Comparison between tunings: Pythagorean, equal-tempered, quarter-comma meantone, and others. For each, the common origin is arbitrarily chosen as C. The degrees are arranged in the order or the cycle of fifths; as in each of these tunings except just intonation all fifths are of the same size, the tunings appear as straight lines, the slope indicating the relative tempering with respect to Pythagorean, which has pure fifths (3:2, 702 cents). The Pythagorean A♭ (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 A♭ and G♯ are at the same level. .mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}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 A♭ and G♯, is much larger than in Pythagorean, and with the flat degree higher than the sharp one.
Comparison between tunings: Pythagorean, equal-tempered, quarter-comma meantone, and others. For each, the common origin is arbitrarily chosen as C. The degrees are arranged in the order or the cycle of fifths; as in each of these tunings except just intonation all fifths are of the same size, the tunings appear as straight lines, the slope indicating the relative tempering with respect to Pythagorean, which has pure fifths (3:2, 702 cents). The Pythagorean A (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 A and G are at the same level. 14-comma meantone produces the "just" major third (5:4, 386 cents, a syntonic comma lower than the Pythagorean one of 408 cents). 13-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 A and G, is much larger than in Pythagorean, and with the flat degree higher than the sharp one.
Comparison of two sets of musical intervals. The equal-tempered intervals are black; the Pythagorean intervals are green.
Comparison of two sets of musical intervals. The equal-tempered intervals are black; the Pythagorean intervals are green.

Below is a list of intervals expressible 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.

Terminology

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 Limit M S
0.00
C[2] 1 : 1 1 : 1 play Unison,[3] monophony,[4] perfect prime,[3] tonic,[5] or fundamental 1, 12 3 M
0.03
65537 : 65536 65537 : 216 play Sixty-five-thousand-five-hundred-thirty-seventh harmonic 65537 S
0.40
C
7
4375 : 4374 54×7 : 2×37 play Ragisma[3][6] 7 S
0.72
E
7
7
7
7
triple flat
+
2401 : 2400 74 : 25×3×52 play Breedsma[3][6] 7 S
1.00
21/1200 21/1200 play Cent[7] 1200
1.20
21/1000 21/1000 play Millioctave 1000
1.95
B++ 32805 : 32768 38×5 : 215 play Schisma[3][5] 5
1.96
3:2÷(27/12) 3 : 219/12 Grad, Werckmeister[8]
3.99
101/1000 21/1000×51/1000 play Savart or eptaméride 301.03
7.71
B
7 upside-down
225 : 224 32×52 : 25×7 play Septimal kleisma,[3][6] marvel comma 7 S
8.11
B
double sharp
15625 : 15552 56 : 26×35 play Kleisma or semicomma majeur[3][6] 5
10.06
A
double sharp
double sharp
++
2109375 : 2097152 33×57 : 221 play Semicomma,[3][6] Fokker's comma[3] 5
10.85
C
43U
160 : 159 25×5 : 3×53 play Difference between 5:3 & 53:32 53 S
11.98
C
29
145 : 144 5×29 : 24×32 play Difference between 29:16 & 9:5 29 S
12.50
21/96 21/96 play Sixteenth tone 96
13.07
B
7 upside-down
7 upside-down
7 upside-down
1728 : 1715 26×33 : 5×73 play Orwell comma[3][9] 7
13.47
C
43
129 : 128 3×43 : 27 play Hundred-twenty-ninth harmonic 43 S
13.79
D
double flat
7
126 : 125 2×32×7 : 53 play Small septimal semicomma,[6] small septimal comma,[3] starling comma 7 S
14.37
C 121 : 120 112 : 23×3×5 play Undecimal seconds comma[3] 11 S
16.67
C[a] 21/72 21/72 play 1 step in 72 equal temperament 72
18.13
C
19U
96 : 95 25×3 : 5×19 play Difference between 19:16 & 6:5 19 S
19.55
D
double flat
--[2]
2048 : 2025 211 : 34×52 play Diaschisma,[3][6] minor comma 5
21.51
C+[2] 81 : 80 34 : 24×5 play Syntonic comma,[3][5][6] major comma, komma, chromatic diesis, or comma of Didymus[3][6][10][11] 5 S
22.64
21/53 21/53 play Holdrian comma, Holder's comma, 1 step in 53 equal temperament 53
23.46
B+++ 531441 : 524288 312 : 219 play Pythagorean comma,[3][5][6][10][11] ditonic comma[3][6] 3
25.00
21/48 21/48 play Eighth tone 48
26.84
C
13
65 : 64 5×13 : 26 play Sixty-fifth harmonic,[5] 13th-partial chroma[3] 13 S
