List of pitch intervals

Comparison between tunings: Pythagorean, equal-tempered, 1/4-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 factorized into 2·5 (and 9 in 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.

List

ColumnLegend
TETX-tone equal temperament (12-tet, etc.).
Limit2-limit intonation, or octave(s).
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.
MMeantone temperament or tuning.
SSuperparticular ratio (no separate color code).
List of musical intervals
CentsNote (from C)Freq. ratioPrime factorsInterval nameTETLimitMS
0.00C[2]1 : 11 : 1 play Unison,[3] monophony,[4] perfect prime,[3] tonic,[5] or fundamental1, 122M
0.03 65537 : 6553665537 : 216 play Largest known Fermat prime, Sixty-five-thousand-five-hundred-thirty-seventh harmonic 65537S
0.40C-4375 : 437454·7 : 2·37 play Ragisma[3][6]7S
0.72E+2401 : 240074 : 25·3·52 play Breedsma[3][6]7S
1.00 21/120021/1200 play Cent[7]1200
1.20 21/100021/1000 play Millioctave1000
1.95 B++32805 : 3276838·5 : 215 play Schisma[3][5]5
3.99 101/100021/1000·51/1000 play Savart or eptaméride301.03
7.71 B225 : 22432·52 : 25·7 play Septimal kleisma,[3][6] marvel comma7S
8.11 B-15625 : 1555256 : 26·35 play Kleisma or semicomma majeur[3][6]5
10.06 A++2109375 : 209715233·57 : 221 play Semicomma,[3][6] Fokker's comma[3]5
10.85 C160 : 15925·5 : 3·53 play Difference between 5:3 & 53:3253S
11.98C145 : 1445·29 : 24·32 play Difference between 29:16 & 9:529S
12.50 21/9621/96 play Sixteenth tone96
13.07B-1728 : 171526·33 : 5·73 play Orwell comma[3][8]7
13.47C129 : 1283·43 : 27 play Hundred-twenty-ninth harmonic43S
13.79 D126 : 1252·32·7 : 53 play Small septimal semicomma,[6] small septimal comma,[3] starling comma7S
14.37 C-121 : 120112 : 23·3·5 play Undecimal seconds comma[3]11S
16.67 C[lower-alpha 1]21/7221/72 play 1 step in 72 equal temperament72
18.13 C96 : 9525·3 : 5·19 play Difference between 19:16 & 6:519S
19.55 D--[2]2048 : 2025211 : 34·52 play Diaschisma,[3][6] minor comma5
21.51 C+[2]81 : 8034 : 24·5 play Syntonic comma,[3][5][6] major comma, komma, chromatic diesis, or comma of Didymus[3][6][9][10]5S
22.64 21/5321/53 play Holdrian comma, Holder's comma, 1 step in 53 equal temperament53
23.46 B+++531441 : 524288312 : 219 play Pythagorean comma,[3][5][6][9][10] ditonic comma[3][6]3
25.00 21/4821/48 play Eighth tone48
26.84 C65 : 645·13 : 26 play Sixty-fifth harmonic,[5] 13th-partial chroma[3]13S
27.26 C-64 : 6326 : 32·7 play Septimal comma,[3][6][10] Archytas' comma,[3] 63rd subharmonic7S
29.2721/4121/41 play 1 step in 41 equal temperament41
31.19 D56 : 5523·7 : 5·11 play Undecimal diesis[3], Ptolemy's enharmonic:[5] difference between (11 : 8) and (7 : 5) tritone11S
33.33C/D[lower-alpha 1]21/3621/36 play Sixth tone36, 72
34.28 C51 : 503·17 : 2·52 play Difference between 17:16 & 25:2417S
34.98 B-50 : 492·52 : 72 play Septimal sixth tone or jubilisma, Erlich's decatonic comma or tritonic diesis[3][6]7S
35.70 D49 : 4872 : 24·3 play Septimal diesis, slendro diesis or septimal 1/6-tone[3]7S
38.05 C46 : 452·23 : 32·5 play Inferior quarter tone,[5] difference between 23:16 & 45:3223S
38.71 21/3121/31 play 1 step in 31 equal temperament31
38.91 C+45 : 4432·5 : 4·11 play Undecimal diesis or undecimal fifth tone 11S
40.00 21/3021/30 play Fifth tone30
