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 [[17-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 [[OverToneSeries|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.

 

[http://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem Størmer's theorem] guarantees 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]]. [http://oeis.org/A002071 A002071 -- OEIS] gives the number of superparticular ratios in each prime limit, [http://oeis.org/A145604 A145604 - OEIS] shows the increment from limit to limit, and [http://oeis.org/A117581 A117581] 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"

! colspan="5" | 2-limit (complete)

| {{Monzo|1}}

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

|-

! colspan="5" | 3-limit (complete)

| 701.995

| {{Monzo|-1 1}}

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

| {{Monzo|2 -1}}

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

| {{Monzo|-3 2}}

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

|-

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

| {{Monzo|-2 0 1}}

| (classic) (5-limit) major third, 5th harmonic (octave reduced)

| (2*3)/5

| {{Monzo|1 1 -1}}

| (classic) (5-limit) minor third

| (2*5)/3²

| {{Monzo|1 -2 1}}

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

| 111.713

| 2⁴/(3*5)

| {{Monzo|4 -1 -1}}

| minor diatonic semitone, 15th subharmonic

| 5²/(2³*3)

| {{Monzo|-3 -1 2}}

| chroma, (classic) chromatic semitone, Zarlinian semitone

| {{Monzo|-4 4 -1}}

| syntonic comma, Didymus comma

|-

! colspan="5" | 7-limit (complete)

| 7/(2*3)

| {{Monzo|-1 -1 0 1 }}

| (septimal) subminor third, septimal minor third, augmented second

| {{Monzo|3 0 0 -1}}

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

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

| {{Monzo|-1 1 1 -1}}

| septimal diatonic semitone

| (3*7)/(2²*5)

| {{Monzo|-2 1 -1 1}}

| minor semitone, large septimal chromatic semitone

| (2²*7)/3³

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

| septimal chroma, small septimal chromatic semitone, Archytas' 1/3-tone

| (2²*3³)/(5*7)

| {{Monzo|2 2 -1 -1}}

| septimal quarter tone, septimal diesis

| 7²/(2⁴*3)

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

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

| 2*(5/7)²

| {{Monzo|1 0 2 -2}}

| septimal sixth-tone, jubilisma, small septimal diesis, tritonic diesis, Erlich's decatonic comma

| 2⁶/(3²*7)

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

| septimal comma, Archytas' comma

| (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

| 7⁴/(2⁵*3*5²)

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

| breedsma

| (5⁴*7)/(2*3⁷)

| {{Monzo|-1 -7 4 1}}

| ragisma

|-

! colspan="5" | 11-limit (complete)

| 11/(2*5)

| {{Monzo|-1 0 -1 0 1}}

| (large) (undecimal) neutral second, 4/5-tone, Ptolemy's second

| (2²*3)/11

| {{Monzo|2 1 0 0 -1}}

| (small) (undecimal) neutral second, 3/4-tone

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

| {{Monzo|1 -1 0 -1 1}}

| undecimal minor semitone

| (3*11)/2⁵

| {{Monzo|-5 1 0 0 1}}

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

| (3/2)²*(5/11)

| {{Monzo|-2 2 1 0 -1}}

| 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}}

| small undecimal comma, mothwellsma

| (2*5/3)²/11)

| {{Monzo|2 -2 2 0 -1}}

| Ptolemy's comma, ptolemisma

| 11²/(2³*3*5)

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

| undecimal seconds comma, biyatisma

| (2⁴*11)/(5²*7)

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

| valinorsma

| 3⁵/(2*11²)

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

| neutral third comma, rastma

| (5*7*11)/(2⁷*3)

| {{Monzo|-7 -1 1 1 1}}

| keenanisma

| (3*7)²/(2³*5*11)

| {{Monzo|-3 2 -1 2 -1}}

| Werckmeister's undecimal septenarian schisma, werckisma

| (2/7)²*3³*5/11

| {{Monzo|2 3 1 -2 -1}}

| Swets' comma, swetisma

| (5*11)²/(2⁴*3²*7)

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

| [11/(5*7)]²*3⁴/2³

| {{Monzo|-3 4 -2 -2 2}}

| Gauss comma, kalisma

|-

! colspan="5" | 13-limit (complete)

| 13/(2²*3)

| {{Monzo|-2 -1 0 0 0 1}}

| tridecimal 2/3-tone

| (2*7)/13

| {{Monzo|1 0 0 1 0 -1}}

| 2/3-tone, trienthird

| (2*13)/5²

| {{Monzo|1 0 -2 0 0 1}}

| tridecimal 1/3-tone

| 3³/(2*13)

| {{Monzo|-1 3 0 0 0 -1}}

| tridecimal comma

| (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}}

| [[The_Biosphere|Biome]] comma, superleap comma

| (3*5*7)/(2³*13)

| {{Monzo|-3 1 1 1 0 -1}}

| small tridecimal comma, animist comma

| (2²*3)²/(11*13)

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

| grossma

| 13²/(2³*3*7)

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

| buzurgisma, dhanvantarisma

| (2*7)²/(3*5*13)

| {{Monzo|2 -1 -1 2 0 -1}}

| marveltwin comma

| (5²*13)/(2²*3⁴)

