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

  • 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

ColumnLegend
TETX-tone equal temperament (12-tet, etc.).
Limit3-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.
MMeantone temperament or tuning.
SSuperparticular ratio (no separate color code).
List of musical intervals
CentsNote (from C)Freq. ratioPrime factorsInterval nameTETLimitMS
0.00
C[2]1 : 11 : 1playⓘ Unison,[3] monophony,[4] perfect prime,[3] tonic,[5] or fundamental1, 123M
0.03
65537 : 6553665537 : 216playⓘ Sixty-five-thousand-five-hundred-thirty-seventh harmonic 65537S
0.40
C7♯−4375 : 437454×7 : 2×37playⓘ Ragisma[3][6]7S
0.72
E7777triple flat+2401 : 240074 : 25×3×52playⓘ Breedsma[3][6]7S
1.00
21/120021/1200playⓘ Cent[7]1200
1.20
21/100021/1000playⓘ Millioctave1000
1.95
B♯++32805 : 3276838×5 : 215playⓘ Schisma[3][5]5
1.96
3:2÷(27/12)3 : 219/12Grad, Werckmeister[8]
3.99
101/100021/1000×51/1000playⓘ Savart or eptaméride301.03
7.71
B7 upside-down♯225 : 22432×52 : 25×7playⓘ Septimal kleisma,[3][6] marvel comma7S
8.11
Bdouble sharp−15625 : 1555256 : 26×35playⓘ Kleisma or semicomma majeur[3][6]5
10.06
Adouble sharpdouble sharp++2109375 : 209715233×57 : 221playⓘ Semicomma,[3][6] Fokker's comma[3]5
10.85
C43U160 : 15925×5 : 3×53playⓘ Difference between 5:3 & 53:3253S
11.98
C29145 : 1445×29 : 24×32playⓘ Difference between 29:16 & 9:529S
12.50
21/9621/96playⓘ Sixteenth tone96
13.07
B7 upside-down7 upside-down7 upside-down−1728 : 171526×33 : 5×73playⓘ Orwell comma[3][9]7
13.47
C43129 : 1283×43 : 27playⓘ Hundred-twenty-ninth harmonic43S
13.79
Ddouble flat7126 : 1252×32×7 : 53playⓘ Small septimal semicomma,[6] small septimal comma,[3] starling comma7S
14.37
C♭↑↑−121 : 120112 : 23×3×5playⓘ Undecimal seconds comma[3]11S
16.67
C↑[lower-alpha 1]21/7221/72playⓘ 1 step in 72 equal temperament72
18.13
C19U96 : 9525×3 : 5×19playⓘ Difference between 19:16 & 6:519S
19.55
Ddouble flat--[2]2048 : 2025211 : 34×52playⓘ Diaschisma,[3][6] minor comma5
21.51
C+[2]81 : 8034 : 24×5playⓘ Syntonic comma,[3][5][6] major comma, komma, chromatic diesis, or comma of Didymus[3][6][10][11]5S
22.64
21/5321/53playⓘ Holdrian comma, Holder's comma, 1 step in 53 equal temperament53
23.46
B♯+++531441 : 524288312 : 219playⓘ Pythagorean comma,[3][5][6][10][11] ditonic comma[3][6]3
25.00
21/4821/48playⓘ Eighth tone48
26.84
C1365 : 645×13 : 26playⓘ Sixty-fifth harmonic,[5] 13th-partial chroma[3]13S
27.26
C7 upside-down−64 : 6326 : 32×7playⓘ Septimal comma,[3][6][11] Archytas' comma,[3] 63rd subharmonic7S
29.27
21/4121/41playⓘ 1 step in 41 equal temperament41
31.19
D7♭↓56 : 5523×7 : 5×11playⓘ Undecimal diesis,[3] Ptolemy's enharmonic:[5] difference between (11 : 8) and (7 : 5) tritone11S
33.33
C/D♭[lower-alpha 1]21/3621/36playⓘ Sixth tone36, 72
34.28
C1751 : 503×17 : 2×52playⓘ Difference between 17:16 & 25:2417S
34.98
B7 upside-down7 upside-down♯-50 : 492×52 : 72playⓘ Septimal sixth tone or jubilisma, Erlich's decatonic comma or tritonic diesis[3][6]7S
35.70
D77♭49 : 4872 : 24×3playⓘ Septimal diesis, slendro diesis or septimal 1/6-tone[3]7S
38.05
C2346 : 452×23 : 32×5playⓘ Inferior quarter tone,[5] difference between 23:16 & 45:3223S
38.71
21/3121/31playⓘ 1 step in 31 equal temperament31
38.91
C↓♯+45 : 4432×5 : 4×11playⓘ Undecimal diesis or undecimal fifth tone 11S
