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List of pitch intervals

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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. 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 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

[edit]
  • The prime limit[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 factored into 2 × 5 (and 9 into 3 × 3). There exists another type of limit, the odd limit, a concept used by Harry Partch (bigger of odd numbers obtained after dividing numerator and denominator by highest possible powers of 2), but it is not used here. The term "limit" was devised by Partch.[1]
  • 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.
  • Superparticular ratios are intervals that can be expressed as the ratio of two consecutive integers.

List

[edit]
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
 C7♯− 4375 : 4374 54×7 : 2×37 playⓘ Ragisma[3][6] 7 S
0.72
 E7777triple 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
B7 upside-down♯ 225 : 224 32×52 : 25×7 playⓘ Septimal kleisma,[3][6] marvel comma 7 S
8.11
Bdouble sharp− 15625 : 15552 56 : 26×35 playⓘ Kleisma or semicomma majeur[3][6] 5
10.06
Adouble sharpdouble sharp++ 2109375 : 2097152 33×57 : 221 playⓘ Semicomma,[3][6] Fokker's comma[3] 5
10.85
C43U 160 : 159 25×5 : 3×53 playⓘ Difference between 5:3 & 53:32 53 S
11.98
 C29 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
 B7 upside-down7 upside-down7 upside-down− 1728 : 1715 26×33 : 5×73 playⓘ Orwell comma[3][9] 7
13.47
 C43 129 : 128 3×43 : 27 playⓘ Hundred-twenty-ninth harmonic 43 S
13.79
Ddouble flat7 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
C19U 96 : 95 25×3 : 5×19 playⓘ Difference between 19:16 & 6:5 19 S
19.55
Ddouble 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
C13 65 : 64 5×13 : 26 playⓘ Sixty-fifth harmonic,[5] 13th-partial chroma[3] 13 S
27.26
C7 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
D7♭↓ 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/D♭[a] 21/36 21/36 playⓘ Sixth tone 36, 72
34.28
C17 51 : 50 3×17 : 2×52 playⓘ Difference between 17:16 & 25:24 17 S
34.98
B7 upside-down7 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
D77♭ 49 : 48 72 : 24×3 playⓘ Septimal diesis, slendro diesis or septimal 1/6-tone[3] 7 S
38.05
C23 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 or Normal Diesis 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
Ddouble 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
D41U7♭ 42 : 41 2×3×7 : 41 playⓘ Lesser 41-limit fifth tone 41 S
42.75
C41 41 : 40 41 : 23×5 playⓘ Greater 41-limit fifth tone 41 S
43.83
C13 upside down♯ 40 : 39 23×5 : 3×13 playⓘ Tridecimal fifth tone 13 S
44.97
C19U13 39 : 38 3×13 : 2×19 playⓘ Superior quarter-tone,[5] novendecimal fifth tone 19 S
46.17
D37U19double flat- 38 : 37 2×19 : 37 playⓘ Lesser 37-limit quarter tone 37 S
47.43
C37♯ 37 : 36 37 : 22×32 playⓘ Greater 37-limit quarter tone 37 S
48.77
C7 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
Chalf sharp/Dthree quarter flat 21/24 21/24 playⓘ Equal-tempered quarter tone 24
50.18
D17 upside down7♭ 35 : 34 5×7 : 2×17 playⓘ ET quarter-tone approximation,[5] lesser 17-limit quarter tone 17 S
50.72
B7 upside-down♯++ 59049 : 57344 310 : 213×7 playⓘ Harrison's comma (10 P5s – 1 H7)[3] 7
51.68
C17↓♯ 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
D31U♭- 32 : 31 25 : 31 playⓘ Inferior quarter-tone,[5] thirty-first subharmonic 31 S
56.55
B2323♯+ 529 : 512 232 : 29 playⓘ Five-hundred-twenty-ninth harmonic 23
56.77
C31 31 : 30 31 : 2×3×5 playⓘ Greater quarter-tone,[5] difference between 31:16 & 15:8 31 S
58.69
C29U♯ 30 : 29 2×3×5 : 29 playⓘ Lesser 29-limit quarter tone 29 S
60.75
C297 upside-down 29 : 28 29 : 22×7 playⓘ Greater 29-limit quarter tone 29 S
62.96
D7♭- 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 18.80
65.34
C13 upside down♯+ 27 : 26 33 : 2×13 playⓘ Chromatic diesis,[12] tridecimal comma[3] 13 S
66.34
D197♭ 133 : 128 7×19 : 27 playⓘ One-hundred-thirty-third harmonic 19
66.67
C↑/C♯[a] 21/18 21/18 playⓘ Third tone 18, 36, 72
67.90
D13double 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] 2⁄7-comma meantone chromatic semitone, augmented unison 5 S
73.68