27.26
C
7 upside-down
64 : 63 26 : 32×7 play Septimal comma,[3][6][11] Archytas' comma,[3] 63rd subharmonic 7 S
29.27
21/41 21/41 play 1 step in 41 equal temperament 41
31.19
D
7
56 : 55 23×7 : 5×11 play Undecimal diesis,[3] Ptolemy's enharmonic:[5] difference between (11 : 8) and (7 : 5) tritone 11 S
33.33
C
Half up arrow.png
/D
Half down arrow.png
Half down arrow.png
[a]
21/36 21/36 play Sixth tone 36, 72
34.28
C
17
51 : 50 3×17 : 2×52 play Difference between 17:16 & 25:24 17 S
34.98
B
7 upside-down
7 upside-down
-
50 : 49 2×52 : 72 play Septimal sixth tone or jubilisma, Erlich's decatonic comma or tritonic diesis[3][6] 7 S
35.70
D
7
7
49 : 48 72 : 24×3 play Septimal diesis, slendro diesis or septimal 1/6-tone[3] 7 S
38.05
C
23
46 : 45 2×23 : 32×5 play Inferior quarter tone,[5] difference between 23:16 & 45:32 23 S
38.71
21/31 21/31 play 1 step in 31 equal temperament 31
38.91
C+ 45 : 44 32×5 : 4×11 play Undecimal diesis or undecimal fifth tone 11 S
40.00
21/30 21/30 play Fifth tone 30
41.06
D
double flat
128 : 125 27 : 53 play Enharmonic diesis or 5-limit limma, minor diesis,[6] diminished second,[5][6] minor diesis or diesis,[3] 125th subharmonic 5
41.72
D
41U
7
42 : 41 2×3×7 : 41 play Lesser 41-limit fifth tone 41 S
42.75
C
41
41 : 40 41 : 23×5 play Greater 41-limit fifth tone 41 S
43.83
C
13 upside down
40 : 39 23×5 : 3×13 play Tridecimal fifth tone 13 S
44.97
C
19U
13
39 : 38 3×13 : 2×19 play Superior quarter-tone,[5] novendecimal fifth tone 19 S
46.17
D
37U
19
double flat
-
38 : 37 2×19 : 37 play Lesser 37-limit quarter tone 37 S
47.43
C
37
37 : 36 37 : 22×32 play Greater 37-limit quarter tone 37 S
48.77
C
7 upside-down
36 : 35 22×32 : 5×7 play Septimal quarter tone, septimal diesis,[3][6] septimal chroma,[2] superior quarter tone[5] 7 S
49.98
246 : 239 3×41 : 239 play Just quarter tone[11] 239
50.00
C
half sharp
/D
three quarter flat
21/24 21/24 play Equal-tempered quarter tone 24
50.18
D
17 upside down
7
35 : 34 5×7 : 2×17 play ET quarter-tone approximation,[5] lesser 17-limit quarter tone 17 S
50.72
B
7 upside-down
++
59049 : 57344 310 : 213×7 play Harrison's comma (10 P5s – 1 H7)[3] 7
51.68
C
17
34 : 33 2×17 : 3×11 play Greater 17-limit quarter tone 17 S
53.27
C 33 : 32 3×11 : 25 play Thirty-third harmonic,[5] undecimal comma, undecimal quarter tone 11 S
54.96
D
31U
-
32 : 31 25 : 31 play Inferior quarter-tone,[5] thirty-first subharmonic 31 S
56.55
B
23
23
+
529 : 512 232 : 29 play Five-hundred-twenty-ninth harmonic 23
56.77
C
31
31 : 30 31 : 2×3×5 play Greater quarter-tone,[5] difference between 31:16 & 15:8 31 S
58.69
C
29U
30 : 29 2×3×5 : 29 play Lesser 29-limit quarter tone 29 S
60.75
C
29
7 upside-down
29 : 28 29 : 22×7 play Greater 29-limit quarter tone 29 S
62.96
D
7
-
28 : 27 22×7 : 33 play Septimal minor second, small minor second, inferior quarter tone[5] 7 S
63.81
(3 : 2)1/11 31/11 : 21/11 play Beta scale step 18.75
65.34
C
13 upside down
+
27 : 26 33 : 2×13 play Chromatic diesis,[12] tridecimal comma[3] 13 S
66.34
D
19
7
133 : 128 7×19 : 27 play One-hundred-thirty-third harmonic 19
66.67
C
Sims flagged arrow down.svg
/C
Half down arrow.png
[a]
21/18 21/18 play Third tone 18, 36, 72
67.90
D
13
double flat
-
26 : 25 2×13 : 52 play Tridecimal third tone, third tone[5] 13 S
70.67
C[2] 25 : 24 52 : 23×3 play Just chromatic semitone or minor chroma,[3] lesser chromatic semitone, small (just) semitone[11] or minor second,[4] minor chromatic semitone,[13] or minor semitone,[5] 27-comma meantone chromatic semitone, augmented unison 5 S
73.68
D
23U
-
24 : 23 23×3 : 23 play Lesser 23-limit semitone 23 S
75.00
21/16 23/48 play 1 step in 16 equal temperament, 3 steps in 48 16, 48
76.96
C
23
+
23 : 22 23 : 2×11 play Greater 23-limit semitone 23 S
78.00
(3 : 2)1/9 31/9 : 21/9 play Alpha scale step 15.39
79.31
67 : 64 67 : 26 play Sixty-seventh harmonic[5] 67
80.54
C
7 upside-down
-
22 : 21 2×11 : 3×7 play Hard semitone,[5] two-fifth tone small semitone 11 S
84.47
D
7
21 : 20 3×7 : 22×5 play Septimal chromatic semitone, minor semitone[3] 7 S
88.80
C
19U
20 : 19 22×5 : 19 play Novendecimal augmented unison 19 S
90.22