40.00
21/3021/30playⓘ Fifth tone30
41.06
Ddouble flat−128 : 12527 : 53playⓘ Enharmonic diesis or 5-limit limma, minor diesis,[6] diminished second,[5][6] minor diesis or diesis,[3] 125th subharmonic5
41.72
D41U7♭42 : 412×3×7 : 41playⓘ Lesser 41-limit fifth tone41S
42.75
C4141 : 4041 : 23×5playⓘ Greater 41-limit fifth tone41S
43.83
C13 upside down♯40 : 3923×5 : 3×13playⓘ Tridecimal fifth tone13S
44.97
C19U1339 : 383×13 : 2×19playⓘ Superior quarter-tone,[5] novendecimal fifth tone19S
46.17
D37U19double flat-38 : 372×19 : 37playⓘ Lesser 37-limit quarter tone37S
47.43
C37♯37 : 3637 : 22×32playⓘ Greater 37-limit quarter tone37S
48.77
C7 upside-down36 : 3522×32 : 5×7playⓘ Septimal quarter tone, septimal diesis,[3][6] septimal chroma,[2] superior quarter tone[5]7S
49.98
246 : 2393×41 : 239playⓘ Just quarter tone[11]239
50.00
Chalf sharp/Dthree quarter flat21/2421/24playⓘ Equal-tempered quarter tone24
50.18
D17 upside down7♭35 : 345×7 : 2×17playⓘ ET quarter-tone approximation,[5] lesser 17-limit quarter tone17S
50.72
B7 upside-down♯++59049 : 57344310 : 213×7playⓘ Harrison's comma (10 P5s – 1 H7)[3]7
51.68
C17↓♯34 : 332×17 : 3×11playⓘ Greater 17-limit quarter tone17S
53.27
C↑33 : 323×11 : 25playⓘ Thirty-third harmonic,[5] undecimal comma, undecimal quarter tone11S
54.96
D31U♭-32 : 3125 : 31playⓘ Inferior quarter-tone,[5] thirty-first subharmonic31S
56.55
B2323♯+529 : 512232 : 29playⓘ Five-hundred-twenty-ninth harmonic23
56.77
C3131 : 3031 : 2×3×5playⓘ Greater quarter-tone,[5] difference between 31:16 & 15:831S
58.69
C29U♯30 : 292×3×5 : 29playⓘ Lesser 29-limit quarter tone29S
60.75
C297 upside-down29 : 2829 : 22×7playⓘ Greater 29-limit quarter tone29S
62.96
D7♭-28 : 2722×7 : 33playⓘ Septimal minor second, small minor second, inferior quarter tone[5]7S
63.81
(3 : 2)1/1131/11 : 21/11playⓘ Beta scale step18.75
65.34
C13 upside down♯+27 : 2633 : 2×13 playⓘ Chromatic diesis,[12] tridecimal comma[3]13S
66.34
D197♭133 : 1287×19 : 27playⓘ One-hundred-thirty-third harmonic19
66.67
C↑/C♯[lower-alpha 1]21/1821/18playⓘ Third tone18, 36, 72
67.90
D13double flat-26 : 252×13 : 52playⓘ Tridecimal third tone, third tone[5]13S
70.67
C♯[2]25 : 2452 : 23×3playⓘ 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 unison5S
73.68
D23U♭-24 : 2323×3 : 23playⓘ Lesser 23-limit semitone23S
75.00
21/1623/48playⓘ 1 step in 16 equal temperament, 3 steps in 4816, 48
76.96
C23↓♯+23 : 2223 : 2×11playⓘ Greater 23-limit semitone23S
78.00
(3 : 2)1/931/9 : 21/9playⓘ Alpha scale step15.39
79.31
67 : 6467 : 26playⓘ Sixty-seventh harmonic[5]67
80.54
C↑7 upside-down-22 : 212×11 : 3×7playⓘ Hard semitone,[5] two-fifth tone small semitone11S
84.47
D7♭21 : 203×7 : 22×5playⓘ Septimal chromatic semitone, minor semitone[3]7S
88.80
C19U♯20 : 1922×5 : 19playⓘ Novendecimal augmented unison19S
90.22
D♭−−[2]256 : 24328 : 35playⓘ Pythagorean minor second or limma,[3][6][11] Pythagorean diatonic semitone, Low Semitone[14]3
92.18
C♯+[2]135 : 12833×5 : 27playⓘ 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 : 1819 : 2×9Novendecimal minor secondplayⓘ 19S
97.36
D↓↓128 : 12127 : 112playⓘ 121st subharmonic,[5][6] undecimal minor second11
98.95
D17 upside down♭18 : 172×32 : 17playⓘ Just minor semitone, Arabic lute index finger[3]17S
100.00
C♯/D♭21/1221/12playⓘ Equal-tempered minor second or semitone12M