41.06 D-128 : 12527 : 53 play Enharmonic diesis or 5-limit limma, minor diesis,[6] diminished second,[5][6] minor diesis or diesis,[3] 125th subharmonic5
41.72 D42 : 412·3·7 : 41 play Lesser 41-limit fifth tone41S
42.75 C41 : 4041 : 23·5 play Greater 41-limit fifth tone41S
43.83 C40 : 3923·5 : 3·13 play Tridecimal fifth tone13S
44.97 C39 : 383·13 : 2·19 play Superior quarter-tone,[5] novendecimal fifth tone19S
46.17 D-38 : 372·19 : 37 play Lesser 37-limit quarter tone37S
47.43 C37 : 3637 : 22·32 play Greater 37-limit quarter tone37S
48.77 C36 : 3522·32 : 5·7 play Septimal quarter tone, septimal diesis,[3][6] septimal comma,[2] superior quarter tone[5]7S
49.98 246 : 2393·41 : 239 play Just quarter tone[10]239
50.00 C/D21/2421/24 play Equal-tempered quarter tone24
50.18 D35 : 345·7 : 2·17 play ET quarter-tone approximation,[5] lesser 17-limit quarter tone17S
50.72 B++59049 : 57344310 : 213·7 play Harrison's comma (10 P5s - 1 H7)[3]7
51.68 C34 : 332·17 : 3·11 play Greater 17-limit quarter tone17S
53.27 C33 : 323·11 : 25 play Thirty-third harmonic,[5] undecimal comma, undecimal quarter tone11S
54.96 D-32 : 3125 : 31 play Inferior quarter-tone,[5] thirty-first subharmonic31S
56.55 B+529 : 512232 : 29 play Five-hundred-twenty-ninth harmonic23
56.77 C31 : 3031 : 2·3·5 play Greater quarter-tone,[5] difference between 31:16 & 15:831S
58.69 C30 : 292·3·5 : 29 play Lesser 29-limit quarter tone29S
60.75 C29 : 2829 : 22·7 play Greater 29-limit quarter tone29S
62.96 D-28 : 2722·7 : 33 play Septimal minor second, small minor second, inferior quarter tone[5]7S
63.81 (3 : 2)1/1131/11 : 21/11 play Beta scale step18.75
65.34 C+27 : 2633 : 2·13  play Chromatic diesis,[11] tridecimal comma[3]13S
66.34 D133 : 1287·19 : 27 play One-hundred-thirty-third harmonic19
66.67 C/C[lower-alpha 1]21/1821/18 play Third tone18, 36, 72
67.90 D-26 : 252·13 : 52 play Tridecimal third tone, third tone[5]13S
70.67 C[2]25 : 2452 : 23·3 play Just chromatic semitone or minor chroma,[3] lesser chromatic semitone, small (just) semitone[10] or minor second,[4] minor chromatic semitone,[12] or minor semitone,[5] 2/7-comma meantone chromatic semitone, augmented unison5S
73.68 D-24 : 2323·3 : 23 play Lesser 23-limit semitone23S
75.00 21/1623/48 play 1 step in 16 equal temperament, 3 steps in 4816, 48
76.96 C+23 : 2223 : 2·11 play Greater 23-limit semitone23S
78.00 (3 : 2)1/931/9 : 21/9 play Alpha scale step15.39
79.31 67 : 6467 : 26 play Sixty-seventh harmonic[5]67
80.54 C-22 : 212·11 : 3·7 play Hard semitone,[5] two-fifth tone small semitone11S
84.47 D21 : 203·7 : 22·5 play Septimal chromatic semitone, minor semitone[3]7S
88.80 C20 : 1922·5 : 19 play Novendecimal augmented unison19S
90.22 D--[2]256 : 24328 : 35 play Pythagorean minor second or limma,[3][6][10] Pythagorean diatonic semitone, Low Semitone[13]3
92.18 C+[2]135 : 12833·5 : 27 play Greater chromatic semitone, chromatic semitone, semitone medius, major chroma or major limma,[3] small limma,[10] major chromatic semitone,[12] limma ascendant[5]5
93.60 D-19 : 1819 : 2·9Novendecimal minor second play 19S
97.36 D↓↓128 : 12127 : 112 play 121st subharmonic[5][6]11
98.95 D18 : 172·32 : 17 play Just minor semitone, Arabic lute index finger[3]17S
100.00 C/D21/1221/12 play Equal-tempered minor second or semitone12M
104.96 C[2]17 : 1617 : 24 play Minor diatonic semitone, just major semitone, overtone semitone,[5] 17th harmonic,[3] limma17S