D−−[2] 256 : 243 28 : 35 play Pythagorean minor second or limma,[3][6][11] Pythagorean diatonic semitone, Low Semitone[14] 3
92.18
C+[2] 135 : 128 33×5 : 27 play Greater chromatic semitone, chromatic semitone, semitone medius, major chroma or major limma,[3] small limma,[11] major chromatic semitone,[13] limma ascendant[5] 5
93.60
D
19
-
19 : 18 19 : 2×9 Novendecimal minor secondplay 19 S
97.36
D↓↓ 128 : 121 27 : 112 play 121st subharmonic,[5][6] undecimal minor second 11
98.95
D
17 upside down
18 : 17 2×32 : 17 play Just minor semitone, Arabic lute index finger[3] 17 S
100.00
C/D 21/12 21/12 play Equal-tempered minor second or semitone 12 M
104.96
C
17
[2]
17 : 16 17 : 24 play Minor diatonic semitone, just major semitone, overtone semitone,[5] 17th harmonic,[3] limma[citation needed] 17 S
111.45
255 (5 : 1)1/25 play Studie II interval (compound just major third, 5:1, divided into 25 equal parts) 25
111.73
D-[2] 16 : 15 24 : 3×5 play Just minor second,[15] just diatonic semitone, large just semitone or major second,[4] major semitone,[5] limma, minor diatonic semitone,[3] diatonic second[16] semitone,[14] diatonic semitone,[11] 16-comma meantone minor second 5 S
113.69
C++ 2187 : 2048 37 : 211 play Apotome[3][11] or Pythagorean major semitone,[6] Pythagorean augmented unison, Pythagorean chromatic semitone, or Pythagorean apotome 3
116.72
(18 : 5)1/19 21/19×32/19 : 51/19 play Secor 10.28
119.44
C
7 upside-down
15 : 14 3×5 : 2×7 play Septimal diatonic semitone, major diatonic semitone,[3] Cowell semitone[5] 7 S
125.00
25/48 25/48 play 5 steps in 48 equal temperament 48
128.30
D
13 upside down
7
14 : 13 2×7 : 13 play Lesser tridecimal 2/3-tone[17] 13 S
130.23
C
23
+
69 : 64 3×23 : 26 play Sixty-ninth harmonic[5] 23
133.24
D 27 : 25 33 : 52 play Semitone maximus, minor second, large limma or Bohlen-Pierce small semitone,[3] high semitone,[14] alternate Renaissance half-step,[5] large limma, acute minor second[citation needed] 5
133.33
C
Half up arrow.png
/D
Half up arrow.png
[a]
21/9 22/18 play Two-third tone 9, 18, 36, 72
138.57
D
13
-
13 : 12 13 : 22×3 play Greater tridecimal 2/3-tone,[17] Three-quarter tone[5] 13 S
150.00
C
three quarter sharp
/D
half flat
23/24 21/8 play Equal-tempered neutral second 8, 24
150.64
D↓[2] 12 : 11 22×3 : 11 play 34 tone or Undecimal neutral second,[3][5] trumpet three-quarter tone,[11] middle finger [between frets][14] 11 S
155.14
D
7
35 : 32 5×7 : 25 play Thirty-fifth harmonic[5] 7
160.90
D−− 800 : 729 25×52 : 36 play Grave whole tone,[3] neutral second, grave major second[citation needed] 5
165.00
D[2] 11 : 10 11 : 2×5 play Greater undecimal minor/major/neutral second, 4/5-tone[6] or Ptolemy's second[3] 11 S
171.43
21/7 21/7 play 1 step in 7 equal temperament 7
175.00
27/48 27/48 play 7 steps in 48 equal temperament 48
179.70
71 : 64 71 : 26 play Seventy-first harmonic[5] 71
180.45
E
double flat
−−−
65536 : 59049 216 : 310 play Pythagorean diminished third,[3][6] Pythagorean minor tone 3
182.40
D-[2] 10 : 9 2×5 : 32 play Small just whole tone or major second,[4] minor whole tone,[3][5] lesser whole tone,[16] minor tone,[14] minor second,[11] half-comma meantone major second 5 S
200.00
D 22/12 21/6 play Equal-tempered major second 6, 12 M
203.91
D[2] 9 : 8 32 : 23 play Pythagorean major second, Large just whole tone or major second[11] (sesquioctavan),[4] tonus, major whole tone,[3][5] greater whole tone,[16] major tone[14] 3 S
215.89
D
29
145 : 128 5×29 : 27 play Hundred-forty-fifth harmonic 29
223.46
E
double flat
[2]
256 : 225 28 : 32×52 play Just diminished third,[16] 225th subharmonic 5
225.00
23/16 29/48 play 9 steps in 48 equal temperament 16, 48
227.79
73 : 64 73 : 26 play Seventy-third harmonic[5] 73
231.17
D
7 upside-down
[2]
8 : 7 23 : 7 play Septimal major second,[4] septimal whole tone[3][5] 7 S
240.00
21/5 21/5 play 1 step in 5 equal temperament 5
247.74
D
13 upside down
15 : 13 3×5 : 13 play Tridecimal 54 tone[3] 13
250.00
D
half sharp
/E
three quarter flat
25/24 25/24 play 5 steps in 24 equal temperament 24
251.34
D
37
37 : 32 37 : 25 play Thirty-seventh harmonic[5] 37
253.08
