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 .

Terminology The prime limitFox, Christopher (2003). "Microtones and Microtonalities", Contemporary Music Review, v. 22, pt. 1–2. (Abingdon, Oxfordshire, UK: Routledge): p. 13. 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 (and 9 into ). 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. 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 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. , the mean of the major third , and the fifth (3:2) is not tempered; and the -comma meantone, where the fifth is tempered to the extent that three ascending fifths produce a pure minor third. (See meantone temperaments). 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 CFonville, John. 1991. "Ben Johnston's Extended Just Intonation: A Guide for Interpreters". Perspectives of New Music 29, no. 2 (Summer): 106–137. 1 : 1 1 : 1 1, 12 3 M 65537 : 65536 65537 : 216 65537 S C− 4375 : 4374 7 S E+ 2401 : 2400 7 S 21/1200 21/1200 1200 21/1000 21/1000 1000 B++ 38×5 : 215 5 3 : 219/12 101/1000 21/1000×51/1000 301.03 B 225 : 224 7 S B− 15625 : 15552 56 : 26×35 5 A++ 2109375 : 2097152 33×57 : 221 5 C 160 : 159 25×5 : 3×53 53 S C 145 : 144 29 S 21/96 21/96 96 B− 1728 : 1715 26×33 : 5×73 7 C 129 : 128 3×43 : 27 43 S D 126 : 125 2×32×7 : 53 7 S C− 121 : 120 112 : 23×3×5 11 S C 21/72 21/72 72 C 96 : 95 25×3 : 5×19 19 S D-- 2048 : 2025 5 C+ 81 : 80 34 : 24×5 5 S 21/53 21/53 53 B+++ 531441 : 524288 312 : 219 3 21/48 21/48 48 C 65 : 64 5×13 : 26 13 S C− 64 : 63 26 : 32×7 7 S 21/41 21/41 41 D 56 : 55 23×7 : 5×11 11 S C/D 21/36 21/36 36, 72 C 51 : 50 17 S B- 50 : 49 2×52 : 72 7 S D 49 : 48 72 : 24×3 7 S C 46 : 45 2×23 : 32×5 23 S 21/31 21/31 31 C 45 : 44 32×5 : 4×11 11 S 21/30 21/30 30 D− 128 : 125 27 : 53 5 D 42 : 41 2×3×7 : 41 41 S C 41 : 40 41 : 23×5 41 S C 40 : 39 23×5 : 3×13 13 S C 39 : 38 3×13 : 2×19 19 S D 38 : 37 2×19 : 37 37 S C 37 : 36 37 : 22×32 37 S C 36 : 35 22×32 : 5×7 7 S 246 : 239 3×41 : 239 239 C/D 21/24 21/24 24 D 35 : 34 5×7 : 2×17 17 S B 59049 : 57344 310 : 213×7 7 C 34 : 33 2×17 : 3×11 17 S C 33 : 32 3×11 : 25 11 S D 32 : 31 25 : 31 31 S B 529 : 512 232 : 29 23 C 31 : 30 31 : 2×3×5 31 S C 30 : 29 2×3×5 : 29 29 S C 29 : 28 29 : 22×7 29 S D 28 : 27 22×7 : 33 7 S (3 : 2)1/11 31/11 : 21/11 18.80 C+ 27 : 26 33 : 2×13 13 S D 133 : 128 7×19 : 27 19 C/C 21/18 21/18 18, 36, 72 D 26 : 25 2×13 : 52 13 S C 25 : 24 52 : 23×3 5 S D 24 : 23 23×3 : 23 23 S 21/16 23/48 16, 48 C 23 : 22 23 : 2×11 23 S (3 : 2)1/9 31/9 : 21/9 15.39 67 : 64 67 : 26 67 C 22 : 21 2×11 : 3×7 11 S D 21 : 20 3×7 : 22×5 7 S C 20 : 19 22×5 : 19 19 S D−− 256 : 243 28 : 35 3 C+ 135 : 128 33×5 : 27 5 D 19 : 18 19 : 2×32 19 S D↓↓ 128 : 121 27 : 112 11 D 18 : 17 2×32 : 17 17 S C/D 21/12 21/12 12 M C 