52 Acute minor third[3] 5 7 11 13 17 19 H
342.48
E 39 : 32 3·13 : 25 Thirty-ninth harmonic[5] 13 17 19 H
342.86
22/7 22/7 2 steps in 7 equal temperament 7
347.41
E-[2] 11 : 9 11 : 32 Undecimal neutral third[3] 11 13 17 19 H
350.00
D/E 27/24 27/24 Equal-tempered neutral third 24
359.47
E[2] 16 : 13 24 : 13 Tridecimal neutral third[3] 13 17 19 H
364.54
79 : 64 79 : 26 Seventy-ninth harmonic[5] H
364.81
E- 100 : 81 22·52 : 34 Grave major third[3] 5 7 11 13 17 19 H
384.36
F-- 8192 : 6561 213 : 38 Pythagorean diminished fourth,[3][6] Pythagorean 'schismatic' third[5] 3 5 7 11 13 17 19 H
386.31
E[2] 5 : 4 5 : 22 Just major third,[3][4][5][9][14] major third,[12] quarter-comma meantone major third 5 7 11 13 17 19 H M S
400.00
E 24/12 21/3 Equal-tempered major third 3, 12 M
407.82
E+[2] 81 : 64 34 : 26 Pythagorean major third,[3][5][6][12][14] ditone 3 5 7 11 13 17 19 H
417.51
F+[2] 14 : 11 2·7 : 11 Undecimal diminished fourth or major third[3] 11 13 17 19 H
427.37
F[2] 32 : 25 25 : 52 Just diminished fourth,[14] diminished fourth[5][11] 5 7 11 13 17 19 H
429.06
41 : 32 41 : 25 Forty-first harmonic[5] H
435.08
E[2] 9 : 7 32 : 7 Septimal major third,[3][5] Bohlen-Pierce third,[3] Super major Third[12] 7 11 13 17 19 H
450.05
83 : 64 83 : 26 Eighty-third harmonic[5] H
454.21
F 13 : 10 13 : 2·5 Tridecimal major third or diminished fourth 13 17 19 H
456.99
E[2] 125 : 96 53 : 25·3 Just augmented third, augmented third[5] 5 7 11 13 17 19 H
470.78
F+[2] 21 : 16 3·7 : 24 Twenty-first harmonic, narrow fourth,[3] septimal fourth,[5] wide augmented third[citation needed] 7 11 13 17 19 H
478.49
E+ 675 : 512 33·52 : 29 Wide augmented third[3] 5 7 11 13 17 19 H
480.00
22/5 22/5 2 steps in 5 equal temperament 5
491.27
E 85 : 64 5·17 : 26 Eighty-fifth harmonic[5] 17 19 H
498.04
F[2] 4 : 3 22 : 3 Perfect fourth,[3][5][14] Pythagorean perfect fourth, Just perfect fourth or diatessaron[4] 3 5 7 11 13 17 19 H S
500.00
F 25/12 25/12 Equal-tempered perfect fourth 12 M
510.51
(3 : 2)8/11 38/11 : 28/11 18.75
511.52
43 : 32 43 : 25 Forty-third harmonic[5] H
514.29
23/7 23/7 3 steps in 7 equal temperament 7
519.55
F+[2] 27 : 20 33 : 22·5 5-limit wolf fourth, acute fourth,[3] imperfect fourth[14] 5 7 11 13 17 19 H
521.51
E+++ 177147 : 131072 311 : 217 Pythagorean augmented third[3][6] (F+ (pitch)) 3 5 7 11 13 17 19 H
531.53
F29+ 87 : 64 3·29 : 26 Eighty-seventh harmonic[5] H
551.32
F[2] 11 : 8 11 : 23 eleventh harmonic,[5] undecimal tritone,[5] lesser undecimal tritone, undecimal semi-augmented fourth[3] 11 13 17 19 H
568.72
F[2] 25 : 18 52 : 2·32 Just augmented fourth[3][5] 5 7 11 13 17 19 H
570.88
89 : 64 89 : 26 Eighty-ninth harmonic[5] H
582.51
G[2] 7 : 5 7 : 5 Lesser septimal tritone, septimal tritone[3][4][5] Huygens' tritone or Bohlen-Pierce fourth,[3] septimal fifth,[9] septimal diminished fifth[15] 7 11 13 17 19 H
588.27
G-- 1024 : 729 210 : 36 Pythagorean diminished fifth,[3][6] low Pythagorean tritone[5] 3 5 7 11 13 17 19 H
590.22
F+[2] 45 : 32 32·5 : 25 Just augmented fourth, just tritone,[4][9] tritone,[6] diatonic tritone,[3] 'augmented' or 'false' fourth,[14] high 5-limit tritone,[5] 1/6-comma meantone augmented fourth 5 7 11 13 17 19 H