D 125 : 108 53 : 22×33 play Semi-augmented whole tone,[3] semi-augmented second[citation needed] 5
262.37
E↓ 64 : 55 26 : 5×11 play 55th subharmonic[5][6] 11
268.80
D
23
13
299 : 256 13×23 : 28 play Two-hundred-ninety-ninth harmonic 23
266.87
E
7
[2]
7 : 6 7 : 2×3 play Septimal minor third[3][4][11] or Sub minor third[14] 7 S
274.58
D[2] 75 : 64 3×52 : 26 play Just augmented second,[16] Augmented tone,[14] augmented second[5][13] 5
275.00
211/48 211/48 play 11 steps in 48 equal temperament 48
289.21
E
13
13 : 11 13 : 11 play Tridecimal minor third[3] 13
294.13
E[2] 32 : 27 25 : 33 play Pythagorean minor third[3][5][6][14][16] semiditone, or 27th subharmonic 3
297.51
E
19
[2]
19 : 16 19 : 24 play 19th harmonic,[3] 19-limit minor third, overtone minor third[5] 19
300.00
D/E 23/12 21/4 play Equal-tempered minor third 4, 12 M
301.85
D
7 upside-down
-
25 : 21[5] 52 : 3×7 play Quasi-equal-tempered minor third, 2nd 7-limit minor third, Bohlen-Pierce second[3][6] 7
310.26
6:5÷(81:80)1/4 22 : 53/4 play Quarter-comma meantone minor third M
311.98
(3 : 2)4/9 34/9 : 24/9 play Alpha scale minor third 3.85
315.64
E[2] 6 : 5 2×3 : 5 play Just minor third,[3][4][5][11][16] minor third,[14] 13-comma meantone minor third 5 M S
317.60
D++ 19683 : 16384 39 : 214 play Pythagorean augmented second[3][6] 3
320.14
E
7
77 : 64 7×11 : 26 play Seventy-seventh harmonic[5] 11
325.00
213/48 213/48 play 13 steps in 48 equal temperament 48
336.13
D
17
7 upside-down
-
17 : 14 17 : 2×7 play Superminor third[18] 17
337.15
E+ 243 : 200 35 : 23×52 play Acute minor third[3] 5
342.48
E
13
39 : 32 3×13 : 25 play Thirty-ninth harmonic[5] 13
342.86
22/7 22/7 play 2 steps in 7 equal temperament 7
342.91
E
7 upside-down
-
128 : 105 27 : 3×5×7 play 105th subharmonic,[5] septimal neutral third[6] 7
347.41
E[2] 11 : 9 11 : 32 play Undecimal neutral third[3][5] 11
350.00
D
three quarter sharp
/E
half flat
27/24 27/24 play Equal-tempered neutral third 24
354.55
E+ 27 : 22 33 : 2×11 play Zalzal's wosta[6] 12:11 X 9:8[14] 11
359.47
E
13 upside down
[2]
16 : 13 24 : 13 play Tridecimal neutral third[3] 13
364.54
79 : 64 79 : 26 play Seventy-ninth harmonic[5] 79
364.81
E− 100 : 81 22×52 : 34 play Grave major third[3] 5
375.00
25/16 215/48 play 15 steps in 48 equal temperament 16, 48
384.36
F−− 8192 : 6561 213 : 38 play Pythagorean diminished fourth,[3][6] Pythagorean 'schismatic' third[5] 3
386.31
E[2] 5 : 4 5 : 22 play Just major third,[3][4][5][11][16] major third,[14] quarter-comma meantone major third 5 M S
397.10
E
23
7
+
161 : 128 7×23 : 27 play One-hundred-sixty-first harmonic 23
400.00
E 24/12 21/3 play Equal-tempered major third 3, 12 M
402.47
E
19
17
323 : 256 17×19 : 28 play Three-hundred-twenty-third harmonic 19
407.82
E+[2] 81 : 64 34 : 26 play Pythagorean major third,[3][5][6][14][16] ditone 3
417.51
F
7
+[2]
14 : 11 2×7 : 11 play Undecimal diminished fourth or major third[3] 11
425.00
217/48 217/48 play 17 steps in 48 equal temperament 48
427.37
F[2] 32 : 25 25 : 52 play Just diminished fourth,[16] diminished fourth,[5][13] 25th subharmonic 5
429.06
E
41
41 : 32 41 : 25 play Forty-first harmonic[5] 41
435.08
E
7 upside-down
[2]
9 : 7 32 : 7 play Septimal major third,[3][5] Bohlen-Pierce third,[3] Super major Third[14] 7
444.77
F↓ 128 : 99 27 : 9×11 play 99th subharmonic[5][6] 11
450.00
E
half sharp
/F
half flat
29/24 29/24 play 9 steps in 24 equal temperament 24
450.05
83 : 64 83 : 26 play Eighty-third harmonic[5] 83
454.21
F
13
13 : 10 13 : 2×5 play Tridecimal major third or diminished fourth 13
456.99
E[2] 125 : 96 53 : 25×3 play Just augmented third, augmented third[5] 5
462.35
E
7 upside-down
7 upside-down
-
64 : 49 26 : 72 play 49th subharmonic[5][6] 7
470.78
F
7
+[2]
21 : 16 3×7 : 24 play Twenty-first harmonic, narrow fourth,[3] septimal fourth,[5] wide augmented third,[citation needed] H7 on G 7
475.00
219/48 219/48 play 19 steps in 48 equal temperament 48
478.49
E+ 675 : 512 33×52 : 29 play Six-hundred-seventy-fifth harmonic, wide augmented third[3] 5
480.00
22/5 22/5 play 2 steps in 5 equal temperament 5
491.27
E
17
85 : 64 5×17 : 26 play Eighty-fifth harmonic[5] 17
498.04