104.96
C17♯[2]17 : 1617 : 24playⓘ Minor diatonic semitone, just major semitone, overtone semitone,[5] 17th harmonic,[3] limma17S
111.45
25√5(5 : 1)1/25playⓘ Studie II interval (compound just major third, 5:1, divided into 25 equal parts)25
111.73
D♭-[2]16 : 1524 : 3×5playⓘ 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 second5S
113.69
C♯++2187 : 204837 : 211playⓘ Apotome[3][11] or Pythagorean major semitone,[6] Pythagorean augmented unison, Pythagorean chromatic semitone, or Pythagorean apotome3
116.72
(18 : 5)1/1921/19×32/19 : 51/19playⓘ Secor10.28
119.44
C7 upside-down♯15 : 143×5 : 2×7playⓘ Septimal diatonic semitone, major diatonic semitone,[3] Cowell semitone[5]7S
125.00
25/4825/48playⓘ 5 steps in 48 equal temperament48
128.30
D13 upside down714 : 132×7 : 13playⓘ Lesser tridecimal 2/3-tone[17]13S
130.23
C23♯+69 : 643×23 : 26playⓘ Sixty-ninth harmonic[5]23
133.24
D♭27 : 2533 : 52playⓘ Semitone maximus, minor second, large limma or Bohlen-Pierce small semitone,[3] high semitone,[14] alternate Renaissance half-step,[5] large limma, acute minor second5
133.33
C♯/D♭[lower-alpha 1]21/922/18playⓘ Two-third tone9, 18, 36, 72
138.57
D13♭-13 : 1213 : 22×3playⓘ Greater tridecimal 2/3-tone,[17] Three-quarter tone[5]13S
150.00
Cthree quarter sharp/Dhalf flat23/2421/8playⓘ Equal-tempered neutral second8, 24
150.64
D↓[2]12 : 1122×3 : 11playⓘ 3⁄4 tone or Undecimal neutral second,[3][5] trumpet three-quarter tone,[11] middle finger [between frets][14]11S
155.14
D735 : 325×7 : 25playⓘ Thirty-fifth harmonic[5]7
160.90
D−−800 : 72925×52 : 36playⓘ Grave whole tone,[3] neutral second, grave major second5
165.00
D↑♭−[2]11 : 1011 : 2×5playⓘ Greater undecimal minor/major/neutral second, 4/5-tone[6] or Ptolemy's second[3]11S
171.43
21/721/7playⓘ 1 step in 7 equal temperament7
175.00
27/4827/48playⓘ 7 steps in 48 equal temperament48
179.70
71 : 6471 : 26playⓘ Seventy-first harmonic[5]71
180.45
Edouble flat−−−65536 : 59049216 : 310playⓘ Pythagorean diminished third,[3][6] Pythagorean minor tone3
182.40
D−[2]10 : 92×5 : 32playⓘ 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 second5S
200.00
D22/1221/6playⓘ Equal-tempered major second6, 12M
203.91
D[2]9 : 832 : 23playⓘ 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]3S
215.89
D29145 : 1285×29 : 27playⓘ Hundred-forty-fifth harmonic29
223.46
Edouble flat−[2]256 : 22528 : 32×52playⓘ Just diminished third,[16] 225th subharmonic5
225.00
23/1629/48playⓘ 9 steps in 48 equal temperament16, 48
227.79
73 : 6473 : 26playⓘ Seventy-third harmonic[5]73
231.17
D7 upside-down−[2]8 : 723 : 7playⓘ Septimal major second,[4] septimal whole tone[3][5]7S
240.00
21/521/5playⓘ 1 step in 5 equal temperament5
247.74
D13 upside down♯15 : 133×5 : 13playⓘ Tridecimal 5⁄4 tone[3]13
250.00
Dhalf sharp/Ethree quarter flat25/2425/24playⓘ 5 steps in 24 equal temperament24
251.34
D37♯37 : 3237 : 25playⓘ Thirty-seventh harmonic[5]37
253.08
D♯−125 : 10853 : 22×33playⓘ Semi-augmented whole tone,[3] semi-augmented second5
262.37
E↓♭64 : 5526 : 5×11playⓘ 55th subharmonic[5][6]11
266.87
E7♭[2]7 : 67 : 2×3playⓘ Septimal minor third[3][4][11] or Sub minor third[14]7S
268.80
D2313299 : 25613×23 : 28playⓘ Two-hundred-ninety-ninth harmonic23
274.58
D♯[2]75 : 643×52 : 26playⓘ Just augmented second,[16] Augmented tone,[14] augmented second[5][13]5