D23U♭- 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
C23↓♯+ 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
D7♭ 21 : 20 3×7 : 22×5 playⓘ Septimal chromatic semitone, minor semitone[3] 7 S
88.80
C19U♯ 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
D19♭- 19 : 18 19 : 2×32 Novendecimal minor secondplayⓘ 19 S
97.36
D↓↓ 128 : 121 27 : 112 playⓘ 121st subharmonic,[5][6] undecimal minor second 11
98.95
D17 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
C17♯[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
25√5 (5 : 1)1/25 playⓘ Studie II interval (compound just major third, 5:1, divided into 25 equal parts) 10.77
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] 1⁄6-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
C7 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
D13 upside down7 14 : 13 2×7 : 13 playⓘ Lesser tridecimal 2/3-tone[17] 13 S
130.23
C23♯+ 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♯/D♭[a] 21/9 22/18 playⓘ Two-third tone 9, 18, 36, 72
138.57
D13♭- 13 : 12 13 : 22×3 playⓘ Greater tridecimal 2/3-tone,[17] Three-quarter tone[5] 13 S
150.00
Cthree quarter sharp/Dhalf flat 23/24 21/8 playⓘ Equal-tempered neutral second 8, 24
150.64
D↓[2] 12 : 11 22×3 : 11 playⓘ 3⁄4 tone or Undecimal neutral second,[3][5] trumpet three-quarter tone,[11] middle finger [between frets][14] 11 S
155.14
D7 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
Edouble 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
D29 145 : 128 5×29 : 27 playⓘ Hundred-forty-fifth harmonic 29
223.46
Edouble 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
D7 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
D13 upside down♯ 15 : 13 3×5 : 13 playⓘ Tridecimal 5⁄4 tone[3] 13
250.00
Dhalf sharp/Ethree quarter flat 25/24 25/24 playⓘ 5 steps in 24 equal temperament 24
251.34
D37♯ 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
266.87
E7♭[2] 7 : 6 7 : 2×3 playⓘ Septimal minor third[3][4][11] or Sub minor third[14] 7 S
268.80
D2313 299 : 256 13×23 : 28 playⓘ Two-hundred-ninety-ninth harmonic 23
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
E13↓♭ 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
E19♭[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
D7 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 15.39
315.64
E♭[2] 6 : 5 2×3 : 5 playⓘ Just minor third,[3][4][5][11][16] minor third,[14] 1⁄3-comma meantone minor third 5 M S
317.60
D♯++ 19683 : 16384 39 : 214 playⓘ Pythagorean augmented second[3][6] 3
320.14
E7♭↑ 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
D177 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
E13♭ 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
E7 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
Dthree quarter sharp/Ehalf 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
E13 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
E237+ 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
E1917 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
F7↓+[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
E41 41 : 32 41 : 25 playⓘ Forty-first harmonic[5] 41
435.08
E7 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 : 32×11 playⓘ 99th subharmonic[5][6] 11
450.00
Ehalf sharp/Fhalf flat 29/24 29/24 playⓘ 9 steps in 24 equal temperament 8, 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
E7 upside-down7 upside-down- 64 : 49 26 : 72 playⓘ 49th subharmonic[5][6] 7
470.78
F7+[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
E17♯ 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
F19+ 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 18.80
511.52
F43 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
F29+ 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
Fhalf sharp/Gthree 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
F13 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
G7♭[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] 1⁄6-comma meantone augmented fourth 5
595.03
G1919♭ 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
G137♭ 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
F23♯+ 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
F31♯+ 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