F[2] 4 : 3 22 : 3 play Perfect fourth,[3][5][16] Pythagorean perfect fourth, Just perfect fourth or diatessaron[4] 3 S
500.00
F 25/12 25/12 play Equal-tempered perfect fourth 12 M
501.42
F
19
+
171 : 128 32×19 : 27 play One-hundred-seventy-first harmonic 19
510.51
(3 : 2)8/11 38/11 : 28/11 play Beta scale perfect fourth 18.75
511.52
F
43
43 : 32 43 : 25 play Forty-third harmonic[5] 43
514.29
23/7 23/7 play 3 steps in 7 equal temperament 7
519.55
F+[2] 27 : 20 33 : 22×5 play 5-limit wolf fourth, acute fourth,[3] imperfect fourth[16] 5
521.51
E+++ 177147 : 131072 311 : 217 play Pythagorean augmented third[3][6] (F+ (pitch)) 3
525.00
27/16 221/48 play 21 steps in 48 equal temperament 16, 48
531.53
F
29
+
87 : 64 3×29 : 26 play Eighty-seventh harmonic[5] 29
536.95
F+ 15 : 11 3×5 : 11 play Undecimal augmented fourth[3] 11
550.00
F
half sharp
/G
three quarter flat
211/24 211/24 play 11 steps in 24 equal temperament 24
551.32
F[2] 11 : 8 11 : 23 play eleventh harmonic,[5] undecimal tritone,[5] lesser undecimal tritone, undecimal semi-augmented fourth[3] 11
563.38
F
13 upside down
+
18 : 13 2×9 : 13 play Tridecimal augmented fourth[3] 13
568.72
F[2] 25 : 18 52 : 2×32 play Just augmented fourth[3][5] 5
570.88
89 : 64 89 : 26 play Eighty-ninth harmonic[5] 89
575.00
223/48 223/48 play 23 steps in 48 equal temperament 48
582.51
G
7
[2]
7 : 5 7 : 5 play Lesser septimal tritone, septimal tritone[3][4][5] Huygens' tritone or Bohlen-Pierce fourth,[3] septimal fifth,[11] septimal diminished fifth[19] 7
588.27
G−− 1024 : 729 210 : 36 play Pythagorean diminished fifth,[3][6] low Pythagorean tritone[5] 3
590.22
F+[2] 45 : 32 32×5 : 25 play Just augmented fourth, just tritone,[4][11] tritone,[6] diatonic tritone,[3] 'augmented' or 'false' fourth,[16] high 5-limit tritone,[5] 16-comma meantone augmented fourth 5
595.03
G
19
19
361 : 256 192 : 28 play Three-hundred-sixty-first harmonic 19
600.00
F/G 26/12 21/2=2 play Equal-tempered tritone 2, 12 M
609.35
G
13
7
91 : 64 7×13 : 26 play Ninety-first harmonic[5] 13
609.78
G[2] 64 : 45 26 : 32×5 play Just tritone,[4] 2nd tritone,[6] 'false' fifth,[16] diminished fifth,[13] low 5-limit tritone,[5] 45th subharmonic 5
611.73
F++ 729 : 512 36 : 29 play Pythagorean tritone,[3][6] Pythagorean augmented fourth, high Pythagorean tritone[5] 3
617.49
F
7 upside-down
[2]
10 : 7 2×5 : 7 play Greater septimal tritone, septimal tritone,[4][5] Euler's tritone[3] 7
625.00
225/48 225/48 play 25 steps in 48 equal temperament 48
628.27
F
23
+
23 : 16 23 : 24 play Twenty-third harmonic,[5] classic diminished fifth[citation needed] 23
631.28
G[2] 36 : 25 22×32 : 52 play Just diminished fifth[5] 5
646.99
F
31
+
93 : 64 3×31 : 26 play Ninety-third harmonic[5] 31
648.68
G↓[2] 16 : 11 24 : 11 play ` undecimal semi-diminished fifth[3] 11
650.00
F
three quarter sharp
/G
half flat
213/24 213/24 play 13 steps in 24 equal temperament 24
665.51
G
43U
47 : 32 47 : 25 play Forty-seventh harmonic[5] 47
675.00
29/16 227/48 play 27 steps in 48 equal temperament 16, 48
678.49
A
double flat
−−−
262144 : 177147 218 : 311 play Pythagorean diminished sixth[3][6] 3
680.45
G− 40 : 27 23×5 : 33 play 5-limit wolf fifth,[5] or diminished sixth, grave fifth,[3][6][11] imperfect fifth,[16] 5
683.83
G
19
95 : 64 5×19 : 26 play Ninety-fifth harmonic[5] 19
684.82
E
23
23
23
double sharp
++
12167 : 8192 233 : 213 play 12167th harmonic 23
685.71
24/7 : 1 play 4 steps in 7 equal temperament
691.20
3:2÷(81:80)1/2 2×51/2 : 3 play Half-comma meantone perfect fifth M
694.79
3:2÷(81:80)1/3 21/3×51/3 : 31/3 play 13-comma meantone perfect fifth M
695.81
3:2÷(81:80)2/7 21/7×52/7 : 31/7 play 27-comma meantone perfect fifth M
696.58
3:2÷(81:80)1/4 51/4 play Quarter-comma meantone perfect fifth M
697.65
3:2÷(81:80)1/5 31/5×51/5 : 21/5 play 15-comma meantone perfect fifth M
698.37
3:2÷(81:80)1/6 31/3×51/6 : 21/3 play 16-comma meantone perfect fifth M
700.00
G 27/12 27/12 play Equal-tempered perfect fifth 12 M
701.89
231/53 231/53 play 53-TET perfect fifth 53
701.96
G[2] 3 : 2 3 : 2 play Perfect fifth,[3][5][16] Pythagorean perfect fifth, Just perfect fifth or diapente,[4] fifth,[14] Just fifth[11] 3 S
702.44