17 : 16 17 : 24 17 S (5 : 1)1/25 10.77 D 16 : 15 24 : 3×5 5 S C++ 2187 : 2048 37 : 211 3 (18 : 5)1/19 21/19×32/19 : 51/19 10.28 C 15 : 14 3×5 : 2×7 7 S 25/48 25/48 48 D 14 : 13 2×7 : 13 13 S C+ 69 : 64 3×23 : 26 23 D 27 : 25 33 : 52 5 C/D 21/9 22/18 9, 18, 36, 72 D 13 : 12 13 : 22×3 13 S C/D 23/24 21/8 8, 24 D↓ 12 : 11 22×3 : 11 11 S D 35 : 32 5×7 : 25 7 D−− 800 : 729 25×52 : 36 5 D− 11 : 10 11 : 2×5 11 S 21/7 21/7 7 27/48 27/48 48 71 : 64 71 : 26 71 E−−− 65536 : 59049 216 : 310 3 D− 10 : 9 2×5 : 32 5 S D 22/12 21/6 6, 12 M D 9 : 8 32 : 23 3 S D 145 : 128 5×29 : 27 29 E− 256 : 225 28 : 32×52 5 23/16 29/48 16, 48 73 : 64 73 : 26 73 D− 8 : 7 23 : 7 7 S 21/5 21/5 5 D 15 : 13 3×5 : 13 13 D/E 25/24 25/24 24 D 37 : 32 37 : 25 37 D− 125 : 108 53 : 22×33 5 E↓ 64 : 55 26 : 5×11 11 E 7 : 6 7 : 2×3 7 S D 299 : 256 13×23 : 28 23 D 75 : 64 3×52 : 26 5 211/48 211/48 48 E 13 : 11 13 : 11 13 E− 32 : 27 25 : 33 3 E 19 : 16 19 : 24 19 D/E 23/12 21/4 4, 12 M D 25 : 21 52 : 3×7 7 6:5÷(81:80)1/4 22 : 53/4 M (3 : 2)4/9 34/9 : 24/9 15.39 E 6 : 5 2×3 : 5 5 M S D++ 19683 : 16384 39 : 214 3 E 77 : 64 7×11 : 26 11 213/48 213/48 48 D 17 : 14 17 : 2×7 17 E+ 243 : 200 35 : 23×52 5 E 39 : 32 3×13 : 25 13 22/7 22/7 7 E 128 : 105 27 : 3×5×7 7 E− 11 : 9 11 : 32 11 D/E 27/24 27/24 24 E 27 : 22 33 : 2×11 11 E 16 : 13 24 : 13 13 79 : 64 79 : 26 79 E− 100 : 81 22×52 : 34 5 25/16 215/48 16, 48 F−− 8192 : 6561 213 : 38 3 E 5 : 4 5 : 22 5 M S E 161 : 128 7×23 : 27 23 E 24/12 21/3 3, 12 M E 323 : 256 17×19 : 28 19 E+ 81 : 64 34 : 26 3 F+ 14 : 11 2×7 : 11 11 217/48 217/48 48 F 32 : 25 25 : 52 5 E 41 : 32 41 : 25 41 E 9 : 7 32 : 7 7 F↓ 128 : 99 27 : 32×11 11 E/F 29/24 29/24 8, 24 83 : 64 83 : 26 83 F 13 : 10 13 : 2×5 13 E 125 : 96 53 : 25×3 5 E 64 : 49 26 : 72 7 F+ 21 : 16 3×7 : 24 7 219/48 219/48 48 E+ 675 : 512 33×52 : 29 5 22/5 22/5 5 E 85 : 64 17 F 4 : 3 22 : 3 3 S F 25/12 25/12 12 M F 171 : 128 32×19 : 27 19 (3 : 2)8/11 38/11 : 28/11 18.80 F 43 : 32 43 : 25 43 23/7 23/7 7 F+ 27 : 20 33 : 22×5 5 E+++ 177147 : 131072 311 : 217 3 27/16 221/48 16, 48 F+ 87 : 64 3×29 : 26 29 F 15 : 11 3×5 : 11 11 F/G 211/24 211/24 24 F 11 : 8 11 : 23 11 F 18 : 13 2×9 : 13 13 F 25 : 18 52 : 2×32 5 89 : 64 89 : 26 89 223/48 223/48 48 G 7 : 5 7 : 5 7 G−− 1024 : 729 210 : 36 3 F+ 45 : 32 32×5 : 25 5 G 361 : 256 192 : 28 19 F/G 26/12 21/2= 2, 12 M G 91 : 64 7×13 : 26 13 G− 64 : 45 26 : 32×5 5 F++ 729 : 512 36 : 29 3 F 10 : 7 2×5 : 7 7 225/48 225/48 48 F+ 23 : 16 23 : 24 23 G 36 : 25 22×32 : 52 5 F+ 93 : 64 3×31 : 26 31 G↓ 16 : 11 24 : 11 11 F/G 213/24 213/24 24 G 47 : 32 47 : 25 47 29/16 227/48 16, 48 A−−− 262144 : 177147 218 : 311 3 G− 40 : 27 23×5 : 33 5 G 95 : 64 5×19 : 26 19 E 12167 : 8192 233 : 213 23 24/7 : 1 7 3:2÷(81:80)1/2 2×51/2 : 3 M 3:2÷(81:80)1/3 21/3×51/3 : 31/3 