Article · Wikipedia archive · Last revised Jun 2, 2026

List of pitch intervals

Below is a list of intervals expressible in terms of a prime limit, completed by a choice of intervals in various equal subdivisions of the octave or of other 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. source ↗
Comparison of two sets of musical intervals. The equal-tempered intervals are black; the Pythagorean intervals are green. source ↗

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

  • The prime limit1 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

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
 C2 1 : 1 1 : 1 playⓘ Unison,3 monophony,4 perfect prime/first,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ⓘ Ragisma36 7 S
0.72
 E7777triple flat+ 2401 : 2400 74 : 25×3×52 playⓘ Breedsma36 7 S
1.00
21/1200 21/1200 playⓘ Cent7 1200
1.20
21/1000 21/1000 playⓘ Millioctave 1000
1.95
B♯++ 32805 : 32768 38×5 : 215 playⓘ Schisma35 5
1.96
3:2÷(27/12) 3 : 219/12 Grad, Werckmeister8
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,36 marvel comma 7 S
8.11
Bdouble sharp− 15625 : 15552 56 : 26×35 playⓘ Kleisma or semicomma majeur36 5
10.06
Adouble sharpdouble sharp++ 2109375 : 2097152 33×57 : 221 playⓘ Semicomma,36 Fokker's comma3 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 comma39 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 comma3 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,36 minor comma 5
21.51
C+2 81 : 80 34 : 24×5 playⓘ Syntonic comma,356 major comma, komma, chromatic diesis, or comma of Didymus361011 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,3561011 ditonic comma, Pythagorean augmented seventh36 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 chroma3 13 S
27.26
C7 upside-down− 64 : 63 26 : 32×7 playⓘ Septimal comma,3611 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 diesis36 7 S
35.70
D77♭ 49 : 48 72 : 24×3 playⓘ Septimal diesis, slendro diesis or septimal 1/6-tone3 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,56 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,36 septimal chroma,2 superior quarter tone5 7 S
49.98
246 : 239 3×41 : 239 playⓘ Just quarter tone11 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 tone5 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 comma3 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 tone5 13 S
70.67
C♯2 25 : 24 52 : 23×3 playⓘ Just chromatic semitone or minor chroma,3 lesser chromatic semitone, small (just) semitone11 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 harmonic5 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 semitone3 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,3611 Pythagorean diatonic semitone, Low Semitone14 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 ascendant5 5
93.60
D19♭- 19 : 18 19 : 2×32 Novendecimal minor secondplayⓘ 19 S
97.36
D↓↓ 128 : 121 27 : 112 playⓘ 121st subharmonic,56 undecimal minor second 11
98.95
D17 upside down♭ 18 : 17 2×32 : 17 playⓘ Just minor semitone, Arabic lute index finger3 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 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 second16 semitone,14 diatonic semitone,11 1⁄6-comma meantone minor second 5 S
113.69
C♯++ 2187 : 2048 37 : 211 playⓘ Apotome311 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 semitone5 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-tone17 13 S
130.23
C23♯+ 69 : 64 3×23 : 26 playⓘ Sixty-ninth harmonic5 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 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 tone5 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,35 trumpet three-quarter tone,11 middle finger [between frets]14 11 S
155.14
D7 35 : 32 5×7 : 25 playⓘ Thirty-fifth harmonic5 7
160.90
D−− 800 : 729 25×52 : 36 playⓘ Grave whole tone,3 neutral second, grave major second 5