224/41 224/41 play 41-TET perfect fifth 41
703.45
217/29 217/29 play 29-TET perfect fifth 29
719.90
97 : 64 97 : 26 play Ninety-seventh harmonic[5] 97
720.00
23/5 : 1 play 3 steps in 5 equal temperament 5
721.51
A
double flat
1024 : 675 210 : 33×52 play Narrow diminished sixth[3] 5
725.00
229/48 229/48 play 29 steps in 48 equal temperament 48
729.22
G
7 upside-down
-
32 : 21 24 : 3×7 play 21st subharmonic,[5][6] septimal diminished sixth 7
733.23
F
23
17
double sharp
+
391 : 256 17×23 : 28 play Three-hundred-ninety-first harmonic 23
737.65
A
7
7
+
49 : 32 7×7 : 25 play Forty-ninth harmonic[5] 7
743.01
A
double flat
192 : 125 26×3 : 53 play Classic diminished sixth[3] 5
750.00
G
half sharp
/A
three quarter flat
215/24 215/24 play 15 steps in 24 equal temperament 24
755.23
G 99 : 64 32×11 : 26 play Ninety-ninth harmonic[5] 11
764.92
A
7
[2]
14 : 9 2×7 : 32 play Septimal minor sixth[3][5] 7
772.63
G 25 : 16 52 : 24 play Just augmented fifth[5][16] 5
775.00
231/48 231/48 play 31 steps in 48 equal temperament 48
781.79
π : 2 play Wallis product
782.49
G
7 upside-down
-[2]
11 : 7 11 : 7 play Undecimal minor sixth,[5] undecimal augmented fifth,[3] Fibonacci numbers 11
789.85
101 : 64 101 : 26 play Hundred-first harmonic[5] 101
792.18
A[2] 128 : 81 27 : 34 play Pythagorean minor sixth,[3][5][6] 81st subharmonic 3
798.40
A
29
7
+
203 : 128 7×29 : 27 play Two-hundred-third harmonic 29
800.00
G/A 28/12 22/3 play Equal-tempered minor sixth 3, 12 M
806.91
G
17
51 : 32 3×17 : 25 play Fifty-first harmonic[5] 17
813.69
A[2] 8 : 5 23 : 5 play Just minor sixth[3][4][11][16] 5
815.64
G++ 6561 : 4096 38 : 212 play Pythagorean augmented fifth,[3][6] Pythagorean 'schismatic' sixth[5] 3
823.80
103 : 64 103 : 26 play Hundred-third harmonic[5] 103
825.00
211/16 233/48 play 33 steps in 48 equal temperament 16, 48
832.18
G
23
+
207 : 128 32×23 : 27 play Two-hundred-seventh harmonic 23
833.09
(51/2+1)/2 φ : 1 play Golden ratio (833 cents scale)
835.19
A+ 81 : 50 34 : 2×52 play Acute minor sixth[3] 5
840.53
A
13
[2]
13 : 8 13 : 23 play Tridecimal neutral sixth,[3] overtone sixth,[5] thirteenth harmonic 13
848.83
A
19
209 : 128 11×19 : 27 play Two-hundred-ninth harmonic 19
850.00
G
three quarter sharp
/A
half flat
217/24 217/24 play Equal-tempered neutral sixth 24
852.59
A↓+[2] 18 : 11 2×32 : 11 play Undecimal neutral sixth,[3][5] Zalzal's neutral sixth 11
857.09
A
7
+
105 : 64 3×5×7 : 26 play Hundred-fifth harmonic[5] 7
857.14
25/7 25/7 play 5 steps in 7 equal temperament 7
862.85
A− 400 : 243 24×52 : 35 play Grave major sixth[3] 5
873.50
A
43U
53 : 32 53 : 25 play Fifty-third harmonic[5] 53
875.00
235/48 235/48 play 35 steps in 48 equal temperament 48
879.86
A↓
7 upside-down
128 : 77 27 : 7×11 play 77th subharmonic[5][6] 11
882.40
B
double flat
−−−
32768 : 19683 215 : 39 play Pythagorean diminished seventh[3][6] 3
884.36
A[2] 5 : 3 5 : 3 play Just major sixth,[3][4][5][11][16] Bohlen-Pierce sixth,[3] 13-comma meantone major sixth 5 M
889.76
107 : 64 107 : 26 play Hundred-seventh harmonic[5] 107
892.54
B
19
19
19
double flat
6859 : 4096 193 : 212 play 6859th harmonic 19
900.00
A 29/12 23/4 play Equal-tempered major sixth 4, 12 M
902.49
A
19U
32 : 19 25 : 19 play 19th subharmonic[5][6] 19
905.87
A+[2] 27 : 16 33 : 24 play Pythagorean major sixth[3][5][11][16] 3
921.82
109 : 64 109 : 26 play Hundred-ninth harmonic[5] 109
925.00
237/48 237/48 play 37 steps in 48 equal temperament 48
925.42
B
double flat
[2]
128 : 75 27 : 3×52 play Just diminished seventh,[16] diminished seventh,[5][13] 75th subharmonic 5
925.79
A
23
19
+
437 : 256 19×23 : 28 play Four-hundred-thirty-seventh harmonic 23
933.13
A
7 upside-down
[2]
12 : 7 22×3 : 7 play Septimal major sixth[3][4][5] 7
937.63
A 55 : 32 5×11 : 25 play Fifty-fifth harmonic[5][20] 11
950.00
A
half sharp
/B
three quarter flat
219/24 219/24 play 19 steps in 24 equal temperament 24
953.30
A
37
+
111 : 64 3×37 : 26 play Hundred-eleventh harmonic[5] 37
955.03
A[2] 125 : 72 53 : 23×32 play Just augmented sixth[5] 5
957.21
(3 : 2)15/11 315/11 : 215/11 play 15 steps in Beta scale 18.75
960.00
24/5 24/5 play 4 steps in 5 equal temperament 5