600.00
F/G 26/12 21/2 Equal-tempered tritone 2, 12 M
609.35
G 91 : 64 7·13 : 26 Ninety-first harmonic[5] 13 17 19 H
609.78
G-[2] 64 : 45 26 : 32·5 Just tritone,[4] 2nd tritone,[6] 'false' fifth,[14] diminished fifth,[11] low 5-limit tritone[5] 5 7 11 13 17 19 H
611.73
F#++ 729 : 512 36 : 29 Pythagorean tritone,[3][6] Pythagorean augmented fourth, high Pythagorean tritone[5] 3 5 7 11 13 17 19 H
617.49
F[2] 10 : 7 2·5 : 7 Greater septimal tritone, septimal tritone,[4][5] Euler's tritone[3] 7 11 13 17 19 H
628.27
F23+ 23 : 16 23 : 24 Twenty-third harmonic,[5] classic diminished fifth[citation needed] H
631.28
G[2] 36 : 25 22·32 : 52 5 7 11 13 17 19 H
646.99
F31+ 93 : 64 3·31 : 26 Ninety-third harmonic[5] H
648.68
G↓[2] 16 : 11 24 : 11 Inversion of eleventh harmonic, undecimal semi-diminished fifth[3] 11 13 17 19 H
665.51
47 : 32 47 : 25 Forty-seventh harmonic[5] H
678.49
A--- 262144 : 177147 218 : 311 Pythagorean diminished sixth[3][6] 3 5 7 11 13 17 19 H
680.45
G- 40 : 27 23·5 : 33 5-limit wolf fifth,[5] or diminished sixth, grave fifth,[3][6][9] imperfect fifth,[14] 5 7 11 13 17 19 H
683.83
G 95 : 64 5·19 : 26 Ninety-fifth harmonic[5] 19 H
691.20
3:2÷(81:80)1/2 2·51/2 : 3 Half-comma meantone perfect fifth M
694.79
3:2÷(81:80)1/3 21/3·51/3 : 31/3 1/3-comma meantone perfect fifth M
695.81
3:2÷(81:80)2/7 21/7·52/7 : 31/7 2/7-comma meantone perfect fifth M
696.58
3:2÷(81:80)1/4 51/4 Quarter-comma meantone perfect fifth M
697.65
3:2÷(81:80)1/5 31/5·51/5 : 21/5 1/5-comma meantone perfect fifth M
698.37
3:2÷(81:80)1/6 31/3·51/6 : 21/3 1/6-comma meantone perfect fifth M
700.00
G 27/12 27/12 Equal-tempered perfect fifth 12 M
701.89
231/53 231/53 53
701.96
G[2] 3 : 2 3 : 2 Perfect fifth,[3][5][14] Pythagorean perfect fifth, Just perfect fifth or diapente,[4] fifth,[12] Just fifth[9] 3 5 7 11 13 17 19 H S
702.44
224/41 224/41 41
703.45
217/29 217/29 29
719.90
97 : 64 97 : 26 Ninety-seventh harmonic[5] H
721.51
A- 1024 : 675 210 : 33·52 Narrow diminished sixth[3] 5 7 11 13 17 19 H
737.65
A+ 49 : 32 7·7 : 25 Forty-ninth harmonic[5] 7 11 13 17 19 H
743.01
A 192 : 125 26·3 : 53 Classic diminished sixth[3] 5 7 11 13 17 19 H
755.23
99 : 64 32·11 : 26 Ninety-ninth harmonic[5] 11 13 17 19 H
764.92
A[2] 14 : 9 2·7 : 32 7 11 13 17 19 H
772.63
G 25 : 16 52 : 24 5 7 11 13 17 19 H
782.49
G-[2] 11 : 7 11 : 7 Undecimal minor sixth,[5] undecimal augmented fifth,[3] pi 11 13 17 19 H
789.85
101 : 64 101 : 26 Hundred-first harmonic[5] H
792.18
A-[2] 128 : 81 27 : 34 Pythagorean minor sixth[3][5][6] 3 5 7 11 13 17 19 H
800.00
G/A 28/12 22/3 Equal-tempered minor sixth 3, 12 M
806.91
G 51 : 32 3·17 : 25 Fifty-first harmonic[5] 17 19 H
813.69
A[2] 8 : 5 23 : 5 5 7 11 13 17 19 H
815.64
G++ 6561 : 4096 38 : 212 Pythagorean augmented fifth,[3][6] Pythagorean 'schismatic' sixth[5] 3 5 7 11 13 17 19 H
823.80
103 : 64 103 : 26 Hundred-third harmonic[5] H
833.09
51/2+1 : 2
833.11
233 : 144 233 : 24·32 Golden ratio approximation (833 cents scale) H
835.19
A+ 81 : 50 34 : 2·52 Acute minor sixth[3] 5 7 11 13 17 19 H
840.53
A[2] 13 : 8 13 : 23 Tridecimal neutral sixth,[3] overtone sixth,[5] thirteenth harmonic 13 17 19 H
850.00
G/A 217/24 217/24 Equal-tempered neutral sixth 24