275.00
211/48211/48playⓘ 11 steps in 48 equal temperament48
289.21
E13↓♭13 : 1113 : 11playⓘ Tridecimal minor third[3]13
294.13
E♭−[2]32 : 2725 : 33playⓘ Pythagorean minor third[3][5][6][14][16] semiditone, or 27th subharmonic3
297.51
E19♭[2]19 : 1619 : 24playⓘ 19th harmonic,[3] 19-limit minor third, overtone minor third[5]19
300.00
D♯/E♭23/1221/4playⓘ Equal-tempered minor third4, 12M
301.85
D7 upside-down♯-25 : 21[5]52 : 3×7playⓘ Quasi-equal-tempered minor third, 2nd 7-limit minor third, Bohlen-Pierce second[3][6]7
310.26
6:5÷(81:80)1/422 : 53/4playⓘ Quarter-comma meantone minor thirdM
311.98
(3 : 2)4/934/9 : 24/9playⓘ Alpha scale minor third3.85
315.64
E♭[2]6 : 52×3 : 5playⓘ Just minor third,[3][4][5][11][16] minor third,[14] 1⁄3-comma meantone minor third5MS
317.60
D♯++19683 : 1638439 : 214playⓘ Pythagorean augmented second[3][6]3
320.14
E7♭↑77 : 647×11 : 26playⓘ Seventy-seventh harmonic[5]11
325.00
213/48213/48playⓘ 13 steps in 48 equal temperament48
336.13
D177 upside-down♯-17 : 1417 : 2×7playⓘ Superminor third[18]17
337.15
E♭+243 : 20035 : 23×52playⓘ Acute minor third[3]5
342.48
E13♭39 : 323×13 : 25playⓘ Thirty-ninth harmonic[5]13
342.86
22/722/7playⓘ 2 steps in 7 equal temperament7
342.91
E7 upside-down♭-128 : 10527 : 3×5×7playⓘ 105th subharmonic,[5] septimal neutral third[6]7
347.41
E↑♭−[2]11 : 911 : 32playⓘ Undecimal neutral third[3][5]11
350.00
Dthree quarter sharp/Ehalf flat27/2427/24playⓘ Equal-tempered neutral third24
354.55
E↓+27 : 2233 : 2×11playⓘ Zalzal's wosta[6] 12:11 X 9:8[14]11
359.47
E13 upside down[2]16 : 1324 : 13playⓘ Tridecimal neutral third[3]13
364.54
79 : 6479 : 26playⓘ Seventy-ninth harmonic[5]79
364.81
E−100 : 8122×52 : 34playⓘ Grave major third[3]5
375.00
25/16215/48playⓘ 15 steps in 48 equal temperament16, 48
384.36
F♭−−8192 : 6561213 : 38playⓘ Pythagorean diminished fourth,[3][6] Pythagorean 'schismatic' third[5]3
386.31
E[2]5 : 45 : 22playⓘ Just major third,[3][4][5][11][16] major third,[14] quarter-comma meantone major third5MS
397.10
E237+161 : 1287×23 : 27playⓘ One-hundred-sixty-first harmonic23
400.00
E24/1221/3playⓘ Equal-tempered major third3, 12M
402.47
E1917323 : 25617×19 : 28playⓘ Three-hundred-twenty-third harmonic19
407.82
E+[2]81 : 6434 : 26playⓘ Pythagorean major third,[3][5][6][14][16] ditone3
417.51
F7↓+[2]14 : 112×7 : 11playⓘ Undecimal diminished fourth or major third[3]11
425.00
217/48217/48playⓘ 17 steps in 48 equal temperament48
427.37
F♭[2]32 : 2525 : 52playⓘ Just diminished fourth,[16] diminished fourth,[5][13] 25th subharmonic5
429.06
E4141 : 3241 : 25playⓘ Forty-first harmonic[5]41
435.08
E7 upside-down[2]9 : 732 : 7playⓘ Septimal major third,[3][5] Bohlen-Pierce third,[3] Super major Third[14]7
444.77
F↓128 : 9927 : 9×11playⓘ 99th subharmonic[5][6]11
450.00
Ehalf sharp/Fhalf flat29/2429/24playⓘ 9 steps in 24 equal temperament24
450.05
83 : 6483 : 26playⓘ Eighty-third harmonic[5]83
454.21
F♭1313 : 1013 : 2×5playⓘ Tridecimal major third or diminished fourth13
456.99
E♯[2]125 : 9653 : 25×3playⓘ Just augmented third, augmented third[5]5
462.35
E7 upside-down7 upside-down-64 : 4926 : 72playⓘ 49th subharmonic[5][6]7
470.78
F7+[2]21 : 163×7 : 24playⓘ Twenty-first harmonic, narrow fourth,[3] septimal fourth,[5] wide augmented third, H7 on G7
475.00