968.83
B
7
[2]
7 : 4 7 : 22 play Septimal minor seventh,[4][5][11] harmonic seventh,[3][11] augmented sixth[citation needed] 7
975.00
213/16 239/48 play 39 steps in 48 equal temperament 16, 48
976.54
A+[2] 225 : 128 32×52 : 27 play Just augmented sixth[16] 5
984.21
113 : 64 113 : 26 play Hundred-thirteenth harmonic[5] 113
996.09
B[2] 16 : 9 24 : 32 play Pythagorean minor seventh,[3] Small just minor seventh,[4] lesser minor seventh,[16] just minor seventh,[11] Pythagorean small minor seventh[5] 3
999.47
B
19
57 : 32 3×19 : 25 play Fifty-seventh harmonic[5] 19
1000.00
A/B 210/12 25/6 play Equal-tempered minor seventh 6, 12 M
1014.59
A
23
+
115 : 64 5×23 : 26 play Hundred-fifteenth harmonic[5] 23
1017.60
B[2] 9 : 5 32 : 5 play Greater just minor seventh,[16] large just minor seventh,[4][5] Bohlen-Pierce seventh[3] 5
1019.55
A+++ 59049 : 32768 310 : 215 play Pythagorean augmented sixth[3][6] 3
1025.00
241/48 241/48 play 41 steps in 48 equal temperament 48
1028.57
26/7 26/7 play 6 steps in 7 equal temperament 7
1029.58
B
29
29 : 16 29 : 24 play Twenty-ninth harmonic,[5] minor seventh[citation needed] 29
1035.00
B↓[2] 20 : 11 22×5 : 11 play Lesser undecimal neutral seventh, large minor seventh[3] 11
1039.10
B+ 729 : 400 36 : 24×52 play Acute minor seventh[3] 5
1044.44
B
13
117 : 64 32×13 : 26 play Hundred-seventeenth harmonic[5] 13
1044.86
B
7 upside-down
-
64 : 35 26 : 5×7 play 35th subharmonic,[5] septimal neutral seventh[6] 7
1049.36
B[2] 11 : 6 11 : 2×3 play 214-tone or Undecimal neutral seventh,[3] undecimal 'median' seventh[5] 11
1050.00
A
three quarter sharp
/B
half flat
221/24 27/8 play Equal-tempered neutral seventh 8, 24
1059.17
59 : 32 59 : 25 play Fifty-ninth harmonic[5] 59
1066.76
B− 50 : 27 2×52 : 33 play Grave major seventh[3] 5
1071.70
B
13
7 upside-down
-
13 : 7 13 : 7 play Tridecimal neutral seventh[21] 13
1073.78
B
7
17
119 : 64 7×17 : 26 play Hundred-nineteenth harmonic[5] 17
1075.00
243/48 243/48 play 43 steps in 48 equal temperament 48
1086.31
C′−− 4096 : 2187 212 : 37 play Pythagorean diminished octave[3][6] 3
1088.27
B[2] 15 : 8 3×5 : 23 play Just major seventh,[3][5][11][16] small just major seventh,[4] 16-comma meantone major seventh 5
1095.04
C
17 upside down
32 : 17 25 : 17 play 17th subharmonic[5][6] 17
1100.00
B 211/12 211/12 play Equal-tempered major seventh 12 M
1102.64
B- 121 : 64 112 : 26 play Hundred-twenty-first harmonic[5] 11
1107.82
C′ 256 : 135 28 : 33×5 play Octave − major chroma,[3] 135th subharmonic, narrow diminished octave[citation needed] 5
1109.78
B+[2] 243 : 128 35 : 27 play Pythagorean major seventh[3][5][6][11] 3
1116.88
61 : 32 61 : 25 play Sixty-first harmonic[5] 61
1125.00
215/16 245/48 play 45 steps in 48 equal temperament 16, 48
1129.33
C′[2] 48 : 25 24×3 : 52 play Classic diminished octave,[3][6] large just major seventh[4] 5
1131.02
B
41
123 : 64 3×41 : 26 play Hundred-twenty-third harmonic[5] 41
1137.04
B
7 upside-down
27 : 14 33 : 2×7 play Septimal major seventh[5] 7
1138.04
C
19
13
247 : 128 13×19 : 27 play Two-hundred-forty-seventh harmonic 19
1145.04
B
31
31 : 16 31 : 24 play Thirty-first harmonic,[5] augmented seventh[citation needed] 31
1146.73
C↓ 64 : 33 26 : 3×11 play 33rd subharmonic[6] 11
1150.00
B
half sharp
/C
half flat
223/24 223/24 play 23 steps in 24 equal temperament 24
1151.23
C
7
35 : 18 5×7 : 2×32 play Septimal supermajor seventh, septimal quarter tone inverted 7
1158.94
B[2] 125 : 64 53 : 26 play Just augmented seventh,[5] 125th harmonic 5
1172.74
C
7
+
63 : 32 32×7 : 25 play Sixty-third harmonic[5] 7
1175.00
247/48 247/48 play 47 steps in 48 equal temperament 48
1178.49
C′− 160 : 81 25×5 : 34 play Octave − syntonic comma,[3] semi-diminished octave[citation needed] 5
1179.59
B
23
253 : 128 11×23 : 27 play Two-hundred-fifty-third harmonic[5] 23
1186.42
127 : 64 127 : 26 play Hundred-twenty-seventh harmonic[5] 127
1200.00
C′ 2 : 1 2 : 1 play Octave[3][11] or diapason[4] 1, 12 3 M S

See also

Notes

References

  1. ^ a b Fox, Christopher (2003). "Microtones and Microtonalities", Contemporary Music Review, v. 22, pt. 1–2. (Abingdon, Oxfordshire, UK: Routledge): p. 13.