111.45 (5 : 1)1/25 play Studie II interval (compound just major third, 5:1, divided into 25 equal parts)25
111.73 D-[2]16 : 1524 : 3·5 play Just minor second,[14] just diatonic semitone, large just semitone or major second,[4] major semitone,[5] limma, minor diatonic semitone,[3] diatonic second[15] semitone,[13] diatonic semitone,[10] 1/6-comma meantone minor second5S
113.69 C++2187 : 204837 : 211 play Apotome[3][10] or Pythagorean major semitone,[6] Pythagorean augmented unison, Pythagorean chromatic semitone, or Pythagorean apotome3
116.72 (18 : 5)1/1921/19·32/19 : 51/19 play Secor10.28
119.44 C15 : 143·5 : 2·7 play Septimal diatonic semitone, major diatonic semitone,[3] Cowell semitone[5]7S
125.00 25/4825/48 play 5 steps in 48 equal temperament48
128.30 D14 : 132·7 : 13 play Lesser tridecimal 2/3-tone[16]13S
130.23 C+69 : 643·23 : 26 play Sixty-ninth harmonic[5]23
133.24 D27 : 2533 : 52 play Semitone maximus, minor second, large limma or Bohlen-Pierce small semitone,[3] high semitone,[13] alternate Renaissance half-step,[5] large limma, acute minor second5
133.33 C/D[lower-alpha 1]21/922/18 play Two-third tone9, 18, 36, 72
138.57 D-13 : 1213 : 22·3 play Greater tridecimal 2/3-tone,[16] Three-quarter tone[5]13S
150.00 C/D23/2421/8 play Equal-tempered neutral second8, 24
150.64 D↓[2]12 : 1122·3 : 11 play 3/4-tone or Undecimal neutral second,[3][5] trumpet three-quarter tone,[10] middle finger [between frets][13]11S
155.14 D35 : 325·7 : 25 play Thirty-fifth harmonic[5]7
160.90 D--800 : 72925·52 : 36 play Grave whole tone,[3] neutral second, grave major second5
165.00 D-[2]11 : 1011 : 2·5 play Greater undecimal minor/major/neutral second, 4/5-tone[6] or Ptolemy's second[3]11S
171.43 21/721/7 play 1 step in 7 equal temperament7
175.00 27/4827/48 play 7 steps in 48 equal temperament48
179.70 71 : 6471 : 26 play Seventy-first harmonic[5]71
180.45 E---65536 : 59049216 : 310 play Pythagorean diminished third,[3][6] Pythagorean minor tone3
182.40 D-[2]10 : 92·5 : 32 play Small just whole tone or major second,[4] minor whole tone,[3][5] lesser whole tone,[15] minor tone,[13] minor second,[10] half-comma meantone major second5S
200.00 D22/1221/6 play Equal-tempered major second6, 12M
203.91 D[2]9 : 832 : 23 play Pythagorean major second, Large just whole tone or major second[10] (sesquioctavan),[4] tonus, major whole tone,[3][5] greater whole tone,[15] major tone[13]3S
215.89 D145 : 1285·29 : 27 play Hundred-forty-fifth harmonic29
223.46 E-[2]256 : 22528 : 32·52 play Just diminished third,[15] 225th subharmonic5
225.00 23/1629/48 play 9 steps in 48 equal temperament16, 48
227.79 73 : 6473 : 26 play Seventy-third harmonic[5]73
231.17 D-[2]8 : 723 : 7 play Septimal major second,[4] septimal whole tone[3][5]7S
240.00 21/521/5 play 1 step in 5 equal temperament5
247.74 D15 : 133·5 : 13 play Tridecimal 5/4 tone[3]13
250.00 25/2425/24 play 5 steps in 24 equal temperament24
251.34 D37 : 3237 : 25 play Thirty-seventh harmonic[5]37
253.08 D-125 : 10853 : 22·33 play Semi-augmented whole tone,[3] semi-augmented second5
262.37 E↓64 : 5526 : 5·11 play 55th subharmonic[5][6]11
268.80 D299 : 25613·23 : 28 play Two-hundred-ninety-ninth harmonic23
266.87 E[2]7 : 67 : 2·3 play Septimal minor third[3][4][10] or Sub minor third[13]7S