219/48219/48playⓘ 19 steps in 48 equal temperament48
478.49
E♯+675 : 51233×52 : 29playⓘ Six-hundred-seventy-fifth harmonic, wide augmented third[3]5
480.00
22/522/5playⓘ 2 steps in 5 equal temperament5
491.27
E17♯85 : 645×17 : 26playⓘ Eighty-fifth harmonic[5]17
498.04
F[2]4 : 322 : 3playⓘ Perfect fourth,[3][5][16] Pythagorean perfect fourth, Just perfect fourth or diatessaron[4]3S
500.00
F25/1225/12playⓘ Equal-tempered perfect fourth12M
501.42
F19+171 : 12832×19 : 27playⓘ One-hundred-seventy-first harmonic19
510.51
(3 : 2)8/1138/11 : 28/11playⓘ Beta scale perfect fourth18.75
511.52
F4343 : 3243 : 25playⓘ Forty-third harmonic[5]43
514.29
23/723/7playⓘ 3 steps in 7 equal temperament7
519.55
F+[2]27 : 2033 : 22×5playⓘ 5-limit wolf fourth, acute fourth,[3] imperfect fourth[16]5
521.51
E♯+++177147 : 131072311 : 217playⓘ Pythagorean augmented third[3][6] (F+ (pitch))3
525.00
27/16221/48playⓘ 21 steps in 48 equal temperament16, 48
531.53
F29+87 : 643×29 : 26playⓘ Eighty-seventh harmonic[5]29
536.95
F↓♯+15 : 113×5 : 11playⓘ Undecimal augmented fourth[3]11
550.00
Fhalf sharp/Gthree quarter flat211/24211/24playⓘ 11 steps in 24 equal temperament24
551.32
F↑[2]11 : 811 : 23playⓘ eleventh harmonic,[5] undecimal tritone,[5] lesser undecimal tritone, undecimal semi-augmented fourth[3]11
563.38
F13 upside down♯+18 : 132×9 : 13playⓘ Tridecimal augmented fourth[3]13
568.72
F♯[2]25 : 1852 : 2×32playⓘ Just augmented fourth[3][5]5
570.88
89 : 6489 : 26playⓘ Eighty-ninth harmonic[5]89
575.00
223/48223/48playⓘ 23 steps in 48 equal temperament48
582.51
G7♭[2]7 : 57 : 5playⓘ 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 : 729210 : 36playⓘ Pythagorean diminished fifth,[3][6] low Pythagorean tritone[5]3
590.22
F♯+[2]45 : 3232×5 : 25playⓘ 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 fourth5
595.03
G1919♭361 : 256192 : 28playⓘ Three-hundred-sixty-first harmonic19
600.00
F♯/G♭26/1221/2=√2playⓘ Equal-tempered tritone2, 12M
609.35
G137♭91 : 647×13 : 26playⓘ Ninety-first harmonic[5]13
609.78
G♭−[2]64 : 4526 : 32×5playⓘ Just tritone,[4] 2nd tritone,[6] 'false' fifth,[16] diminished fifth,[13] low 5-limit tritone,[5] 45th subharmonic5
611.73
F♯++729 : 51236 : 29playⓘ Pythagorean tritone,[3][6] Pythagorean augmented fourth, high Pythagorean tritone[5]3
617.49
F♯7 upside-down[2]10 : 72×5 : 7playⓘ Greater septimal tritone, septimal tritone,[4][5] Euler's tritone[3]7
625.00
225/48225/48playⓘ 25 steps in 48 equal temperament48
628.27
F23♯+23 : 1623 : 24playⓘ Twenty-third harmonic,[5] classic diminished fifth23
631.28
G♭[2]36 : 2522×32 : 52playⓘ Just diminished fifth[5]5
646.99
F31♯+93 : 643×31 : 26playⓘ Ninety-third harmonic[5]31
648.68
G↓[2]16 : 1124 : 11playⓘ ` undecimal semi-diminished fifth[3]11
650.00
Fthree quarter sharp/Ghalf flat213/24213/24playⓘ 13 steps in 24 equal temperament24
665.51
G43U47 : 3247 : 25playⓘ Forty-seventh harmonic[5]47
675.00
29/16227/48playⓘ 27 steps in 48 equal temperament16, 48
678.49
Adouble flat−−−262144 : 177147218 : 311playⓘ Pythagorean diminished sixth[3][6]3
680.45
G−40 : 2723×5 : 33playⓘ 5-limit wolf fifth,[5] or diminished sixth, grave fifth,[3][6][11] imperfect fifth,[16]5
683.83
G1995 : 645×19 : 26playⓘ Ninety-fifth harmonic[5]19
684.82
E232323double sharp++12167 : 8192233 : 213playⓘ 12167th harmonic23
685.71
