List of superparticular intervals: Difference between revisions

[[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).

== List of superparticular intervals ==

|-

|-

| [[2/1]]

| 2/1

| {{Monzo| 1 }}

|-

| [[3/2]]

| 203.910

| 3²/2³

| S3

|-

| [[5/4]]

| {{Monzo| -2 0 1 }}

| Classic(al)/just major third, octave-reduced 5th harmonic

|-

| [[6/5]]

| {{Monzo| 1 1 -1 }}

| Classic(al)/just minor third

|-

| [[10/9]]

| {{Monzo| 1 -2 1 }}

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

|-

| [[16/15]]

| Syntonic comma, Didymus comma

| S9

|-

| [[7/6]]

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

| (Septimal) subminor third, septimal minor third

|-

| [[8/7]]

| 2³/7

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

|-

| [[15/14]]

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

| Septimal major semitone, septimal diatonic semitone

|-

| [[21/20]]

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

| Septimal minor semitone, large septimal chroma

|-

| [[28/27]]

| (2²×7)/3³

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

|-

| [[36/35]]

| 48.770

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

| S6

|-

| 7²/(2⁴×3)

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

| S7

|-

| 2×(5/7)²

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

|-

| [[64/63]]

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

| Starling comma, septimal semicomma

|-

| [[225/224]]

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

| Ragisma

|-

| [[11/10]]

| 11/(2×5)

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

|-

| [[12/11]]

| (2²×3)/11

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

|-

| [[22/21]]

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

| Undecimal minor semitone

|-

| [[33/32]]

| (3×11)/2⁵

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

|-

| [[45/44]]

| (3/2)²×(5/11)

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

|-

| [[55/54]]

| (5×11)/(2×3³)

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

|-

| [[56/55]]

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

| Undecimal tritonic comma, konbini comma

|-

| [[99/98]]

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

| Mothwellsma, small undecimal comma

|-

| [[100/99]]

| 17.399

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

| Ptolemisma, Ptolemy's comma

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

| Valinorsma

|-

| [[243/242]]

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

| Rastma, neutral thirds comma

|-

| [[385/384]]

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

| Keenanisma

|-

| [[441/440]]

| [[540/539]]

| 3.2090

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

| Swetisma, Swets' comma

|-

| [[3025/3024]]

| 0.57240

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

| Lehmerisma

| [[9801/9800]]

| 0.17665

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

| Kalisma, Gauss comma

| S99

|-

| [[13/12]]

| 13/(2²×3)

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

|-

| [[14/13]]

| (2×7)/13

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

|-

| [[26/25]]

| (2×13)/5²

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

|-

| [[27/26]]

| 3³/(2×13)

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

|-

| [[40/39]]

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

| Tridecimal minor diesis

|-

| [[65/64]]

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

| Wilsorma, 13th-partial chroma

|-

| [[66/65]]

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

| Winmeanma

|-

| [[78/77]]

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

| Negustma

|-

| [[91/90]]

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

| Biome comma, superleap comma

|-

| [[105/104]]

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

| Animist comma, small tridecimal comma

|-

| [[144/143]]

| [[325/324]]

| 5.3351

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

| Marveltwin comma

|-

| [[351/350]]

| 4.9393

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

| Ratwolfsma

|-

| [[352/351]]

| 4.9253

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

|-

| [[364/363]]

| 4.7627

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

|-

| [[625/624]]

| [[676/675]]

| 2.5629

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

| Island comma

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

| Sinbadma

|-

| [[1716/1715]]

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

| Lummic comma

|-

| [[2080/2079]]

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

| Ibnsinma

|-

| [[4096/4095]]

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

| Schismina, tridecimal schisma

|-

| [[4225/4224]]

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

| Leprechaun comma

|-

| [[6656/6655]]

| 0.26012

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

| Jacobin comma

|-

| 0.16260

| (2×11)³/((3×13)²×7)

