List of superparticular intervals: Difference between revisions

This is a list of [[superparticular]] [[interval]]s ordered by [[prime limit]]. It reaches to the 127-limit and is complete up 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).

! width="10%" | [[Ratio]]

! width="5%" | Meta<ref name="ssp">Denoted by S-expressions, where s''k'' is defined as (''k''/(''k'' - 1))/((''k'' + 1)/''k''). See [[square superparticular]] for details.</ref>

| [[2/1]]

| Octave, duple, 2nd harmonic, diapason

 

=== 3-limit ===

| [[3/2]]

| 3/2

| {{Monzo| -1 1 }}

| Perfect fifth, octave-reduced 3rd harmonic, diapente

| [[4/3]]

| 2²/3

| {{Monzo| 2 -1 }}

| Perfect fourth, octave-reduced 3rd subharmonic, diatessaron

| [[9/8]]

| 203.910

| 3²/2³

| Pythagorean whole tone, Pythagorean major second, <br>major whole tone, octave-reduced 9th harmonic, harmonic ninth

{| class="wikitable center-6" style="width:100%"

! width="10%" | [[Cent]]s

! width="15%" | [[Monzo]]

| 386.314

| [[6/5]]

| Classic(al)/just minor third

| [[10/9]]

| 182.404

|-

| 111.731

| 2⁴/(3×5)

| 70.672

| [[81/80]]

| 21.506

| (Septimal) subminor third, septimal minor third

| 2³/7

| (3×7)/(2²×5)

| (2²×7)/3³

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

| (5⁴×7)/(2×3⁷)

{| class="wikitable center-6" style="width:100%"

! width="10%" | [[Cent]]s

| [[55/54]]

| 31.767

| (5×11)/(2×3³)

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

| Telepathma, eleventyfive comma, <br>undecimal diasecundal comma

|  

| [[56/55]]

| (2³×7)/(5×11)

| [[99/98]]

| 17.576

| (3/7)²×(11/2)

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

| Mothwellsma, small undecimal comma

|  

| [[100/99]]

| ((2×5)/3)²/11

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

| [[121/120]]

| 14.376

| 11²/(2³×3×5)

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

| Biyatisma, undecimal seconds comma

| S11

| [[176/175]]

| 9.8646

| (2⁴×11)/(5²×7)

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

| Valinorsma

|  

| [[243/242]]

| 7.1391

| 3⁵/(2×11²)

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

| Rastma, neutral thirds comma

|  

| [[385/384]]

| (5×7×11)/(2⁷×3)

| [[441/440]]

| 3.9302

| (3×7)²/(2³×5×11)

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

| Werckisma, Werckmeister's undecimal septenarian schisma

| S21

| [[540/539]]

| 3.2090

| (2/7)²×((3³×5)/11)

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

| Swetisma, Swets' comma

|  

| [[3025/3024]]

| (5×11)²/(2⁴×3³×7)

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

| [[9801/9800]]

| 0.17665

| ((3²×11)/(5×7))²/2³

 

{| class="wikitable center-6" style="width:100%"

! width="10%" | [[Ratio]]

! width="10%" | [[Cent]]s

! width="15%" | Factorization

! width="15%" | [[Monzo]]

! width="45%" | Name(s)

| [[13/12]]

| 138.573

| 13/(2²×3)

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

| Large tridecimal 2/3-tone, <br>tridecimal neutral second

|  

| [[14/13]]

| 128.298

| (2×7)/13

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

| Small tridecimal 2/3-tone, trienthird

|  

| [[26/25]]

| 67.900

| (2×13)/5²

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

| Large tridecimal 1/3-tone

|  

| [[27/26]]

| 3³/(2×13)

| [[40/39]]

| 43.831

| (2³×5)/(3×13)

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

| Tridecimal minor diesis

|  

| [[65/64]]

| 26.841

| (5×13)/2⁶

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

| Wilsorma, 13th-partial chroma

|  

| [[66/65]]

| 26.432

| (2×3×11)/(5×13)

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

| Winmeanma

|  

| [[78/77]]

| 22.339

| (2×3×13)/(7×11)