24/7 : 1playⓘ 4 steps in 7 equal temperament
691.20
3:2÷(81:80)1/22×51/2 : 3playⓘ Half-comma meantone perfect fifthM
694.79
3:2÷(81:80)1/321/3×51/3 : 31/3playⓘ 1⁄3-comma meantone perfect fifthM
695.81
3:2÷(81:80)2/721/7×52/7 : 31/7playⓘ 2⁄7-comma meantone perfect fifthM
696.58
3:2÷(81:80)1/451/4playⓘ Quarter-comma meantone perfect fifthM
697.65
3:2÷(81:80)1/531/5×51/5 : 21/5playⓘ 1⁄5-comma meantone perfect fifthM
698.37
3:2÷(81:80)1/631/3×51/6 : 21/3playⓘ 1⁄6-comma meantone perfect fifthM
700.00
G27/1227/12playⓘ Equal-tempered perfect fifth12M
701.89
231/53231/53playⓘ 53-TET perfect fifth53
701.96
G[2]3 : 23 : 2playⓘ Perfect fifth,[3][5][16] Pythagorean perfect fifth, Just perfect fifth or diapente,[4] fifth,[14] Just fifth[11]3S
702.44
224/41224/41playⓘ 41-TET perfect fifth41
703.45
217/29217/29playⓘ 29-TET perfect fifth29
719.90
97 : 6497 : 26playⓘ Ninety-seventh harmonic[5]97
720.00
23/5 : 1playⓘ 3 steps in 5 equal temperament5
721.51
Adouble flat−1024 : 675210 : 33×52playⓘ Narrow diminished sixth[3]5
725.00
229/48229/48playⓘ 29 steps in 48 equal temperament48
729.22
G7 upside-down-32 : 2124 : 3×7playⓘ 21st subharmonic,[5][6] septimal diminished sixth7
733.23
F2317double sharp+391 : 25617×23 : 28playⓘ Three-hundred-ninety-first harmonic23
737.65
A77♭+49 : 327×7 : 25playⓘ Forty-ninth harmonic[5]7
743.01
Adouble flat192 : 12526×3 : 53playⓘ Classic diminished sixth[3]5
750.00
Ghalf sharp/Athree quarter flat215/24215/24playⓘ 15 steps in 24 equal temperament24
755.23
G↑99 : 6432×11 : 26playⓘ Ninety-ninth harmonic[5]11
764.92
A7♭[2]14 : 92×7 : 32playⓘ Septimal minor sixth[3][5]7
772.63
G♯25 : 1652 : 24playⓘ Just augmented fifth[5][16]5
775.00
231/48231/48playⓘ 31 steps in 48 equal temperament48
781.79
π : 2playⓘ Wallis product
782.49
G7 upside-down↑-[2]11 : 711 : 7playⓘ Undecimal minor sixth,[5] undecimal augmented fifth,[3] Fibonacci numbers11
789.85
101 : 64101 : 26playⓘ Hundred-first harmonic[5]101
792.18
A♭−[2]128 : 8127 : 34playⓘ Pythagorean minor sixth,[3][5][6] 81st subharmonic3
798.40
A297♭+203 : 1287×29 : 27playⓘ Two-hundred-third harmonic29
800.00
G♯/A♭28/1222/3playⓘ Equal-tempered minor sixth3, 12M
806.91
G17♯51 : 323×17 : 25playⓘ Fifty-first harmonic[5]17
813.69
A♭[2]8 : 523 : 5playⓘ Just minor sixth[3][4][11][16]5
815.64
G♯++6561 : 409638 : 212playⓘ Pythagorean augmented fifth,[3][6] Pythagorean 'schismatic' sixth[5]3
823.80
103 : 64103 : 26playⓘ Hundred-third harmonic[5]103
825.00
211/16233/48playⓘ 33 steps in 48 equal temperament16, 48
832.18
G23♯+207 : 12832×23 : 27playⓘ Two-hundred-seventh harmonic23
833.09
(51/2+1)/2φ : 1playⓘ Golden ratio (833 cents scale)
835.19
A♭+81 : 5034 : 2×52playⓘ Acute minor sixth[3]5
840.53
A13♭[2]13 : 813 : 23playⓘ Tridecimal neutral sixth,[3] overtone sixth,[5] thirteenth harmonic13
848.83
A19♭↑209 : 12811×19 : 27playⓘ Two-hundred-ninth harmonic19
850.00
Gthree quarter sharp/Ahalf flat217/24217/24playⓘ Equal-tempered neutral sixth24
852.59
A↓+[2]18 : 112×32 : 11playⓘ Undecimal neutral sixth,[3][5] Zalzal's neutral sixth11
857.09
A7+105 : 643×5×7 : 26playⓘ Hundred-fifth harmonic[5]7
857.14
25/725/7playⓘ 5 steps in 7 equal temperament7
862.85
A−400 : 24324×52 : 35playⓘ Grave major sixth[3]5
873.50
A43U53 : 3253 : 25playⓘ Fifty-third harmonic[5]53
875.00
235/48235/48playⓘ 35 steps in 48 equal temperament48