  2. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj ak al am an ao ap aq ar as at au av aw ax ay az ba bb bc bd be bf bg bh bi Fonville, John. 1991. "Ben Johnston's Extended Just Intonation: A Guide for Interpreters". Perspectives of New Music 29, no. 2 (Summer): 106–137.
  3. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj ak al am an ao ap aq ar as at au av aw ax ay az ba bb bc bd be bf bg bh bi bj bk bl bm bn bo bp bq br bs bt bu bv bw bx by bz ca cb cc cd ce cf cg ch ci cj ck cl cm cn co cp cq cr cs ct cu cv cw cx cy cz da db dc dd de df dg dh di "List of intervals", Huygens-Fokker Foundation. The Foundation uses "classic" to indicate "just" or leaves off any adjective, as in "major sixth".
  4. ^ a b c d e f g h i j k l m n o p q r s t u v w x Partch, Harry (1979). Genesis of a Music. pp. 68–69. ISBN 978-0-306-80106-8.
  5. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj ak al am an ao ap aq ar as at au av aw ax ay az ba bb bc bd be bf bg bh bi bj bk bl bm bn bo bp bq br bs bt bu bv bw bx by bz ca cb cc cd ce cf cg ch ci cj ck cl cm cn co cp cq cr cs ct cu cv cw cx cy cz da db dc dd de df dg dh di dj dk dl dm dn do dp dq dr ds dt du dv dw dx dy dz "Anatomy of an Octave", Kyle Gann (1998). Gann leaves off "just" but includes "5-limit". He uses "median" for "neutral".
  6. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag ah ai aj ak al am an ao ap aq ar as at au av aw ax ay Haluška, Ján (2003). The Mathematical Theory of Tone Systems, pp. xxv–xxix. ISBN 978-0-8247-4714-5.
  7. ^ Ellis, Alexander J.; Hipkins, Alfred J. (1884). "Tonometrical Observations on Some Existing Non-Harmonic Musical Scales". Proceedings of the Royal Society of London. 37 (232–234): 368–385. doi:10.1098/rspl.1884.0041. JSTOR 114325. S2CID 122407786.
  8. ^ "Logarithmic Interval Measures", Huygens-Fokker Foundation. Accessed 2015-06-06.
  9. ^ "Orwell Temperaments", Xenharmony.org.
  10. ^ a b Partch 1979, p. 70
  11. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab Alexander John Ellis (March 1885). On the musical scales of various nations, p. 488. Journal of the Society of Arts, vol. XXXII, no. 1688
  12. ^ William Smythe Babcock Mathews (1895). Pronouncing Dictionary and Condensed Encyclopedia of Musical Terms, p. 13. ISBN 1-112-44188-3.
  13. ^ a b c d e f Anger, Joseph Humfrey (1912). A Treatise on Harmony, with Exercises, Volume 3, pp. xiv–xv. W. Tyrrell.
  14. ^ a b c d e f g h i j k l m n o 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. [ISBN unspecified]
  15. ^ A. R. Meuss (2004). Intervals, Scales, Tones and the Concert Pitch C. Temple Lodge Publishing. p. 15. ISBN 1902636465.
  16. ^ a b c d e f g h i j k l m n o p q r s t u v w x y 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".
  17. ^ a b "13th-harmonic", 31et.com.
  18. ^ Brabner, John H. F. (1884). The National Encyclopaedia, vol. 13, p. 182. London. [ISBN unspecified]
  19. ^ Sabat, Marc and von Schweinitz, Wolfgang (2004). "The Extended Helmholtz-Ellis JI Pitch Notation" [PDF], NewMusicBox. Accessed: 15 March 2014.
  20. ^ Hermann L. F. von Helmholtz (2007). On the Sensations of Tone, p. 456. ISBN 978-1-60206-639-7.
  21. ^ "Gallery of Just Intervals", Xenharmonic Wiki.