165.00
D↑♭−2 11 : 10 11 : 2×5 playⓘ Greater undecimal minor/major/neutral second, 4/5-tone6 or Ptolemy's second3 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 harmonic5 71
180.45
Edouble flat−−− 65536 : 59049 216 : 310 playⓘ Pythagorean diminished third,36 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,35 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
D2 9 : 8 32 : 23 playⓘ Pythagorean major second, Large just whole tone or major second11 (sesquioctavan),4 tonus, major whole tone,35 greater whole tone,16 major tone14 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 harmonic5 73
231.17
D7 upside-down−2 8 : 7 23 : 7 playⓘ Septimal major second,4 septimal whole tone35 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 tone3 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 harmonic5 37
253.08
D♯− 125 : 108 53 : 22×33 playⓘ Semi-augmented whole tone,3 semi-augmented second 5
262.37
E↓♭ 64 : 55 26 : 5×11 playⓘ 55th subharmonic56 11
266.87
E7♭2 7 : 6 7 : 2×3 playⓘ Septimal minor third3411 or Sub minor third14 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 second513 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 third3 13
294.13
E♭−2 32 : 27 25 : 33 playⓘ Pythagorean minor third3561416 semiditone, or 27th subharmonic 3
297.51
E19♭2 19 : 16 19 : 24 playⓘ 19th harmonic,3 19-limit minor third, overtone minor third5 19
300.00
D♯/E♭ 23/12 21/4 playⓘ Equal-tempered minor third 4, 12 M
301.85
D7 upside-down♯- 25 : 215 52 : 3×7 playⓘ Quasi-equal-tempered minor third, 2nd 7-limit minor third, Bohlen-Pierce second36 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,3451116 minor third,14 1⁄3-comma meantone minor third 5 M S
317.60
D♯++ 19683 : 16384 39 : 214 playⓘ Pythagorean augmented second36 3
320.14
E7♭↑ 77 : 64 7×11 : 26 playⓘ Seventy-seventh harmonic5 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 third18 17
337.15
E♭+ 243 : 200 35 : 23×52 playⓘ Acute minor third3 5
342.48
E13♭ 39 : 32 3×13 : 25 playⓘ Thirty-ninth harmonic5 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 third6 7
347.41
E↑♭−2 11 : 9 11 : 32 playⓘ Undecimal neutral third35 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 wosta6 12:11 X 9:814 11
359.47
E13 upside down2 16 : 13 24 : 13 playⓘ Tridecimal neutral third3 13
364.54
79 : 64 79 : 26 playⓘ Seventy-ninth harmonic5 79
364.81
E− 100 : 81 22×52 : 34 playⓘ Grave major third3 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,36 Pythagorean 'schismatic' third5 3
386.31
E2 5 : 4 5 : 22 playⓘ Just major third,3451116 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,3561416 ditone 3
417.51
F7↓+2 14 : 11 2×7 : 11 playⓘ Undecimal diminished fourth or major third3 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,513 25th subharmonic 5
429.06
E41 41 : 32 41 : 25 playⓘ Forty-first harmonic5 41
435.08
E7 upside-down2 9 : 7 32 : 7 playⓘ Septimal major third,35 Bohlen-Pierce third,3 Super major Third14 7
444.77
F↓ 128 : 99 27 : 32×11 playⓘ 99th subharmonic56 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 harmonic5 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 third5 5
462.35
E7 upside-down7 upside-down- 64 : 49 26 : 72 playⓘ 49th subharmonic56 7
470.78
F7+2 21 : 16 3×7 : 24 playⓘ Twenty-first harmonic, narrow fourth,3 septimal fourth,5 wide augmented third, 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 third3 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 harmonic5 17
498.04
F2 4 : 3 22 : 3 playⓘ Perfect fourth,3516 Pythagorean perfect fourth, Just perfect fourth or diatessaron4 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 harmonic5 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 fourth16 5
521.51
E♯+++ 177147 : 131072 311 : 217 playⓘ Pythagorean augmented third36 (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 harmonic5 29