852.59
A↓[2] 18 : 11 2·32 : 11 Undecimal neutral sixth,[3][5] Zalzal's neutral sixth 11 13 17 19 H
857.10
A+ 105 : 64 3·5·7 : 26 Hundred-fifth harmonic[5] 7 11 13 17 19 H
857.14
25/7 25/7 5 steps in 7 equal temperament 7
862.85
A- 400 : 243 24·52 : 35 Grave major sixth[3] 5 7 11 13 17 19 H
873.51
53 : 32 53 : 25 Fifty-third harmonic[5] H
882.40
B--- 32768 : 19683 215 : 39 Pythagorean diminished seventh[3][6] 3 5 7 11 13 17 19 H
884.36
A[2] 5 : 3 5 : 3 Just major sixth,[3][4][5][9][14] Bohlen-Pierce sixth,[3] 1/3-comma meantone major sixth 5 7 11 13 17 19 H M
889.76
107 : 64 107 : 26 Hundred-seventh harmonic[5] H
900.00
A 29/12 23/4 Equal-tempered major sixth 4, 12 M
905.87
A+[2] 27 : 16 33 : 24 Pythagorean major sixth[3][5][9][14] 3 5 7 11 13 17 19 H
921.82
109 : 64 109 : 26 Hundred-ninth harmonic[5] H
925.42
B-[2] 128 : 75 27 : 3·52 Just diminished seventh,[14] diminished seventh[5][11] 5 7 11 13 17 19 H
933.13
A[2] 12 : 7 22·3 : 7 7 11 13 17 19 H
937.63
A 55 : 32 5·11 : 25 Fifty-fifth harmonic[5] 11 13 17 19 H
953.30
111 : 64 3·37 : 26 Hundred-eleventh harmonic[5] H
955.03
A[2] 125 : 72 53 : 23·32 5 7 11 13 17 19 H
957.21
(3 : 2)15/11 315/11 : 215/11 15 steps in Beta scale 18.75
960.00
24/5 24/5 4 steps in 5 equal temperament 5
968.83
B[2] 7 : 4 7 : 22 Septimal minor seventh,[4][5][9] harmonic seventh,[3][9] augmented sixth[citation needed] 7 11 13 17 19 H
976.54
A+[2] 225 : 128 32·52 : 27 5 7 11 13 17 19 H
984.22
113 : 64 113 : 26 Hundred-thirteenth harmonic[5] H
996.09
B-[2] 16 : 9 24 : 32 Pythagorean minor seventh,[3] Small just minor seventh,[4] lesser minor seventh,[14] just minor seventh,[9] Pythagorean small minor seventh[5] 3 5 7 11 13 17 19 H
999.47
B 57 : 32 3·19 : 25 Fifty-seventh harmonic[5] 19 H
1000.00
A/B 210/12 25/6 Equal-tempered minor seventh 6, 12 M
1014.59
A23+ 115 : 64 5·23 : 26 Hundred-fifteenth harmonic[5] H
1017.60
B[2] 9 : 5 32 : 5 Greater just minor seventh,[14] large just minor seventh,[4][5] Bohlen-Pierce seventh[3] 5 7 11 13 17 19 H
1019.55
A+++ 59049 : 32768 310 : 215 Pythagorean augmented sixth[3][6] 3 5 7 11 13 17 19 H
1028.57
26/7 26/7 6 steps in 7 equal temperament 7
1029.58
B29 29 : 16 29 : 24 Twenty-ninth harmonic,[5] minor seventh[citation needed] H
1035.00
B↓[2] 20 : 11 22·5 : 11 Lesser undecimal neutral seventh, large minor seventh[3] 11 13 17 19 H
1039.10
B+ 729 : 400 36 : 24·52 Acute minor seventh[3] 5 7 11 13 17 19 H
1044.44
A 117 : 64 32·13 : 26 Hundred-seventeenth harmonic[5] 13 17 19 H
1049.36
B-[2] 11 : 6 11 : 2·3 21/4-tone or Undecimal neutral seventh,[3] undecimal 'median' seventh[5] 11 13 17 19 H
1050.00
A/B 221/24 27/8 Equal-tempered neutral seventh 8, 24
1059.17
59 : 32 59 : 25 Fifty-ninth harmonic[5] H
1066.76
B- 50 : 27 2·52 : 33 Grave major seventh[3] 5 7 11 13 17 19 H
1073.78
B 119 : 64 7·17 : 26 Hundred-nineteenth harmonic[5] 17 19 H
1086.31
C-- 4096 : 2187 212 : 37 Pythagorean diminished octave[3][6] 3 5 7 11 13 17 19 H
1088.27
B[2] 15 : 8 3·5 : 23 Just major seventh,[3][5][9][14] small just major seventh,[4] 1/6-comma meantone major seventh 5 7 11 13 17 19 H
1100.00
B 211/12 211/12 Equal-tempered major seventh 12 M
1102.64