879.86
A↓7 upside-down128 : 7727 : 7×11playⓘ 77th subharmonic[5][6]11
882.40
Bdouble flat−−−32768 : 19683215 : 39playⓘ Pythagorean diminished seventh[3][6]3
884.36
A[2]5 : 35 : 3playⓘ Just major sixth,[3][4][5][11][16] Bohlen-Pierce sixth,[3] 1⁄3-comma meantone major sixth5M
889.76
107 : 64107 : 26playⓘ Hundred-seventh harmonic[5]107
892.54
B191919double flat6859 : 4096193 : 212playⓘ 6859th harmonic19
900.00
A29/1223/4playⓘ Equal-tempered major sixth4, 12M
902.49
A19U32 : 1925 : 19playⓘ 19th subharmonic[5][6]19
905.87
A+[2]27 : 1633 : 24playⓘ Pythagorean major sixth[3][5][11][16]3
921.82
109 : 64109 : 26playⓘ Hundred-ninth harmonic[5]109
925.00
237/48237/48playⓘ 37 steps in 48 equal temperament48
925.42
Bdouble flat−[2]128 : 7527 : 3×52playⓘ Just diminished seventh,[16] diminished seventh,[5][13] 75th subharmonic5
925.79
A2319+437 : 25619×23 : 28playⓘ Four-hundred-thirty-seventh harmonic23
933.13
A7 upside-down[2]12 : 722×3 : 7playⓘ Septimal major sixth[3][4][5]7
937.63
A↑55 : 325×11 : 25playⓘ Fifty-fifth harmonic[5][20]11
950.00
Ahalf sharp/Bthree quarter flat219/24219/24playⓘ 19 steps in 24 equal temperament24
953.30
A37♯+111 : 643×37 : 26playⓘ Hundred-eleventh harmonic[5]37
955.03
A♯[2]125 : 7253 : 23×32playⓘ Just augmented sixth[5]5
957.21
(3 : 2)15/11315/11 : 215/11playⓘ 15 steps in Beta scale18.75
960.00
24/524/5playⓘ 4 steps in 5 equal temperament5
968.83
B7♭[2]7 : 47 : 22playⓘ Septimal minor seventh,[4][5][11] harmonic seventh,[3][11] augmented sixth7
975.00
213/16239/48playⓘ 39 steps in 48 equal temperament16, 48
976.54
A♯+[2]225 : 12832×52 : 27playⓘ Just augmented sixth[16]5
984.21
113 : 64113 : 26playⓘ Hundred-thirteenth harmonic[5]113
996.09
B♭−[2]16 : 924 : 32playⓘ 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 : 323×19 : 25playⓘ Fifty-seventh harmonic[5]19
1000.00
A♯/B♭210/1225/6playⓘ Equal-tempered minor seventh6, 12M
1014.59
A23♯+115 : 645×23 : 26playⓘ Hundred-fifteenth harmonic[5]23
1017.60
B♭[2]9 : 532 : 5playⓘ Greater just minor seventh,[16] large just minor seventh,[4][5] Bohlen-Pierce seventh[3]5
1019.55
A♯+++59049 : 32768310 : 215playⓘ Pythagorean augmented sixth[3][6]3
1025.00
241/48241/48playⓘ 41 steps in 48 equal temperament48
1028.57
26/726/7playⓘ 6 steps in 7 equal temperament7
1029.58
B29♭29 : 1629 : 24playⓘ Twenty-ninth harmonic,[5] minor seventh29
1035.00
B↓[2]20 : 1122×5 : 11playⓘ Lesser undecimal neutral seventh, large minor seventh[3]11
1039.10
B♭+729 : 40036 : 24×52playⓘ Acute minor seventh[3]5
1044.44
B13♭117 : 6432×13 : 26playⓘ Hundred-seventeenth harmonic[5]13
1044.86
B7 upside-down♭-64 : 3526 : 5×7playⓘ 35th subharmonic,[5] septimal neutral seventh[6]7
1049.36
B↑♭−[2]11 : 611 : 2×3playⓘ 21⁄4-tone or Undecimal neutral seventh,[3] undecimal 'median' seventh[5]11
1050.00
Athree quarter sharp/Bhalf flat221/2427/8playⓘ Equal-tempered neutral seventh8, 24
1059.17
59 : 3259 : 25playⓘ Fifty-ninth harmonic[5]59
1066.76
B−50 : 272×52 : 33playⓘ Grave major seventh[3]5
1071.70
B137 upside-down♭-13 : 713 : 7playⓘ Tridecimal neutral seventh[21]13
1073.78
B717119 : 647×17 : 26playⓘ Hundred-nineteenth harmonic[5]17
1075.00
243/48243/48playⓘ 43 steps in 48 equal temperament48
1086.31
C′♭−−4096 : 2187212 : 37playⓘ Pythagorean diminished octave[3][6]3
1088.27