M 3:2÷(81:80)2/7 21/7×52/7 : 31/7 M 3:2÷(81:80)1/4 51/4 M 3:2÷(81:80)1/5 31/5×51/5 : 21/5 M 3:2÷(81:80)1/6 31/3×51/6 : 21/3 M G 27/12 27/12 12 M 231/53 231/53 53 G 3 : 2 3 : 2 3 S 224/41 224/41 41 217/29 217/29 29 97 : 64 97 : 26 97 23/5 : 1 5 A− 1024 : 675 210 : 33×52 5 229/48 229/48 48 G 32 : 21 24 : 3×7 7 F 391 : 256 17×23 : 28 23 A+ 49 : 32 7×7 : 25 7 A 192 : 125 26×3 : 53 5 G/A 215/24 215/24 8, 24 G 99 : 64 32×11 : 26 11 A 14 : 9 2×7 : 32 7 G 25 : 16 52 : 24 5 231/48 231/48 48 : 2 G- 11 : 7 11 : 7 11 101 : 64 101 : 26 101 A− 128 : 81 27 : 34 3 A 203 : 128 7×29 : 27 29 G/A 28/12 22/3 3, 12 M G 51 : 32 3×17 : 25 17 A 8 : 5 23 : 5 5 G++ 6561 : 4096 38 : 212 3 103 : 64 103 : 26 103 211/16 233/48 16, 48 G 207 : 128 32×23 : 27 23 (51/2+1)/2 : 1 A+ 81 : 50 34 : 2×52 5 A 13 : 8 13 : 23 13 A 209 : 128 11×19 : 27 19 G/A 217/24 217/24 24 A↓+ 18 : 11 2×32 : 11 11 A+ 105 : 64 3×5×7 : 26 7 25/7 25/7 7 A− 400 : 243 24×52 : 35 5 A 53 : 32 53 : 25 53 235/48 235/48 48 A↓ 128 : 77 27 : 7×11 11 B−−− 32768 : 19683 215 : 39 3 A 5 : 3 5 : 3 5 M 107 : 64 107 : 26 107 B 6859 : 4096 193 : 212 19 A 29/12 23/4 4, 12 M A 32 : 19 25 : 19 19 A+ 27 : 16 33 : 24 3 109 : 64 109 : 26 109 237/48 237/48 48 B− 128 : 75 27 : 3×52 5 A 437 : 256 19×23 : 28 23 A 12 : 7 22×3 : 7 7 A 55 : 32 5×11 : 25 11 A/B 219/24 219/24 24 A 111 : 64 3×37 : 26 37 A 125 : 72 53 : 23×32 5 (3 : 2)15/11 315/11 : 215/11 18.80 24/5 24/5 5 B 7 : 4 7 : 22 7 213/16 239/48 16, 48 A+ 225 : 128 32×52 : 27 5 113 : 64 113 : 26 113 B− 16 : 9 24 : 32 3 B 57 : 32 3×19 : 25 19 A/B 210/12 25/6 6, 12 M A+ 115 : 64 5×23 : 26 23 B 9 : 5 32 : 5 5 A+++ 59049 : 32768 310 : 215 3 241/48 241/48 48 26/7 26/7 7 B 29 : 16 29 : 24 29 B↓ 20 : 11 22×5 : 11 11 B+ 729 : 400 36 : 24×52 5 B 117 : 64 32×13 : 26 13 B 64 : 35 26 : 5×7 7 B− 11 : 6 11 : 2×3 11 A/B 221/24 27/8 8, 24 59 : 32 59 : 25 59 B− 50 : 27 2×52 : 33 5 B 13 : 7 13 : 7 13 B 119 : 64 7×17 : 26 17 243/48 243/48 48 C′−− 4096 : 2187 212 : 37 3 B 15 : 8 3×5 : 23 5 C 32 : 17 25 : 17 17 B 211/12 211/12 12 M B- 121 : 64 112 : 26 11 C′− 256 : 135 28 : 33×5 5 B+ 243 : 128 35 : 27 3 61 : 32 61 : 25 61 215/16 245/48 16, 48 C′ 48 : 25 24×3 : 52 5 B 123 : 64 3×41 : 26 41 B 27 : 14 33 : 2×7 7 C 247 : 128 13×19 : 27 19 B 31 : 16 31 : 24 31 C↓ 64 : 33 26 : 3×11 11 B/C 223/24 223/24 24 C 35 : 18 5×7 : 2×32 7 B 125 : 64 53 : 26 5 C+ 63 : 32 32×7 : 25 7 247/48 247/48 48 C′− 160 : 81 25×5 : 34 5 B 253 : 128 11×23 : 27 23 127 : 64 127 : 26 127 C′ 2 : 1 2 : 1 1, 12 3 M S

See also List of chord progressions List of meantone intervals List of musical scales and modes

Notes

References

External links "Names of seven-limit commas", XenHarmony.org. (Archived copy) "List of Overtones", Xenharmonic Wiki. "All Known Musical Intervals" (by Dale Pond), Svpvril.com."