B- 121 : 64 112 : 26 Hundred-twenty-first harmonic[5] 11 13 17 19 H
1107.82
C'- 256 : 135 28 : 33·5 Octave − major chroma,[3] narrow diminished octave[citation needed] 5 7 11 13 17 19 H
1109.78
B+[2] 243 : 128 35 : 27 Pythagorean major seventh[3][5][6][9] 3 5 7 11 13 17 19 H
1116.89
61 : 32 61 : 25 Sixty-first harmonic[5] H
1129.33
C'[2] 48 : 25 24·3 : 52 Classic diminished octave,[3][6] large just major seventh[4] 5 7 11 13 17 19 H
1131.02
123 : 64 3·41 : 26 Hundred-twenty-third harmonic[5] H
1137.04
B 27 : 14 33 : 2·7 Septimal major seventh[5] 7 11 13 17 19 H
1145.04
B31 31 : 16 31 : 24 Thirty-first harmonic,[5] augmented seventh[citation needed] H
1158.94
B[2] 125 : 64 53 : 26 Just augmented seventh,[5] 125th harmonic 5 7 11 13 17 19 H
1172.74
C+ 63 : 32 32·7 : 25 Sixty-third harmonic[5] 7 11 13 17 19 H
1178.49
C'- 160 : 81 25·5 : 34 Octave − syntonic comma,[3] semi-diminished octave[citation needed] 5 7 11 13 17 19 H
1186.42
127 : 64 127 : 26 Hundred-twenty-seventh harmonic[5] H
1200.00
C' 2 : 1 2 : 1 Octave[3][9] or diapason[4] 1, 12 3 5 7 11 13 17 19 H M S
1223.46
B+++ 531441 : 262144 312 : 218 Pythagorean augmented seventh[3][6] 3 5 7 11 13 17 19 H
1525.86
21/2+1 Silver ratio
1901.96
G' 3 : 1 3 : 1 Tritave or just perfect twelfth 3 5 7 11 13 17 19 H
2400.00
C" 4 : 1 22 : 1 Fifteenth or two octaves 1, 12 3 5 7 11 13 17 19 H M
3986.31
E''' 10 : 1 5·2 : 1 Decade, compound just major third 5 7 11 13 17 19 H M

## References

1. ^ Fox, Christopher (2003). Microtones and Microtonalities, p.13. Taylor & Francis.
2. Fonville, John. 1991. "Ben Johnston's Extended Just Intonation: A Guide for Interpreters". Perspectives of New Music 29, no. 2 (Summer): 106–37.
3. "List of intervals", Huygens-Fokker Foundation. The Foundation uses "classic" to indicate "just" or leaves off any adjective, as in "major sixth".
4. Partch, Harry (1979). Genesis of a Music, p.68-69. ISBN 978-0-306-80106-8.
5. "Anatomy of an Octave", KyleGann.com. Gann leaves off "just" but includes "5-limit".
6. Haluška, Ján (2003). The Mathematical Theory of Tone Systems, p.xxv-xxix. ISBN 978-0-8247-4714-5.
7. ^ "Orwell Temperaments", Xenharmony.org.
8. ^ a b Partch (1979), p.70.
9. Alexander John Ellis (1885). On the musical scales of various nations, p.488. s.n.
10. ^ William Smythe Babcock Mathews (1895). Pronouncing dictionary and condensed encyclopedia of musical terms, p.13. ISBN 1-112-44188-3.
11. Anger, Joseph Humfrey (1912). A treatise on harmony, with exercises, Volume 3, p.xiv-xv. W. Tyrrell.
12. Hermann Ludwig F. von Helmholtz (Alexander John Ellis, trans.) (1875). "Additions by the translator", On the sensations of tone as a physiological basis for the theory of music, p.644. No ISBN specified.
13. ^ A. R. Meuss (2004). Intervals, Scales, Tones and the Concert Pitch C. Temple Lodge Publishing. p. 15. ISBN 1902636465.
14. Paul, Oscar (1885). A manual of harmony for use in music-schools and seminaries and for self-instruction, p.165. Theodore Baker, trans. G. Schirmer. Paul uses "natural" for "just".
15. ^ Sabat, Marc and von Schweinitz, Wolfgang (2004). "The Extended Helmholtz-Ellis JI Pitch Notation" [PDF], NewMusicBox.org. Accessed: 04:12, 15 March 2014 (UTC).