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

| Negustma

|  

| [[91/90]]

| 19.130

| (7×13)/(2×3²×5)

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

| Biome comma, superleap comma

|  

| [[105/104]]

| 16.567

| (3×5×7)/(2³×13)

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

| Animist comma, small tridecimal comma

|  

| [[144/143]]

| 12.064

| (2²×3)²/(11×13)

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

| Grossma

| S12

| [[169/168]]

| 10.274

| 13²/(2³×3×7)

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

| Buzurgisma, dhanvantarisma

| S13

| [[196/195]]

| 8.8554

| (2×7)²/(3×5×13)

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

| Mynucuma

| S14

| [[325/324]]

| 5.3351

| (5/(2×3²))²×13

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

| Marveltwin comma

|  

| [[351/350]]

| 4.9393

| (3³×13)/(2×5²×7)

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

| Ratwolfsma

|  

| [[352/351]]

| 4.9253

| (2⁵×11)/(3³×13)

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

| Major minthma, major gentle comma

|  

| [[364/363]]

| 4.7627

| (2/11)²×((7×13)/3)

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

| Minor minthma, minor gentle comma

|  

| (5/2)⁴/(3×13)

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

| [[676/675]]

| 2.5629

| ((2×13)/5)²/3³

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

| Island comma

| S26

| [[729/728]]

| 2.3764

| (3²/2)³/(7×13)

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

| Squbema

| S27

| [[1001/1000]]

| 1.7304

| (7×11×13)/(2×5)³

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

| Sinbadma

|  

| [[1716/1715]]

| 1.0092

| (2²×3×11×13)/(5×7³)

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

| Lummic comma

|  

| [[2080/2079]]

| 0.83252

| (2⁵×5×13)/(3³×7×11)

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

| Ibnsinma, sinaisma

|  

| [[4096/4095]]

| 0.42272

| (2⁶/3)²/(5×7×13)

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

| Minisma

| S64

| [[4225/4224]]

| 0.40981

| (5×13)²/(2⁷×3×11)

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

| Leprechaun comma

| S65

| [[6656/6655]]

| 0.26012

| (2³/11)³×(13/5)

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

| Jacobin comma

|  

| [[Harmonisma|10648/10647]]

| 0.16260

| (2×11)³/((3×13)²×7)

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

| Harmonisma

|  

| [[Chalmersia|123201/123200]]

| 0.014052

| (3/2)⁶×(13/5)²/(7×11)

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

 

! width="10%" | [[Cent]]s

! width="15%" | Factorization

| [[17/16]]

| 104.955

| 17/2⁴

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

| Large septendecimal semitone, <br>octave-reduced 17th harmonic

|  

| [[18/17]]

| 98.955

| (2×3²)/17

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

| Small septendecimal semitone, <br>Arabic lute index finger

|  

| [[34/33]]

| 51.682

| (2×17)/(3×11)

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

| Large septendecimal 1/4-tone

|  

| [[35/34]]

| 50.184

| (5×7)/(2×17)

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

| Small septendecimal 1/4-tone

|  

| [[51/50]]

| 34.283

| (3×17)/(2×5²)

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

| Large septendecimal 1/6-tone

|  

| [[52/51]]

| 33.617

| (2²×13)/(3×17)

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

| Small septendecimal 1/6-tone

|  

| [[85/84]]

| 20.488

| (5×17)/(2²×3×7)

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

| Monk comma

|  

| [[120/119]]

| (2³×3×5)/(7×17)

| [[136/135]]

| 12.777

| (2/3)³×(17/5)

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

| Diatisma, septendecimal major second comma

|  

| [[154/153]]

| 11.278

| (2×7×11)/(3²×17)

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

| Augustma

|  

| [[170/169]]

| 10.214

| (2×5×17)/13²

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

| Major naiadma

|  

| [[221/220]]

| 7.8514

| (13×17)/(2²×5×11)

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

| Minor naiadma

|  

| [[256/255]]

| 6.7759

| 2⁸/(3×5×17)

