List of superparticular intervals: Difference between revisions

This '''list of superparticular intervals''' ordered by prime limit. It reaches to the 101-limit and is complete up to the [[23-limit]].

[[Superparticular]] numbers are ratios of the form (''n'' + 1)/''n'', or 1 + 1/''n'', where ''n'' is a whole number other than 1. They appear frequently in [[just intonation]] and [[harmonic series]] music. Adjacent tones in the harmonic series are separated by superparticular intervals: for instance, the 20th and 21st by the superparticular ratio [[21/20]]. As the overtones get closer together, the superparticular intervals get smaller and smaller. Thus, an examination of the superparticular intervals is an examination of some of the simplest small intervals in rational tuning systems. Indeed, many but not all common [[comma]]s are superparticular ratios.

The list below is ordered by [[harmonic limit]], or the largest prime involved in the prime factorization. [[36/35]], for instance, is an interval of the [[7-limit]], as it factors to (2²×3²)/(5×7), while 37/36 would belong to the 37-limit.

 

[[Wikipedia:Størmer's theorem|Størmer's theorem]] states that, in each limit, there are only a finite number of superparticular ratios. Many of the sections below are complete. For example, there is no 3-limit superparticular ratio other than [[2/1]], [[3/2]], [[4/3]], and [[9/8]]. [[OEIS: A002071]] gives the number of superparticular ratios in each prime limit, [[OEIS: A145604]] shows the increment from limit to limit, and [[OEIS: A117581]] gives the largest numerator for each prime limit (with some exceptions, such as the 23-limit, where the largest value is smaller than that of a smaller prime limit, in this case the 19-limit).

 

 

{| class="wikitable"

! Ratio

! Cents

! Factorization

! [[Monzo]]

! Name(s)

! Meta

! colspan="6" | 2-limit (complete)

| {{monzo|1}}

| octave, duple; ''after [[octave reduction]]:'' (perfect) unison, unity, perfect prime, tonic

|

|-

! colspan="6" | 3-limit (complete)

| {{monzo|-1 1}}

| [[perfect fifth]], 3rd harmonic (octave reduced), diapente

| {{monzo|2 -1}}

| perfect fourth, 3rd subharmonic (octave reduced), diatessaron

| 3/2 to 2/1

| {{monzo|-3 2}}

| (Pythagorean) (whole) tone, Pythagorean major second, major whole tone, 9th harmonic or harmonic ninth (octave reduced)

| 4/3 to 3/2

|-

! colspan="6" | 5-limit (complete)

| {{monzo|-2 0 1}}

| classic/just major third, 5th harmonic (octave reduced)

|

| (2*3)/5

| {{monzo|1 1 -1}}

| classic/just minor third

|

| (2*5)/3²

| {{monzo|1 -2 1}}

| classic (whole) tone, classic major second, minor whole tone

|

| 2⁴/(3*5)

| {{monzo|4 -1 -1}}

| classic/just diatonic semitone, 15th subharmonic

| 5/4 to 4/3

| 5²/(2³*3)

| {{monzo|-3 -1 2}}

| classic/just chromatic semitone, chroma, Zarlinian semitone

| 6/5 to 5/4

| {{monzo|-4 4 -1}}

| syntonic comma, Didymus comma

| 10/9 to 9/8

|-

! colspan="6" | 7-limit (complete)

| 7/(2*3)

| {{monzo|-1 -1 0 1 }}

| (septimal) subminor third, septimal minor third

|

| {{monzo|3 0 0 -1}}

| (septimal) supermajor second, septimal whole tone, 7th subharmonic

|

| (3*5)/(2*7)

| {{monzo|-1 1 1 -1}}

| septimal major semitone, septimal diatonic semitone

|

| (3*7)/(2²*5)

| {{monzo|-2 1 -1 1}}

| septimal minor semitone, large septimal chroma

|

| (2²*7)/3³

| {{monzo|2 -3 0 1}}

| septimal 1/3-tone, small septimal chroma, (septimal) subminor second, septimal minor second, trienstonic comma