274.58 D[2]75 : 643·52 : 26 play Just augmented second,[15] Augmented tone,[13] augmented second[5][12]5
275.00 211/48211/48 play 11 steps in 48 equal temperament48
289.21 E13 : 1113 : 11 play Tridecimal minor third[3]13S
294.13 E-[2]32 : 2725 : 33 play Pythagorean minor third[3][5][6][13][15] semiditone, or 27th subharmonic3
297.51 E[2]19 : 1619 : 24 play 19th harmonic,[3] 19-limit minor third, overtone minor third[5]19
300.00 D/E23/1221/4 play Equal-tempered minor third4, 12M
301.85 D-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)1/422 : 53/4 play Quarter-comma meantone minor thirdM
311.98 (3 : 2)4/934/9 : 24/9 play Alpha scale minor third3.85
315.64 E[2]6 : 52·3 : 5 play Just minor third,[3][4][5][10][15] minor third,[13] 1/3-comma meantone minor third5MS
317.60 D++19683 : 1638439 : 214 play Pythagorean augmented second[3][6]3
320.14 E77 : 647·11 : 26 play Seventy-seventh harmonic[5]11
325.00 213/48213/48 play 13 steps in 48 equal temperament48
336.13 D-17 : 1417 : 2·7 play Superminor third[17]17
337.15 E+243 : 20035 : 23·52 play Acute minor third[3]5
342.48 E39 : 323·13 : 25 play Thirty-ninth harmonic[5]13
342.86 22/722/7 play 2 steps in 7 equal temperament7
342.91 E-128 : 10527 : 3·5·7 play 105th subharmonic,[5] septimal neutral third[6]7
347.41 E-[2]11 : 911 : 32 play Undecimal neutral third[3][5]11
350.00 D/E27/2427/24 play Equal-tempered neutral third24
354.55 E+27 : 2233 : 2·11 play Zalzal's wosta[6] 12:11 X 9:8[13]11
359.47 E[2]16 : 1324 : 13 play Tridecimal neutral third[3]13
364.54 79 : 6479 : 26 play Seventy-ninth harmonic[5]79
364.81 E-100 : 8122·52 : 34 play Grave major third[3]5
375.00 25/16215/48 play 15 steps in 48 equal temperament16, 48
384.36 F--8192 : 6561213 : 38 play Pythagorean diminished fourth,[3][6] Pythagorean 'schismatic' third[5]3
386.31 E[2]5 : 45 : 22 play Just major third,[3][4][5][10][15] major third,[13] quarter-comma meantone major third5MS
397.10 E+161 : 1287·23 : 27 play One-hundred-sixty-first harmonic23
400.00 E24/1221/3 play Equal-tempered major third3, 12M
402.47 E323 : 25617·19 : 28 play Three-hundred-twenty-third harmonic19
407.82 E+[2]81 : 6434 : 26 play Pythagorean major third,[3][5][6][13][15] ditone3
417.51 F+[2]14 : 112·7 : 11 play Undecimal diminished fourth or major third[3]11
425.00 217/48217/48 play 17 steps in 48 equal temperament48
427.37 F[2]32 : 2525 : 52 play Just diminished fourth,[15] diminished fourth,[5][12] 25th subharmonic5
429.06 E41 : 3241 : 25 play Forty-first harmonic[5]41
435.08 E[2]9 : 732 : 7 play Septimal major third,[3][5] Bohlen-Pierce third,[3] Super major Third[13]7
444.77 F↓128 : 9927 : 9·11 play 99th subharmonic[5][6]11
450.00 29/2429/24 play 9 steps in 24 equal temperament24
450.05 83 : 6483 : 26 play Eighty-third harmonic[5]83
454.21 F13 : 1013 : 2·5 play Tridecimal major third or diminished fourth13
456.99 E[2]125 : 9653 : 25·3 play Just augmented third, augmented third[5]5
462.35 E-64 : 4926 : 72 play 49th subharmonic[5][6]7
470.78 F+[2]21 : 163·7 : 24 play Twenty-first harmonic, narrow fourth,[3] septimal fourth,[5] wide augmented third, H7 on G7
475.00 219/48219/48 play 19 steps in 48 equal temperament48
478.49 E+675 : 51233·52 : 29 play Six-hundred-seventy-fifth harmonic, wide augmented third[3]5
480.00 22/522/5 play 2 steps in 5 equal temperament5