Fthree quarter sharp/Ghalf flat 213/24 213/24 playⓘ 13 steps in 24 equal temperament 24
665.51
G43U 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
Adouble 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
G19 95 : 64 5×19 : 26 playⓘ Ninety-fifth harmonic[5] 19
684.82
E232323double sharp++ 12167 : 8192 233 : 213 playⓘ 12167th harmonic 23
685.71
24/7 : 1 playⓘ 4 steps in 7 equal temperament 7
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ⓘ 1⁄3-comma meantone perfect fifth M
695.81
3:2÷(81:80)2/7 21/7×52/7 : 31/7 playⓘ 2⁄7-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ⓘ 1⁄5-comma meantone perfect fifth M
698.37
3:2÷(81:80)1/6 31/3×51/6 : 21/3 playⓘ 1⁄6-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
Adouble 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
G7 upside-down- 32 : 21 24 : 3×7 playⓘ 21st subharmonic,[5][6] septimal diminished sixth 7
733.23
F2317double sharp+ 391 : 256 17×23 : 28 playⓘ Three-hundred-ninety-first harmonic 23
737.65
A77♭+ 49 : 32 7×7 : 25 playⓘ Forty-ninth harmonic[5] 7
743.01
Adouble flat 192 : 125 26×3 : 53 playⓘ Classic diminished sixth[3] 5
750.00
Ghalf sharp/Athree quarter flat 215/24 215/24 playⓘ 15 steps in 24 equal temperament 8, 24
755.23
G↑ 99 : 64 32×11 : 26 playⓘ Ninety-ninth harmonic[5] 11
764.92
A7♭[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
G7 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
A297♭+ 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
G17♯ 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
G23♯+ 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
A13♭[2] 13 : 8 13 : 23 playⓘ Tridecimal neutral sixth,[3] overtone sixth,[5] thirteenth harmonic 13
848.83
A19♭↑ 209 : 128 11×19 : 27 playⓘ Two-hundred-ninth harmonic 19
850.00
Gthree quarter sharp/Ahalf 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
A7+ 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
A43U 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
Bdouble 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] 1⁄3-comma meantone major sixth 5 M
889.76
107 : 64 107 : 26 playⓘ Hundred-seventh harmonic[5] 107
892.54
B191919double 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
A19U 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
Bdouble flat−[2] 128 : 75 27 : 3×52 playⓘ Just diminished seventh,[16] diminished seventh,[5][13] 75th subharmonic 5
925.79
A2319+ 437 : 256 19×23 : 28 playⓘ Four-hundred-thirty-seventh harmonic 23
933.13
A7 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
Ahalf sharp/Bthree quarter flat 219/24 219/24 playⓘ 19 steps in 24 equal temperament 24
953.30
A37♯+ 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 18.80
960.00
24/5 24/5 playⓘ 4 steps in 5 equal temperament 5
968.83
B7♭[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
B19♭ 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
A23♯+ 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
B29♭ 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
B13♭ 117 : 64 32×13 : 26 playⓘ Hundred-seventeenth harmonic[5] 13
1044.86
B7 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ⓘ 21⁄4-tone or Undecimal neutral seventh,[3] undecimal 'median' seventh[5] 11
1050.00
Athree quarter sharp/Bhalf 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
B137 upside-down♭- 13 : 7 13 : 7 playⓘ Tridecimal neutral seventh[21] 13
1073.78
B717 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] 1⁄6-comma meantone major seventh 5
1095.04
C17 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
B41 123 : 64 3×41 : 26 playⓘ Hundred-twenty-third harmonic[5] 41
1137.04
B7 upside-down 27 : 14 33 : 2×7 playⓘ Septimal major seventh[5] 7
1138.04
C1913♭ 247 : 128 13×19 : 27 playⓘ Two-hundred-forty-seventh harmonic 19
1145.04
B31 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
Bhalf sharp/Chalf flat 223/24 223/24 playⓘ 23 steps in 24 equal temperament 24
1151.23
C7 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
C7+ 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
B23↑ 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

[edit]

Notes

[edit]
  1. ^ a b c d Maneri-Sims notation

References

[edit]
  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.
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