B[2]15 : 83×5 : 23playⓘ Just major seventh,[3][5][11][16] small just major seventh,[4] 1⁄6-comma meantone major seventh5
1095.04
C17 upside down♭32 : 1725 : 17playⓘ 17th subharmonic[5][6]17
1100.00
B211/12211/12playⓘ Equal-tempered major seventh12M
1102.64
B↑↑♭-121 : 64112 : 26playⓘ Hundred-twenty-first harmonic[5]11
1107.82
C′♭−256 : 13528 : 33×5playⓘ Octave − major chroma,[3] 135th subharmonic, narrow diminished octave5
1109.78
B+[2]243 : 12835 : 27playⓘ Pythagorean major seventh[3][5][6][11]3
1116.88
61 : 3261 : 25playⓘ Sixty-first harmonic[5]61
1125.00
215/16245/48playⓘ 45 steps in 48 equal temperament16, 48
1129.33
C′♭[2]48 : 2524×3 : 52playⓘ Classic diminished octave,[3][6] large just major seventh[4]5
1131.02
B41123 : 643×41 : 26playⓘ Hundred-twenty-third harmonic[5]41
1137.04
B7 upside-down27 : 1433 : 2×7playⓘ Septimal major seventh[5]7
1138.04
C1913♭247 : 12813×19 : 27playⓘ Two-hundred-forty-seventh harmonic19
1145.04
B3131 : 1631 : 24playⓘ Thirty-first harmonic,[5] augmented seventh31
1146.73
C↓64 : 3326 : 3×11playⓘ 33rd subharmonic[6]11
1150.00
Bhalf sharp/Chalf flat223/24223/24playⓘ 23 steps in 24 equal temperament24
1151.23
C735 : 185×7 : 2×32playⓘ Septimal supermajor seventh, septimal quarter tone inverted 7
1158.94
B♯[2]125 : 6453 : 26playⓘ Just augmented seventh,[5] 125th harmonic5
1172.74
C7+63 : 3232×7 : 25playⓘ Sixty-third harmonic[5]7
1175.00
247/48247/48playⓘ 47 steps in 48 equal temperament48
1178.49
C′−160 : 8125×5 : 34playⓘ Octave − syntonic comma,[3] semi-diminished octave5
1179.59
B23↑253 : 12811×23 : 27playⓘ Two-hundred-fifty-third harmonic[5]23
1186.42
127 : 64127 : 26playⓘ Hundred-twenty-seventh harmonic[5]127
1200.00
C′2 : 12 : 1playⓘ Octave[3][11] or diapason[4]1, 123MS

See also

Notes

References

  1. 1 2 Fox, Christopher (2003). "Microtones and Microtonalities", Contemporary Music Review, v. 22, pt. 1–2. (Abingdon, Oxfordshire, UK: Routledge): p. 13.
  2. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 Fonville, John. 1991. "Ben Johnston's Extended Just Intonation: A Guide for Interpreters". Perspectives of New Music 29, no. 2 (Summer): 106–137.
  3. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 "List of intervals", Huygens-Fokker Foundation. The Foundation uses "classic" to indicate "just" or leaves off any adjective, as in "major sixth".
  4. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Partch, Harry (1979). Genesis of a Music. pp. 68–69. ISBN 978-0-306-80106-8.
  5. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 "Anatomy of an Octave", Kyle Gann (1998). Gann leaves off "just" but includes "5-limit". He uses "median" for "neutral".
  6. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 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. 1 2 Partch 1979, p. 70
  11. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 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. 1 2 3 4 5 6 Anger, Joseph Humfrey (1912). A Treatise on Harmony, with Exercises, Volume 3, pp. xiv–xv. W. Tyrrell.
  14. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 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. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 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. 1 2 "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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