491.27 E85 : 645·17 : 26 play Eighty-fifth harmonic[5]17
498.04 F[2]4 : 322 : 3 play Perfect fourth,[3][5][15] Pythagorean perfect fourth, Just perfect fourth or diatessaron[4]3S
500.00 F25/1225/12 play Equal-tempered perfect fourth12M
501.42 F+171 : 12832·19 : 27 play One-hundred-seventy-first harmonic19
510.51 (3 : 2)8/1138/11 : 28/11 play Beta scale perfect fourth18.75
511.52 F43 : 3243 : 25 play Forty-third harmonic[5]43
514.29 23/723/7 play 3 steps in 7 equal temperament7
519.55 F+[2]27 : 2033 : 22·5 play 5-limit wolf fourth, acute fourth,[3] imperfect fourth[15]5
521.51 E+++177147 : 131072311 : 217 play Pythagorean augmented third[3][6] (F+ (pitch))3
525.00 27/16221/48 play 21 steps in 48 equal temperament16, 48
531.53 F+87 : 643·29 : 26 play Eighty-seventh harmonic[5]29
536.95 F+15 : 113·5 : 11 play Undecimal augmented fourth[3]11
550.00 211/24211/24 play 11 steps in 24 equal temperament24
551.32 F[2]11 : 811 : 23 play eleventh harmonic,[5] undecimal tritone,[5] lesser undecimal tritone, undecimal semi-augmented fourth[3]11
563.38 F+18 : 132·9 : 13 play Tridecimal augmented fourth[3]13
568.72 F[2]25 : 1852 : 2·32 play Just augmented fourth[3][5]5
570.88 89 : 6489 : 26 play Eighty-ninth harmonic[5]89
575.00 223/48223/48 play 23 steps in 48 equal temperament48
582.51 G[2]7 : 57 : 5 play Lesser septimal tritone, septimal tritone[3][4][5] Huygens' tritone or Bohlen-Pierce fourth,[3] septimal fifth,[10] septimal diminished fifth[18]7
588.27 G--1024 : 729210 : 36 play Pythagorean diminished fifth,[3][6] low Pythagorean tritone[5]3
590.22 F+[2]45 : 3232·5 : 25 play Just augmented fourth, just tritone,[4][10] tritone,[6] diatonic tritone,[3] 'augmented' or 'false' fourth,[15] high 5-limit tritone,[5] 1/6-comma meantone augmented fourth5
595.03 G361 : 256192 : 28 play Three-hundred-sixty-first harmonic19
600.00 F/G26/1221/2=2 play Equal-tempered tritone2, 12M
609.35 G91 : 647·13 : 26 play Ninety-first harmonic[5]13
609.78 G-[2]64 : 4526 : 32·5 play Just tritone,[4] 2nd tritone,[6] 'false' fifth,[15] diminished fifth,[12] low 5-limit tritone,[5] 45th subharmonic5
611.73 F++729 : 51236 : 29 play Pythagorean tritone,[3][6] Pythagorean augmented fourth, high Pythagorean tritone[5]3
617.49 F[2]10 : 72·5 : 7 play Greater septimal tritone, septimal tritone,[4][5] Euler's tritone[3]7
625.00 225/48225/48 play 25 steps in 48 equal temperament48
628.27 F+23 : 1623 : 24 play Twenty-third harmonic,[5] classic diminished fifth23
631.28 G[2]36 : 2522·32 : 52 play Just diminished fifth[5]5
646.99 F+93 : 643·31 : 26 play Ninety-third harmonic[5]31
648.68 G↓[2]16 : 1124 : 11 play ` undecimal semi-diminished fifth[3]11
650.00 213/24213/24 play 13 steps in 24 equal temperament24
665.51 G47 : 3247 : 25 play Forty-seventh harmonic[5]47
675.00 29/16227/48 play 27 steps in 48 equal temperament16, 48
678.49 A---262144 : 177147218 : 311 play Pythagorean diminished sixth[3][6]3
680.45 G-40 : 2723·5 : 33 play 5-limit wolf fifth,[5] or diminished sixth, grave fifth,[3][6][10] imperfect fifth,[15]5
683.83 G95 : 645·19 : 26 play Ninety-fifth harmonic[5]19
684.82 E++12167 : 8192233 : 213 play 12167th harmonic23
685.71 24/7 : 1 play 4 steps in 7 equal temperament
691.20 3:2÷(81:80)1/22·51/2 : 3 play Half-comma meantone perfect fifthM
694.79 3:2÷(81:80)1/321/3·51/3 : 31/3 play 1/3-comma meantone perfect fifthM