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

| Charisma, charic comma, <br>septendecimal kleisma

| S16

| [[273/272]]

| 6.3532

| (3×7×13)/(2⁴×17)

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

| Tannisma, prototannisma

|  

| [[289/288]]

| 6.0008

| (17/3)²/2⁵

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

| Semitonisma

| S17

| [[375/374]]

| 4.6228

| (3×5³)/(2×11×17)

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

| Ursulisma

|  

| [[442/441]]

| 3.9213

| (2×13×17)/(3×7)²

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

| Seminaiadma

|  

| [[561/560]]

| 3.0887

| (3×11×17)/(2⁴×5×7)

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

| Monardisma, tsaharuk comma

| [[595/594]]

| 2.9121

| (5×7×17)/(2×3³×11)

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

| Dakotisma

|  

| [[715/714]]

| 2.4230

| (5×11×13)/(2×3×7×17)

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

| September comma, septembrisma

|  

| [[833/832]]

| 2.0796

| (7²×17)/(2⁶×13)

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

| Horizma, horizon comma

|  

| [[936/935]]

| 1.8506

| (2³×3²×13)/(5×11×17)

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

| Ainisma, ainic comma

|  

| [[1089/1088]]

| 1.5905

| (3×11)²/(2⁶×17)

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

| Twosquare comma

| S33

| [[1156/1155]]

| 1.4983

| (2×17)²/(3×5×7×11)

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

| Quadrantonisma

| S34

| [[1225/1224]]

| 1.4138

| (5×7)²/(2³×3²×17)

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

| Noellisma

| S35

| [[1275/1274]]

| 1.3584

| (3×5²×17)/(2×7²×13)

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

| Cimbrisma

|  

| [[1701/1700]]

| 1.0181

| (3⁵×7)/((2×5)²×17)

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

| Palingenetic comma, palingenesis

|  

| [[2058/2057]]

| 0.84143

| (2×3×7³)/(11²×17)

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

| Xenisma

|  

| [[2431/2430]]

| 0.71230

| (11×13×17)/(2×3⁵×5)

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

| Heptacircle comma

|  

| [[2500/2499]]

| 0.69263

| (2×5²)²/(3×7²×17)

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

| Sperasma

| S50

| [[2601/2600]]

| 0.66573

| (3×17)²/(2³×5²×13)

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

| Sextantonisma

| S51

| [[4914/4913]]

| 0.35234

| (2×3³×7×13)/17³

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

| Baladisma

|  

| [[5832/5831]]

| 0.29688

| (2×3²)³/(7³×17)

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

| Chlorisma

|  

| [[Flashma|12376/12375]]

| 0.13989

| (2³×7×13×17)/(3²×5³×11)

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

| Flashma

|  

| [[Sparkisma|14400/14399]]

| 0.12023

| (2³×3×5)²/(7×11²×17)

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

| Sparkisma

| S120

| [[28561/28560]]

| 0.060616

| (13/2)⁴/(3×5×7×17)

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

| Pisanoisma

| S169

| [[E-shaped comma|31213/31212]]

| (7⁴×13)/(2²×3³×17²)

| [[Lateral comma|37180/37179]]

| 0.046564

| (2²×5×11×13²)/(3⁷×17)

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

| Lateral comma

|  

| [[Scintillisma|194481/194480]]

| 0.0089018

| (3×7)⁴/(2⁴×5×11×13×17)

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

| Scintillisma

| S441

| [[Aksial comma|336141/336140]]

| 0.0051503

| (3²×13³×17)/(2²×5×7⁵)

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

| Aksial comma

|}

 

| [[19/18]]

| 93.603

| 19/(2×3²)

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

| Large undevicesimal semitone

|  

| [[20/19]]

| 88.801

| (2²×5)/19

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

| Small undevicesimal semitone

|  

| [[39/38]]

| 44.970

| (3×13)/(2×19)

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

| Undevicesimal diesis, <br>undevicesimal 2/9-tone

|  

| [[57/56]]

| 30.642

| (3×19)/(2³×7)

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

| Hendrix comma

|  

| [[76/75]]

| 22.931