536.95
F↓♯+ 15 : 11 3×5 : 11 playⓘ Undecimal augmented fourth3 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 fourth3 11
563.38
F13 upside down♯+ 18 : 13 2×9 : 13 playⓘ Tridecimal augmented fourth3 13
568.72
F♯2 25 : 18 52 : 2×32 playⓘ Just augmented fourth35 5
570.88
89 : 64 89 : 26 playⓘ Eighty-ninth harmonic5 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 tritone345 Huygens' tritone or Bohlen-Pierce fourth,3 septimal fifth,11 septimal diminished fifth19 7
588.27
G♭−− 1024 : 729 210 : 36 playⓘ Pythagorean diminished fifth,36 low Pythagorean tritone5 3
590.22
F♯+2 45 : 32 32×5 : 25 playⓘ Just augmented fourth, just tritone,411 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 harmonic5 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,36 Pythagorean augmented fourth, high Pythagorean tritone5 3
617.49
F♯7 upside-down2 10 : 7 2×5 : 7 playⓘ Greater septimal tritone, septimal tritone,45 Euler's tritone3 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 23
631.28
G♭2 36 : 25 22×32 : 52 playⓘ Just diminished fifth5 5
646.99
F31♯+ 93 : 64 3×31 : 26 playⓘ Ninety-third harmonic5 31
648.68
G↓2 16 : 11 24 : 11 playⓘ Undecimal semi-diminished fifth3 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 harmonic5 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 sixth36 3
680.45
G− 40 : 27 23×5 : 33 playⓘ 5-limit wolf fifth,5 or diminished sixth, grave fifth,3611 imperfect fifth,16 5
683.83
G19 95 : 64 5×19 : 26 playⓘ Ninety-fifth harmonic5 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
G2 3 : 2 3 : 2 playⓘ Perfect fifth,3516 Pythagorean perfect fifth, Just perfect fifth or diapente,4 fifth,14 Just fifth11 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 harmonic5 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 sixth3 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,56 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 harmonic5 7
743.01
Adouble flat 192 : 125 26×3 : 53 playⓘ Classic diminished sixth3 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 harmonic5 11
764.92
A7♭2 14 : 9 2×7 : 32 playⓘ Septimal minor sixth35 7
772.63
G♯ 25 : 16 52 : 24 playⓘ Just augmented fifth516 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 Lucas numbers 11
789.85
101 : 64 101 : 26 playⓘ Hundred-first harmonic5 101
792.18
A♭−2 128 : 81 27 : 34 playⓘ Pythagorean minor sixth,356 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 harmonic5 17
813.69
A♭2 8 : 5 23 : 5 playⓘ Just minor sixth341116 5
815.64
G♯++ 6561 : 4096 38 : 212 playⓘ Pythagorean augmented fifth,36 Pythagorean 'schismatic' sixth5 3
823.80
103 : 64 103 : 26 playⓘ Hundred-third harmonic5 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 sixth3 5
840.53
A13♭2 13 : 8 13 : 23 playⓘ Tridecimal neutral sixth,3Fibonacci numbers, 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,35 Zalzal's neutral sixth 11
857.09
A7+ 105 : 64 3×5×7 : 26 playⓘ Hundred-fifth harmonic5 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 sixth3 5
873.50
A43U 53 : 32 53 : 25 playⓘ Fifty-third harmonic5 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 subharmonic56 11
882.40
Bdouble flat−−− 32768 : 19683 215 : 39 playⓘ Pythagorean diminished seventh36 3
884.36
A2 5 : 3 5 : 3 playⓘ Just major sixth,3451116 Bohlen-Pierce sixth,3 1⁄3-comma meantone major sixth 5 M
889.76
107 : 64 107 : 26 playⓘ Hundred-seventh harmonic5 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 subharmonic56 19
905.87
A+2 27 : 16 33 : 24 playⓘ Pythagorean major sixth351116 3
921.82
109 : 64 109 : 26 playⓘ Hundred-ninth harmonic5 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,513 75th subharmonic 5
925.79
A2319+ 437 : 256 19×23 : 28 playⓘ Four-hundred-thirty-seventh harmonic 23
933.13
A7 upside-down2 12 : 7 22×3 : 7 playⓘ Septimal major sixth345 7