695.81 3:2÷(81:80)2/721/7·52/7 : 31/7 play 2/7-comma meantone perfect fifthM
696.58 3:2÷(81:80)1/451/4 play Quarter-comma meantone perfect fifthM
697.65 3:2÷(81:80)1/531/5·51/5 : 21/5 play 1/5-comma meantone perfect fifthM
698.37 3:2÷(81:80)1/631/3·51/6 : 21/3 play 1/6-comma meantone perfect fifthM
700.00 G27/1227/12 play Equal-tempered perfect fifth12M
701.89 231/53231/53 play 53-TET perfect fifth53
701.96 G[2]3 : 23 : 2 play Perfect fifth,[3][5][15] Pythagorean perfect fifth, Just perfect fifth or diapente,[4] fifth,[13] Just fifth[10]3S
702.44 224/41224/41 play 41-TET perfect fifth41
703.45 217/29217/29 play 29-TET perfect fifth29
719.90 97 : 6497 : 26 play Ninety-seventh harmonic[5]97
720.00 23/5 : 1 play 3 steps in 5 equal temperament5
721.51 A-1024 : 675210 : 33·52 play Narrow diminished sixth[3]5
725.00 229/48229/48 play 29 steps in 48 equal temperament48
729.22 G-32 : 2124 : 3·7 play 21st subharmonic[5][6], septimal diminished sixth7
733.23 F+391 : 25617·23 : 28 play Three-hundred-ninety-first harmonic23
737.65 A+49 : 327·7 : 25 play Forty-ninth harmonic[5]7
743.01 A192 : 12526·3 : 53 play Classic diminished sixth[3]5
750.00 215/24215/24 play 15 steps in 48 equal temperament24
755.23 G99 : 6432·11 : 26 play Ninety-ninth harmonic[5]11
764.92 A[2]14 : 92·7 : 32 play Septimal minor sixth[3][5]7
772.63 G25 : 1652 : 24 play Just augmented fifth[5][15]
775.00 231/48231/48 play 31 steps in 48 equal temperament48
781.79 π : 2 play Wallis product
782.49 G-[2]11 : 711 : 7 play Undecimal minor sixth,[5] undecimal augmented fifth,[3] Fibonacci numbers11
789.85 101 : 64101 : 26 play Hundred-first harmonic[5]101
792.18 A-[2]128 : 8127 : 34 play Pythagorean minor sixth,[3][5][6] 81st subharmonic3
798.40 A+203 : 1287·29 : 27 play Two-hundred-third harmonic29
800.00 G/A28/1222/3 play Equal-tempered minor sixth3, 12M
806.91 G51 : 323·17 : 25 play Fifty-first harmonic[5]17
813.69 A[2]8 : 523 : 5 play Just minor sixth[3][4][10][15]5
815.64 G++6561 : 409638 : 212 play Pythagorean augmented fifth,[3][6] Pythagorean 'schismatic' sixth[5]3
823.80 103 : 64103 : 26 play Hundred-third harmonic[5]103
825.00 211/16233/48 play 33 steps in 48 equal temperament16, 48
832.18 G+207 : 12832·23 : 27 play Two-hundred-seventh harmonic23
833.09 51/2+1 : 2  : 2 play Golden ratio (833 cents scale)
833.11 233 : 144233 : 24·32 play Golden ratio approximation (833 cents scale) 233
835.19 A+81 : 5034 : 2·52 play Acute minor sixth[3]5
840.53 A[2]13 : 813 : 23 play Tridecimal neutral sixth,[3] overtone sixth,[5] thirteenth harmonic13
848.83 A209 : 12811·19 : 27 play Two-hundred-ninth harmonic19
850.00 G/A217/24217/24 play Equal-tempered neutral sixth24
852.59 A↓+[2]18 : 112·32 : 11 play Undecimal neutral sixth,[3][5] Zalzal's neutral sixth11
857.09 A+105 : 643·5·7 : 26 play Hundred-fifth harmonic[5]7
857.14 25/725/7 play 5 steps in 7 equal temperament7
862.85 A-400 : 24324·52 : 35 play Grave major sixth[3]5
873.50 A53 : 3253 : 25 play Fifty-third harmonic[5]53
875.00 235/48235/48 play 35 steps in 48 equal temperament48
879.86 A↓128 : 7727 : 7·11 play 77th subharmonic[5][6]11
882.40 B---32768 : 19683215 : 39 play Pythagorean diminished seventh[3][6]3
884.36 A[2]5 : 35 : 3 play Just major sixth,[3][4][5][10][15] Bohlen-Pierce sixth,[3] 1/3-comma meantone major sixth5M