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

| Harmonisma

|-

| 0.014052

| (3/2)⁶×(13/5)²/(7×11)

| Chalmersia

| S351

|-

| [[17/16]]

| 17/2⁴

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

|-

| [[18/17]]

| (2×3²)/17

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

|-

| [[34/33]]

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

| Large septendecimal 1/4-tone

|-

| [[35/34]]

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

| Small septendecimal 1/4-tone

|-

| [[51/50]]

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

| Large septendecimal 1/6-tone

|-

| [[52/51]]

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

| Small septendecimal 1/6-tone

|-

| [[85/84]]

| (5×17)/(2²×3×7)

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

|-

| [[120/119]]

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

| Lynchisma

|-

| [[136/135]]

| 12.777

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

|-

| 11.278

| (2×7×11)/(3²×17)

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

|  

|-

| 10.214

| (2×5×17)/13²

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

|  

|-

| 7.8514

| (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 }}

| S16

|-

| (3×7×13)/(2⁴×17)

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

|-

| [[289/288]]

|  

|-

| 3.9213

| (2×13×17)/(3×7)²

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

|  

|-

| 3.0887

| (3×11×17)/(2⁴×5×7)

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

|  

|-

| [[595/594]]

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

| September comma, septembrisma

|-

| [[833/832]]

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

| Horizma, horizon comma

|-

| [[936/935]]

| (2³×3²×13)/(5×11×17)

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

|-

| [[1089/1088]]

| S35

|-

| 1.3584

| (3×5²×17)/(2×7²×13)

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

|  

|-

| [[1701/1700]]

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

| Palingenetic comma, palingenesis

|-

| [[2058/2057]]

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

| Xenisma

|-

| 0.71230

| (11×13×17)/(2×3⁵×5)

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

|  

|-

| [[2500/2499]]

| S51

|-

| 0.35234

| (2×3³×7×13)/17³

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

|  

|-

| [[5832/5831]]

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

| Chlorisma

|-

| [[Flashma|12376/12375]]

| (2³×7×13×17)/(3²×5³×11)

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

|-

| [[Sparkisma|14400/14399]]

| S120

|-

| 0.060616

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

| S169

|-

| 0.055466

| (7⁴×13)/(2²×3³×17²)

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

|  

|-

| 0.046564

| (2²×5×11×13²)/(3⁷×17)

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

|  

|-

| (3×7)⁴/(2⁴×5×11×13×17)

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

| S441

|-

| (3²×13³×17)/(2²×5×7⁵)

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

|  

|-

| [[19/18]]

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

| Large undevicesimal semitone

|-

| [[20/19]]

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

| Small undevicesimal semitone

|-

| [[39/38]]

| (3×13)/(2×19)

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

|-

| [[57/56]]

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

| Hendrix comma

|-

| [[76/75]]

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

| Large undevicesimal 1/9-tone

|-

| [[77/76]]

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

| Small undevicesimal 1/9-tone

|-

| [[96/95]]

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

| 19th-partial chroma

|-

| [[133/132]]

| (7×19)/(2²×3×11)

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

|  

|-

| [[153/152]]

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

| Ganassisma, Ganassi's comma

|-

| [[171/170]]

| (3²×19)/(2×5×17)

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

|  

|-

| 9.1358

| (2×5×19)/(3³×7)

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

|  

|-

| 8.3033

| (11×19)/(2⁴×13)

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

| Yama comma

|-

| 8.2637

| (2×3×5×7)/(11×19)

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

| Spleen comma

|-

| 6.0639

| (2×11×13)/(3×5×19)

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

|  

|-

| [[324/323]]

| (2×3²)²/(17×19)

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

| S18

|-

| 5.0547

| 7³/(2×3²×19)

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

|  

|-

| 4.8023

| 19²/(2³×3²×5)

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

| S19

|-

| 4.3335

| (2²×5)²/(3×7×19)

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

| S20

|-

| 3.8007

| (2³×3×19)/(5×7×13)