|

| (2²*3³)/(5*7)

| {{monzo|2 2 -1 -1}}

| septimal 1/4-tone, septimal diesis

| 7/6 to 6/5

| 7²/(2⁴*3)

| {{monzo|-4 -1 0 2}}

| slendro diesis, large septimal diesis, large septimal 1/6-tone

| 8/7 to 7/6

| 2*(5/7)²

| {{monzo|1 0 2 -2}}

| jubilisma, small septimal diesis, small septimal 1/6-tone, tritonic diesis, Erlich's decatonic comma

|

| 2⁶/(3²*7)

| {{monzo|6 -2 0 -1}}

| septimal comma, Archytas' comma

| 9/8 to 8/7

| (2*3²*7)/5³

| {{monzo|1 2 -3 1}}

| starling comma, septimal semicomma

|

| (3*5)²/(2⁵*7)

| {{monzo|-5 2 2 -1}}

| marvel comma, septimal kleisma

| 16/15 to 15/14

| 7⁴/(2⁵*3*5²)

| {{monzo|-5 -1 -2 4}}

| breedsma

| 50/49 to 49/48

| (5⁴*7)/(2*3⁷)

| {{monzo|-1 -7 4 1}}

| ragisma

|

|-

! colspan="6" | 11-limit (complete)

| 11/(2*5)

| {{monzo|-1 0 -1 0 1}}

| (large) undecimal neutral second, undecimal submajor second, Ptolemy's second

|

| (2²*3)/11

| {{monzo|2 1 0 0 -1}}

| (small) undecimal neutral second

|

| (2*11)/(3*7)

| {{monzo|1 -1 0 -1 1}}

| undecimal minor semitone

|

| (3*11)/2⁵

| {{monzo|-5 1 0 0 1}}

| undecimal 1/4-tone, undecimal diesis, al-Farabi's 1/4-tone, 33rd harmonic (octave reduced)

|

| (3/2)²*(5/11)

| {{monzo|-2 2 1 0 -1}}

| undecimal 1/5-tone

|

| (5*11)/(2*3³)

| {{monzo|-1 -3 1 0 1}}

| undecimal diasecundal comma, eleventyfive comma

|

| (2³*7)/(5*11)

| {{monzo|3 0 -1 1 -1}}

| undecimal tritonic comma, konbini comma

|

| (3/7)²*(11/2)

| {{monzo|-1 2 0 -2 1}}

| mothwellsma, small undecimal comma

|

| (2*5/3)²/11)

| {{monzo|2 -2 2 0 -1}}

| ptolemisma, Ptolemy's comma

| 11/10 to 10/9

| 11²/(2³*3*5)

| {{monzo|-3 -1 -1 0 2}}

| biyatisma, undecimal seconds comma

| 12/11 to 11/10

| (2⁴*11)/(5²*7)

| {{monzo|4 0 -2 -1 1}}

| valinorsma

|

| 3⁵/(2*11²)

| {{monzo|-1 5 0 0 -2}}

| rastma, neutral thirds comma

|

| (5*7*11)/(2⁷*3)

| {{monzo|-7 -1 1 1 1}}

| keenanisma

|

| (3*7)²/(2³*5*11)

| {{monzo|-3 2 -1 2 -1}}

| werckisma, Werckmeister's undecimal septenarian schisma

| 22/21 to 21/20

| (2/7)²*3³*5/11

| {{monzo|2 3 1 -2 -1}}

| swetisma, Swets' comma

|

| (5*11)²/(2⁴*3²*7)

| {{monzo|-4 -3 2 -1 2}}

| lehmerisma

| 56/55 to 55/54

| (11/(5*7))²*3⁴/2³

| {{monzo|-3 4 -2 -2 2}}

| kalisma, Gauss comma

| 100/99 to 99/98

|-

! colspan="6" | 13-limit (complete)

| 13/(2²*3)