889.76 107 : 64107 : 26 play Hundred-seventh harmonic[5]107
892.54 B6859 : 4096193 : 212 play 6859th harmonic19
900.00 A29/1223/4 play Equal-tempered major sixth4, 12M
902.49 A32 : 1925 : 19 play 19th subharmonic[5][6]19
905.87 A+[2]27 : 1633 : 24 play Pythagorean major sixth[3][5][10][15]3
921.82 109 : 64109 : 26 play Hundred-ninth harmonic[5]109
925.00 237/48237/48 play 37 steps in 48 equal temperament48
925.42 B-[2]128 : 7527 : 3·52 play Just diminished seventh,[15] diminished seventh,[5][12] 75th subharmonic5
925.79 A+437 : 25619·23 : 28 play Four-hundred-thirty-seventh harmonic23
933.13 A[2]12 : 722·3 : 7 play Septimal major sixth[3][4][5]7
937.63 A55 : 325·11 : 25 play Fifty-fifth harmonic[5][19]11
950.00 219/24219/24 play 19 steps in 24 equal temperament24
953.30 A+111 : 643·37 : 26 play Hundred-eleventh harmonic[5]37
955.03 A[2]125 : 7253 : 23·32 play Just augmented sixth[5]5
957.21 (3 : 2)15/11315/11 : 215/11 play 15 steps in Beta scale18.75
960.00 24/524/5 play 4 steps in 5 equal temperament5
968.83 B[2]7 : 47 : 22 play Septimal minor seventh,[4][5][10] harmonic seventh,[3][10] augmented sixth7
975.00 213/16239/48 play 39 steps in 48 equal temperament16, 48
976.54 A+[2]225 : 12832·52 : 27 play Just augmented sixth[15]5
984.21 113 : 64113 : 26 play Hundred-thirteenth harmonic[5]113
996.09B-[2]16 : 924 : 32 play Pythagorean minor seventh,[3] Small just minor seventh,[4] lesser minor seventh,[15] just minor seventh,[10] Pythagorean small minor seventh[5]3
999.47 B57 : 323·19 : 25 play Fifty-seventh harmonic[5]19
1000.00A/B210/1225/6 play Equal-tempered minor seventh6, 12M
1014.59 A+115 : 645·23 : 26 play Hundred-fifteenth harmonic[5]23
1017.60B[2]9 : 532 : 5 play Greater just minor seventh,[15] large just minor seventh,[4][5] Bohlen-Pierce seventh[3]5
1019.55 A+++59049 : 32768310 : 215 play Pythagorean augmented sixth[3][6]3
1025.00 241/48241/48 play 41 steps in 48 equal temperament48
1028.57 26/726/7 play 6 steps in 7 equal temperament7
1029.58 B29 : 1629 : 24 play Twenty-ninth harmonic,[5] minor seventh29
1035.00B↓[2]20 : 1122·5 : 11 play Lesser undecimal neutral seventh, large minor seventh[3]11
1039.10 B+729 : 40036 : 24·52 play Acute minor seventh[3]5
1044.44 B117 : 6432·13 : 26 play Hundred-seventeenth harmonic[5]13
1044.86 B-64 : 3526 : 5·7 play 35th subharmonic,[5] septimal neutral seventh[6]7
1049.36B-[2]11 : 611 : 2·3 play 21/4-tone or Undecimal neutral seventh,[3] undecimal 'median' seventh[5]11
1050.00 A/B221/2427/8 play Equal-tempered neutral seventh8, 24
1059.17 59 : 3259 : 25 play Fifty-ninth harmonic[5]59
1066.76 B-50 : 272·52 : 33 play Grave major seventh[3]5
1071.70 B-13 : 713 : 7 play Tridecimal neutral seventh[20]13
1073.78 B119 : 647·17 : 26 play Hundred-nineteenth harmonic[5]17
1075.00 243/48243/48 play 43 steps in 48 equal temperament48
1086.31 C--4096 : 2187212 : 37 play Pythagorean diminished octave[3][6]3
1088.27 B[2]15 : 83·5 : 23 play Just major seventh,[3][5][10][15] small just major seventh,[4] 1/6-comma meantone major seventh5
1095.04 C32 : 1725 : 17 play 17th subharmonic[5][6]17
1100.00 B211/12211/12 play Equal-tempered major seventh12M
1102.64 B-121 : 64112 : 26 play Hundred-twenty-first harmonic[5]11
1107.82 C'-256 : 13528 : 33·5 play Octave − major chroma,[3] 135th subharmonic, narrow diminished octave5