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

|  

|-

| 3.6409

| (2²×7×17)/(5²×19)

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

|  

|-

| 3.5010

| (3²×5×11)/(2×13×19)

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

|  

|-

| [[513/512]]

| (3³×19)/2⁹

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

|-

| 1.7875

| (3×17×19)/(2³×11²)

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

|  

|-

| [[1216/1215]]

| {{Monzo| 6 -5 -1 0 0 0 0 1 }}

| Password comma, Eratosthenes' comma

|-

| 1.3012

| 11³/(2×5×7×19)

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

|  

|-

| [[1445/1444]]

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

| Aureusma

|-

| [[1521/1520]]

| S39

|-

| 1.1245

| (2²×5×7×11)/(3⁴×19)

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

|  

|-

| [[1729/1728]]

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

| Ramanujanisma

|-

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

|  

|-

| (2⁷×19)/(11×13×17)

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

| Blumeyer comma

|-

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

|  

|-

| (2³×7)²/(3×5×11×19)

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

| S56

|-

| (2×5³×13)/(3×19)²

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

|  

|-

| (2³×3×5²×7)/(13×17×19)

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

|  

|-

| (2²×19)²/(3×5²×7×11)

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

| S76

|-

| (7×11)²/(2³×3×13×19)

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

| S77

|-

| (3²×5×7×19)/(2⁵×11×17)

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

|  

|-

| (5²×13×19)/(2×3²×7³)

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

|  

|-

| (2²×5×7³)/19³

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

|  

|-

| 10241/10240

| (7²×11×19)/(2¹¹×5)

| {{Monzo| -11 0 -1 2 1 0 0 1 }}

|  

|-

| 10830/10829

| (2×3×5×19²)/(7²×13×17)

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

|  

|-

| (2²×3⁵×13)/(5×7×19²)

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

|  

|-

| 13377/13376

| (3×7³×13)/(2⁶×11×19)

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

|  

|-

| 14080/14079

| (2⁸×5×11)/(3×13×19²)

| {{Monzo| 8 -1 1 0 1 -1 0 -2 }}

|  

|-

| 14365/14364

| (5×13²×17)/(2²×3³×7×19)

|  

|-

| 23409/23408

|  

| S153

|-

| 27456/27455

|  

|-

| 28900/28899

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

|  

|-

| 43681/43680

| (11×19)²/(2⁵×3×5×7×13)

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

|-

| 89376/89375

| (2⁵×3×7²×19)/(5⁴×11×13)

| {{Monzo| 5 1 -4 2 -1 -1 0 1 }}

|  

|-

| 104976/104975

| (2×3²)⁴/(5²×13×17×19)

|  

| S324

|-

| {{Monzo| 9 -3 -3 -2 0 0 1 1 }}

| Decimillisma

|-

| 228096/228095

| {{Monzo| 8 4 -1 -4 1 0 0 -1 }}

|  

|-

| 601426/601425

| (2×7²×17×19²)/(3⁷×5²×11)

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

|  

|-

| (2²×7×11³×17)/(3³×5×13×19²)

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

|  

|-

| 709632/709631

| (2¹⁰×3²×7×11)/(13³×17×19)

| {{Monzo| 10 2 0 1 1 -3 -1 -1 }}

|  

|-

| 5909761/5909760

| (11×13×17)²/(2⁸×3⁵×5×19)

| {{Monzo| -8 -5 -1 0 2 2 2 -1 }}

| S2431

|-

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

|  

|-

| [[23/22]]

| 23/(2×11)

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

|-

| [[24/23]]

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

| Small vicesimotertial semitone

|-

| [[46/45]]

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

| Vicesimotertial 1/5-tone

|-

| [[69/68]]

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

| Large vicesimotertial 1/8-tone

|-

| [[70/69]]

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

| Small vicesimotertial 1/8-tone

|-

| [[92/91]]

| (2²×23)/(7×13)

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

|  

|-

| 15.120

| (5×23)/(2×3×19)

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