937.63
A↑ 55 : 32 5×11 : 25 playⓘ Fifty-fifth harmonic520 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 harmonic5 37
955.03
A♯2 125 : 72 53 : 23×32 playⓘ Just augmented sixth5 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,4511 harmonic seventh,311 augmented sixth 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 sixth16 5
984.21
113 : 64 113 : 26 playⓘ Hundred-thirteenth harmonic5 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 seventh5 3
999.47
B19♭ 57 : 32 3×19 : 25 playⓘ Fifty-seventh harmonic5 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 harmonic5 23
1017.60
 B♭2 9 : 5 32 : 5 playⓘ Greater just minor seventh,16 large just minor seventh,45 Bohlen-Pierce seventh3 5
1019.55
A♯+++ 59049 : 32768 310 : 215 playⓘ Pythagorean augmented sixth36 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 29
1035.00
 B↓2 20 : 11 22×5 : 11 playⓘ Lesser undecimal neutral seventh, large minor seventh3 11
1039.10
B♭+ 729 : 400 36 : 24×52 playⓘ Acute minor seventh3 5
1044.44
B13♭ 117 : 64 32×13 : 26 playⓘ Hundred-seventeenth harmonic5 13
1044.86
B7 upside-down♭- 64 : 35 26 : 5×7 playⓘ 35th subharmonic,5 septimal neutral seventh6 7
1049.36
 B↑♭−2 11 : 6 11 : 2×3 playⓘ 21⁄4-tone or Undecimal neutral seventh,3 undecimal 'median' seventh5 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 harmonic5 59
1066.76
B− 50 : 27 2×52 : 33 playⓘ Grave major seventh3 5
1071.70
B137 upside-down♭- 13 : 7 13 : 7 playⓘ Tridecimal neutral seventh21 13
1073.78
B717 119 : 64 7×17 : 26 playⓘ Hundred-nineteenth harmonic5 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 octave36 3
1088.27
B2 15 : 8 3×5 : 23 playⓘ Just major seventh,351116 small just major seventh,4 1⁄6-comma meantone major seventh 5
1095.04
C17 upside down♭ 32 : 17 25 : 17 playⓘ 17th subharmonic56 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 harmonic5 11
1107.82
C′♭− 256 : 135 28 : 33×5 playⓘ Octave − major chroma,3 135th subharmonic, narrow diminished octave 5
1109.78
B+2 243 : 128 35 : 27 playⓘ Pythagorean major seventh35611 3
1116.88
61 : 32 61 : 25 playⓘ Sixty-first harmonic5 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,36 large just major seventh4 5
1131.02
B41 123 : 64 3×41 : 26 playⓘ Hundred-twenty-third harmonic5 41
1137.04
B7 upside-down 27 : 14 33 : 2×7 playⓘ Septimal major seventh5 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 31
1146.73
C↓ 64 : 33 26 : 3×11 playⓘ 33rd subharmonic6 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 harmonic5 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 5
1179.59
B23↑ 253 : 128 11×23 : 27 playⓘ Two-hundred-fifty-third harmonic5 23
1186.42
127 : 64 127 : 26 playⓘ Hundred-twenty-seventh harmonic5 127
1200.00
C′ 2 : 1 2 : 1 playⓘ Octave,311 perfect eighth or diapason4 1, 12 3 M S
See also

See also

Notes

Notes

References

References

  1. Fox, Christopher (2003). "Microtones and Microtonalities", Contemporary Music Review, v. 22, pt. 1–2. (Abingdon, Oxfordshire, UK: Routledge): p. 13.
  2. Fonville, John. 1991. "Ben Johnston's Extended Just Intonation: A Guide for Interpreters". Perspectives of New Music 29, no. 2 (Summer): 106–137.
  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. Hachette Books. pp. 68–69. ISBN 978-0-306-80106-8.
  5. "Anatomy of an Octave", Kyle Gann (1998). Gann leaves off "just" but includes "5-limit". He uses "median" for "neutral".
  6. 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. Partch 1979, p. 70
  11. 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. Anger, Joseph Humfrey (1912). A Treatise on Harmony, with Exercises, Volume 3, pp. xiv–xv. W. Tyrrell.
  14. 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. 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. "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.
External links