| {{monzo|-2 -1 0 0 0 1}}

| (large) tridecimal 2/3-tone, tridecimal neutral second

|

| (2*7)/13

| {{monzo|1 0 0 1 0 -1}}

| (small) tridecimal 2/3-tone, trienthird

|

| (2*13)/5²

| {{monzo|1 0 -2 0 0 1}}

| (large) tridecimal 1/3-tone

|

| 3³/(2*13)

| {{monzo|-1 3 0 0 0 -1}}

| (small) tridecimal 1/3-tone

|

| (2³*5)/(3*13)

| {{monzo|3 -1 1 0 0 -1}}

| tridecimal minor diesis

|

| (5*13)/2⁶

| {{monzo|-6 0 1 0 0 1}}

| wilsorma, 13th-partial chroma

|

| (2*3*11)/(5*13)

| {{monzo|1 1 -1 0 1 -1}}

| winmeanma

|

| (2*3*13)/(7*11)

| {{monzo|1 1 0 -1 -1 1}}

| negustma

|

| (7*13)/(2*3²*5)

| {{monzo|-1 -2 -1 1 0 1}}

| [[Biome comma]], superleap comma

|

| (3*5*7)/(2³*13)

| {{monzo|-3 1 1 1 0 -1}}

| animist comma, small tridecimal comma

|

| (2²*3)²/(11*13)

| {{monzo|4 2 0 0 -1 -1}}

| grossma

| 13/12 to 12/11

| 13²/(2³*3*7)

| {{monzo|-3 -1 0 -1 0 2}}

| buzurgisma, dhanvantarisma

| 14/13 to 13/12

| (2*7)²/(3*5*13)

| {{monzo|2 -1 -1 2 0 -1}}

| [[Mynucumic_chords|mynucuma]]

| 15/14 to 14/13

| (5²*13)/(2²*3⁴)

| {{monzo|-2 -4 2 0 0 1}}

| [[Marveltwin|marveltwin comma]]

|

| (3/5)²*13/(2*7)

| {{monzo|-1 3 -2 -1 0 1}}

| ratwolfsma

|

| (2⁵*11)/(3²*13)

| {{monzo|5 -3 0 0 1 -1}}

| minthma

|

| (2/11)²*7*13/3

| {{monzo|2 -1 0 1 -2 1}}

| gentle comma

|

| (5/2)⁴/(3*13)

| {{monzo|-4 -1 4 0 0 -1}}

| tunbarsma

| 26/25 to 25/24

| (2*13/5)²/3³

| {{monzo|2 -3 -2 0 0 2}}

| island comma

| 27/26 to 26/25

| (3²/2)³/(7*13)

| {{monzo|-3 6 0 -1 0 -1}}

| squbema

| 28/27 to 27/26

| 7*11*13/(2*5)³

| {{monzo|-3 0 -3 1 1 1}}

| sinbadma

|

| 2²*3*11*13/(5*7³)

| {{monzo|2 1 -1 -3 1 1}}

| lummic comma

|

| 2⁵*5*13/(3³*7*11)

| {{monzo|5 -3 1 -1 -1 1}}

| ibnsinma

|

| (2⁶/3)²/(5*7*13)

| {{monzo|12 -2 -1 -1 0 -1}}

| schismina, tridecimal schisma

| 65/64 to 64/63

| (5*13)²/(2⁷*3*11)

| {{monzo|-7 -1 2 0 -1 2}}

| leprechaun comma

| 66/65 to 65/64

| (2³/11)³*13/5

| {{monzo|9 0 -1 0 -3 1}}

| jacobin comma

|

| [[10648/10647]]

| (2*11)³/((3*13)²*7)

| {{monzo|3 -2 0 -1 3 -2}}

| harmonisma

|

| [[123201/123200]]

| (3/2)⁶*(13/5)²/(7*11)

| {{monzo|-6 6 -2 -1 -1 2}}

| chalmersia

| 352/351 to 351/350

|-

! colspan="6" | 17-limit (complete)