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

| (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

|  

| {{Monzo|-4 -1 4 0 0 -1}}

| tunbarsma

|  

| {{Monzo|2 -3 -2 0 0 2}}

| island comma

|  

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

| squbema

|  

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

| tridecimal schisma, Sagittal schismina

|  

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

| leprechaun comma

| [[10648/10647]]

| [[123201/123200]]

|  

| {{Monzo|-6 6 -2 -1 -1 2}}

| chalmersia

|-

! colspan="5" | 17-limit (complete)

| {{Monzo|-4 0 0 0 0 0 1}}

| 17th harmonic (octave reduced)

| (2*3²)/17

| {{Monzo|1 2 0 0 0 0 -1}}

| Arabic lute index finger

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

| {{Monzo|1 -1 0 0 -1 0 1}}

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

| {{Monzo|-1 0 1 1 0 0 -1}}

| septendecimal 1/4-tone

| (3*17)/(2*5²)

| {{Monzo|-1 1 -2 0 0 0 1}}

| 17th-partial chroma

| (2²*13)/(3*17)

| {{Monzo|2 -1 0 0 0 1 -1}}

| (5*17)/(2²*3*7)

| {{Monzo|-2 -1 1 -1 0 0 1}}

| 120/119

| (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}}

| 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}}

| 256/255

| (2⁸)/(3*5*17)

| {{Monzo|8 -1 -1 0 0 0 -1}}

| 255th subharmonic

| 273/272

| (3*7*13)/(2⁴*17)

| {{Monzo|-4 1 0 1 0 1 -1}}

| 289/288

| {{Monzo|-5 -2 0 0 0 0 2}}

|  

| 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}}

| 715/714

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

| {{Monzo|-1 -1 1 -1 1 1 -1}}

| 833/832

| (7²*17)/(2⁶*13)

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

| 936/935

| (2³*3²*13)/(5*11*17)

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

| 1089/1088

| (3²*11²)/(2⁶*17)

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

| twosquare comma

| 1156/1155

| (2²*17²)/(3*5*7*11)

| {{Monzo|2 -1 -1 -1 -1 0 2}}

|  

| 1225/1224

| (5²*7²)/(2³*3²*17)

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

|  

| 1275/1274

| (3*5²*17)/(2*7²*13)

| {{Monzo|-1 1 2 -2 0 -1 1}}

| 1701/1700

| (3⁵*7)/[(2*5)²*17]

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

| 2058/2057

| 0.8414

| (2*3*7³)/(11²*17)

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

| xenisma

| 2431/2430

| 0.7123

| (11*13*17)/(2*3⁵*5)

| {{Monzo|-1 -5 -1 0 1 1 1}}

| 2500/2499

| 0.6926

| (2²*5⁴)/(3*7²*17)

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

|  

| 2601/2600

| 0.6657

| (3²*17²)/(2³*5²*13)

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

|  

| 4914/4913

| 0.3523

| (2*3³*7*13)/(17³)

| {{Monzo|1 3 0 1 0 1 -3}}

| 5832/5831

| 0.2969

| (2³*3⁶)/(7³*17)

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

| 12376/12375

| 0.1399

| (2³*7*13*17)/(3²*5³*11)

| {{Monzo|3 -2 -3 1 -1 1 1}}

| 14400/14399

| 0.1202

| (2⁶*3²*5²)/(7*11²*17)

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

|  

| 28561/28560

| 0.0606

| (13⁴)/(2⁴*3*5*7*17)

| {{Monzo|-4 -1 -1 -1 0 4 -1}}

|  

| 31213/31212

| 0.0555

| (7⁴*13)/(2²*3³*17²)

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

| 37180/37179

| 0.0466

| (2²*5*11*13²)/(3⁷*17)

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

| 194481/194480

| 0.0089

| (3⁴*7⁴)/(2⁴*5*11*13*17)

| {{Monzo|-4 4 -1 4 -1 -1 -1}}

| scintillisma

| 336141/336140

| 0.0052

| (3²*13³*17)/(2²*5*7⁵)

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

|-

! colspan="5" | 19-limit (incomplete)

| 19/(2*3²)

| {{Monzo|-1 -2 0 0 0 0 0 1}}

| 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}}

| (3*19)/(2³*7)

| {{Monzo|-3 1 0 -1 0 0 0 1}}

| (2²*19)/(3*5²)

| {{Monzo|2 -1 -2 0 0 0 0 1}}

| (7*11)/(2²*19)

| {{Monzo|-2 0 0 1 1 0 0 -1}}

| (2⁵*3)/(5*19)

| {{Monzo|5 1 -1 0 0 0 0 -1}}

| 133/132

| (19*7)/(2²*3*11)

| {{Monzo|-2 -1 0 1 -1 0 0 1}}

| 153/152

| (3²*17)/(2³*19)

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

| 171/170

| (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}}

| 210/209

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

| {{Monzo|1 1 1 1 -1 0 0 -1}}

| 286/285

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

| {{Monzo|1 -1 -1 0 1 1 0 -1}}

| 324/323

| (2²*3⁴)/(17*19)

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

|  

| 343/342

| 7⁴/(2*3³*19)

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

| 361/360

| 19²/(2³*3²*5)

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

|  

| 400/399

| (2⁴*5²)/(3*7*19)

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

|  

| 456/455

| (2³*3*19)/(5*7*13)

| {{Monzo|3 1 -1 -1 0 -1 0 1}}

| 476/475

| (2²*7*17)/(5²*19)

| {{Monzo|2 0 -2 1 0 0 1 -1}}

| 495/494

| 3.501

| (3²*5*11)/(2*13*19)

| {{Monzo|-1 2 1 0 1 -1 0 -1}}

| 513/512

| 3.378

| (3³*19)/2⁹

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

| 513th harmonic

! colspan="5" | 23-limit (incomplete)

| [[23/22]]

| 76.956