1109.78 B+[2]243 : 12835 : 27 play Pythagorean major seventh[3][5][6][10]3
1116.88 61 : 3261 : 25 play Sixty-first harmonic[5]61
1125.00 215/16245/48 play 45 steps in 48 equal temperament16, 48
1129.33 C'[2]48 : 2524·3 : 52 play Classic diminished octave,[3][6] large just major seventh[4]5
1131.02 B123 : 643·41 : 26 play Hundred-twenty-third harmonic[5]41
1137.04 B27 : 1433 : 2·7 play Septimal major seventh[5]7
1138.04 C247 : 12813·19 : 27 play Two-hundred-forty-seventh harmonic19
1145.04 B31 : 1631 : 24 play Thirty-first harmonic,[5] augmented seventh31
1146.73 C↓64 : 3326 : 3·11 play 33rd subharmonic[6]11
1150.00 223/24223/24 play 23 steps in 24 equal temperament24
1151.23 C35 : 185·7 : 2·32 play Septimal supermajor seventh, septimal quarter tone inverted 7
1158.94 B[2]125 : 6453 : 26 play Just augmented seventh,[5] 125th harmonic5
1172.74 C+63 : 3232·7 : 25 play Sixty-third harmonic[5]7
1175.00 247/48247/48 play 47 steps in 48 equal temperament48
1178.49 C'-160 : 8125·5 : 34 play Octave − syntonic comma,[3] semi-diminished octave5
1179.59 B253 : 12811·23 : 27 play Two-hundred-fifty-third harmonic[5]23
1186.42 127 : 64127 : 26 play Hundred-twenty-seventh harmonic[5]127
1200.00 C'2 : 12 : 1 play Octave[3][10] or diapason[4]1, 122MS
1223.46 B+++531441 : 262144312 : 218 play Pythagorean augmented seventh[3][6]3
1525.86 21/2+1 play Silver ratio
1901.96 G'3 : 13 : 1 play Tritave or just perfect twelfth3
2400.00 C"4 : 122 : 1 play Fifteenth or two octaves1, 122M
3986.31 E'''10 : 15·2 : 1 play Decade, compound just major third5M

See also

Notes

  1. 1 2 3 4 Maneri-Sims notation

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): 106–37.
  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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 "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, p.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", KyleGann.com. 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 52 Haluška, Ján (2003). The Mathematical Theory of Tone Systems, p.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.
  8. "Orwell Temperaments", Xenharmony.org.
  9. 1 2 Partch (1979), p.70.
  10. 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 (1885). On the musical scales of various nations, p.488. s.n.
  11. William Smythe Babcock Mathews (1895). Pronouncing dictionary and condensed encyclopedia of musical terms, p.13. ISBN 1-112-44188-3.
  12. 1 2 3 4 5 6 Anger, Joseph Humfrey (1912). A treatise on harmony, with exercises, Volume 3, p.xiv-xv. W. Tyrrell.
  13. 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. No ISBN specified.
  14. A. R. Meuss (2004). Intervals, Scales, Tones and the Concert Pitch C. Temple Lodge Publishing. p. 15. ISBN 1902636465.
  15. 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".
  16. 1 2 "13th-harmonic", 31et.com.
  17. Brabner, John H. F. (1884). The National Encyclopaedia, Vol.13, p.182. London. [ISBN unspecified]
  18. Sabat, Marc and von Schweinitz, Wolfgang (2004). "The Extended Helmholtz-Ellis JI Pitch Notation" [PDF], NewMusicBox.org. Accessed: 04:12, 15 March 2014 (UTC).
  19. Hermann L. F Von Helmholtz (2007). On the Sensations of Tone, p.456. ISBN 978-1-60206-639-7.
  20. "Gallery of Just Intervals", Xenharmonic.wikispaces.com.
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