| {{monzo|-4 0 0 0 0 0 1}}

| large septendecimal semitone, 17th harmonic (octave reduced)

|

| (2*3²)/17

| {{monzo|1 2 0 0 0 0 -1}}

| small septendecimal semitone, Arabic lute index finger

|

| (2*17)/(3*11)

| {{monzo|1 -1 0 0 -1 0 1}}

| large septendecimal 1/4-tone

|

| (5*7)/(2*17)

| {{monzo|-1 0 1 1 0 0 -1}}

| small septendecimal 1/4-tone

|

| (3*17)/(2*5²)

| {{monzo|-1 1 -2 0 0 0 1}}

| large septendecimal 1/6-tone

|

| (2²*13)/(3*17)

| {{monzo|2 -1 0 0 0 1 -1}}

| small septendecimal 1/6-tone

|

| (5*17)/(2²*3*7)

| {{monzo|-2 -1 1 -1 0 0 1}}

| septendecimal comma (?)

|

| (2³*3*5)/(7*17)

| {{monzo|3 1 1 -1 0 0 -1}}

| 136/135

| (2/3)³*17/5

| {{monzo|3 -3 -1 0 0 0 1}}

| septendecimal major second comma

|

| 154/153

| (2*7*11)/(3²*17)

| {{monzo|1 -2 0 1 1 0 -1}}

| 170/169

| (2*5*17)/13²

| {{monzo|1 0 1 0 0 -2 1}}

| 221/220

| (13*17)/(2²*5*11)

| {{monzo|-2 0 -1 0 -1 1 1}}

| (2⁸)/(3*5*17)

| {{monzo|8 -1 -1 0 0 0 -1}}

| septendecimal kleisma, 255th subharmonic

| 17/16 to 16/15

| (3*7*13)/(2⁴*17)

| {{monzo|-4 1 0 1 0 1 -1}}

| tannisma

|

| {{monzo|-5 -2 0 0 0 0 2}}

| septendecimal 6-cent comma

| 18/17 to 17/16

| 375/374

| (3*5³)/(2*11*17)

| {{monzo|-1 1 3 0 -1 0 -1}}

| 442/441

| (2*13*17)/(3*7)²

| {{monzo|1 -2 0 -2 0 1 1}}

| 561/560

| (3*11*17)/(2⁴*5*7)

| {{monzo|-4 1 -1 -1 1 0 1}}

| 595/594

| (5*7*17)/(2*3³*11)

| {{monzo|-1 -3 1 1 -1 0 1}}

| (5*11*13)/(2*3*7*17)

| {{monzo|-1 -1 1 -1 1 1 -1}}

| September comma, septembrisma, septendecimal bridge comma

|

| (7²*17)/(2⁶*13)

| {{monzo|-6 0 0 2 0 -1 1}}

| horizon comma

|

| (2³*3²*13)/(5*11*17)

| {{monzo|3 2 -1 0 -1 1 -1}}

| ainos comma, ainma

|

| (3²*11²)/(2⁶*17)

| {{monzo|-6 2 0 0 2 0 -1}}

| twosquare comma

| 34/33 to 33/32

| (2²*17²)/(3*5*7*11)

| {{monzo|2 -1 -1 -1 -1 0 2}}

| septendecimal 1/4-tones comma

| 35/34 to 34/33

| (5²*7²)/(2³*3²*17)

| {{monzo|-3 -2 2 2 0 0 -1}}

| noellisma

| 36/35 to 35/34

| 1275/1274

| (3*5²*17)/(2*7²*13)

| {{monzo|-1 1 2 -2 0 -1 1}}

| (3⁵*7)/[(2*5)²*17]

| {{monzo|-2 5 -2 1 0 0 -1}}

| palingenesis comma, palingenetic comma, palingenesma

|

| 2058/2057

| (2*3*7³)/(11²*17)

| {{monzo|1 1 0 3 -2 0 -1}}

| xenisma

|

| 2431/2430

| (11*13*17)/(2*3⁵*5)

