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).

! [[Ratio]]

! [[Cent]]s

| {{Monzo| 1 }}

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

| {{Monzo| -1 1 }}

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

| {{Monzo| 2 -1 }}

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

| {{Monzo| -2 0 1 }}

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

| {{Monzo| 1 1 -1 }}

| Classic(al)/just minor third

| {{Monzo| 1 -2 1 }}

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

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

| Classic(al)/just diatonic semitone, 15th subharmonic

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

| Classic(al)/just chromatic semitone, chroma, Zarlinian semitone

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

| Syntonic comma, Didymus comma

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

| (Septimal) subminor third, septimal minor third

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

| (Septimal) supermajor second, septimal whole tone, octave-reduced 7th subharmonic

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

| Septimal major semitone, septimal diatonic semitone

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

| Septimal minor semitone, large septimal chroma

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

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

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

| Septimal 1/4-tone, septimal diesis

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

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

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

| Jubilisma, tritonic diesis, small septimal diesis, small septimal 1/6-tone

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

| Septimal comma, Archytas' comma

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

| Starling comma, septimal semicomma

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

| Marvel comma, septimal kleisma

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

| Breedsma

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

| Ragisma

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

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

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

| (Small) undecimal neutral second

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

| Undecimal minor semitone

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

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

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

| Undecimal 1/5-tone

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

| Telepathma, eleventyfive comma, undecimal diasecundal comma

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

| Undecimal tritonic comma, konbini comma

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

| Mothwellsma, small undecimal comma

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

| Ptolemisma, Ptolemy's comma

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

| Biyatisma, undecimal seconds comma

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

| Valinorsma

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

| Rastma, neutral thirds comma

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

| Keenanisma

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

| Werckisma, Werckmeister's undecimal septenarian schisma

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

| Swetisma, Swets' comma

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

| Lehmerisma

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

| Kalisma, Gauss comma

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

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

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

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

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

| (Large) tridecimal 1/3-tone

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

| (Small) tridecimal 1/3-tone

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

| Tridecimal minor diesis

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

| Wilsorma, 13th-partial chroma

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

| Winmeanma

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

| Negustma

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

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

| Animist comma, small tridecimal comma

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

| Grossma

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

| Buzurgisma, dhanvantarisma

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

| Mynucuma

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

| Marveltwin comma

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

| Ratwolfsma

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

| Minthma

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

| Sinbadma

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

| Lummic comma

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

| Ibnsinma

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

| Schismina, tridecimal schisma

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

| Leprechaun comma

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

| Jacobin comma

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

| Harmonisma

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

| Chalmersia

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

| Large septendecimal semitone, octave-reduced 17th harmonic

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

| Small septendecimal semitone, Arabic lute index finger

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

| Large septendecimal 1/4-tone

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

| Small septendecimal 1/4-tone

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

| Large septendecimal 1/6-tone

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

| Small septendecimal 1/6-tone

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

| Septendecimal comma (?)

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

| Lynchisma

| [[136/135]]

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

| Septendecimal major second comma

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

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

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

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

| Septendecimal kleisma, octave-reduced 255th subharmonic

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

| Tannisma

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

| Semitonisma

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

| Ursulisma

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

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

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

| Dakotisma

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

| September comma, septembrisma

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

| Horizma, horizon comma

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

| Ainos comma, ainma

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

| Twosquare comma

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

| Quadrantonisma

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

| Noellisma

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

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

| Palingenetic comma, palingenesis

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

| Xenisma

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

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

| Sperasma

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

| Sextantonisma

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

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

| Chlorisma

| [[Flashma|12376/12375]]

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

| [[Sparkisma|14400/14399]]

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

| Sparkisma

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

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

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

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

| Scintillisma

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

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

| Large undevicesimal semitone

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

| Small undevicesimal semitone

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

| Undevicesimal 2/9-tone

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

| Hendrix comma

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

| Large undevicesimal 1/9-tone

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

| Small undevicesimal 1/9-tone

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

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

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

| Ganassisma, Ganassi's comma

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

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

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

| Yama comma

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

| Spleen comma

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

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

| Nusu comma

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

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

| Go comma

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

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

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

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

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

| Undevicesimal comma, undevicesimal schisma, Boethius' comma, 513th harmonic

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

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

| Password comma, Eratosthenes' comma

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

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

| Aureusma

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

| Pinkanberry

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

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

| Ramanujanisma

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

| {{Monzo| 4 8 -2 0 0 0 -1 -1 -1 }}

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

| Decimillisma

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

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

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

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

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

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

| Greater vicesimotertial semitone

| Small vicesimotertial semitone

| Vicesimotertial 1/5-tone

| Large vicesimotertial 1/8-tone