| {{monzo|-1 -5 -1 0 1 1 1}}

| 2500/2499

| (2²*5⁴)/(3*7²*17)

| {{monzo|2 -1 4 -2 0 0 -1}}

|  

| 51/50 to 50/49

| (3²*17²)/(2³*5²*13)

| {{monzo|-3 2 -2 0 0 -1 2}}

| septendecimal 1/6-tones comma

| 52/51 to 51/50

| 4914/4913

| (2*3³*7*13)/(17³)

| {{monzo|1 3 0 1 0 1 -3}}

| (2³*3⁶)/(7³*17)

| {{monzo|3 6 0 -3 0 0 -1}}

| chlorisma

|

| 12376/12375

| (2³*7*13*17)/(3²*5³*11)

| {{monzo|3 -2 -3 1 -1 1 1}}

| flashma

|

| 14400/14399

| (2⁶*3²*5²)/(7*11²*17)

| {{monzo|6 2 2 -1 -2 0 -1}}

| sparkisma

| 121/120 to 120/119

| 28561/28560

| (13⁴)/(2⁴*3*5*7*17)

| {{monzo|-4 -1 -1 -1 0 4 -1}}

|  

| 170/169 to 169/168

| 31213/31212

| (7⁴*13)/(2²*3³*17²)

| {{monzo|-2 -3 0 4 0 1 -2}}

| 37180/37179

| (2²*5*11*13²)/(3⁷*17)

| {{monzo|2 -7 1 0 1 2 -1}}

| 194481/194480

| 0.008902

| (3⁴*7⁴)/(2⁴*5*11*13*17)

| {{monzo|-4 4 -1 4 -1 -1 -1}}

| scintillisma

| 442/441 to 441/440

| 336141/336140

| 0.005150

| (3²*13³*17)/(2²*5*7⁵)

| {{monzo|-2 2 -1 -5 0 3 1}}

|

|-

! colspan="6" | 19-limit (complete)

| 19/(2*3²)

| {{monzo|-1 -2 0 0 0 0 0 1}}

| large undevicesimal semitone

|

| (2²*5)/19

| {{monzo|2 0 1 0 0 0 0 -1}}

| small undevicesimal semitone

|

| (3*13)/(2*19)

| {{monzo|-1 1 0 0 0 1 0 -1}}

| undevicesimal 2/9-tone

|

| (3*19)/(2³*7)

| {{monzo|-3 1 0 -1 0 0 0 1}}

| hendrix comma

|

| (2²*19)/(3*5²)

| {{monzo|2 -1 -2 0 0 0 0 1}}

| large undevicesimal 1/9-tone

|

| (7*11)/(2²*19)

| {{monzo|-2 0 0 1 1 0 0 -1}}

| small undevicesimal 1/9-tone

|

| (2⁵*3)/(5*19)

| {{monzo|5 1 -1 0 0 0 0 -1}}

|

| (19*7)/(2²*3*11)

| {{monzo|-2 -1 0 1 -1 0 0 1}}

| (3²*17)/(2³*19)

| {{monzo|-3 2 0 0 0 0 1 -1}}

| ganassisma, Ganassi's comma

|

| (3²*19)/(2*5*17)

| {{monzo|-1 2 -1 0 0 0 -1 1}}

| 190/189

| (2*5*19)/(3³*7)

| {{monzo|1 -3 1 -1 0 0 0 1}}

| 209/208

| (11*19)/(2⁴*13)

| {{monzo|-4 0 0 0 1 -1 0 1}}

| yama comma

|

| 210/209

| (2*3*5*7)/(11*19)

| {{monzo|1 1 1 1 -1 0 0 -1}}

| spleen comma

|

| 286/285

| (2*11*13)/(3*5*19)

| {{monzo|1 -1 -1 0 1 1 0 -1}}

| (2²*3⁴)/(17*19)

| {{monzo|2 4 0 0 0 0 -1 -1}}

| nusu comma

| 19/18 to 18/17