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31edo

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//Thirty-one tone equal temperament//, also called //31-tET//, //31-EDO//, //31-et//, or //tricesimoprimal meantone temperament//, is the scale derived by dividing the octave into 31 [[equal|equally]] large steps. The term 'Tricesimoprimal' was first used by [[Adriaan Fokker]]. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]]. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). It's not bad for an 11-limit approximation, as it is consistent through it, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablance in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.

31edo is the 11th [[prime numbers|prime]] edo, following [[29edo]] and coming before [[37edo]].

For more encyclopedic info, see [[http://en.wikipedia.org/wiki/31_equal_temperament|Wikipedia's article]].

=Quick index to intervals and linear temperaments= 
||~ Generator ||~ Cents ||~ Temperaments ||
|| 1\31 || 38.71 || [[Slender]] ||
|| 2\31 || 77.42 || [[Valentine]]/[[Lupercalia]] ||
|| 3\31 || 116.13 || [[Miracle]] ||
|| 4\31 || 154.84 || [[Nusecond]] ||
|| 5\31 || 193.55 || [[Luna]]/[[Hemithird]]/[[Hemiwürschmidt]] ||
|| 6\31 || 232.26 || [[Cynder]]/[[Mothra]] ||
|| 7\31 || 270.97 || [[Orson]]/[[Orwell]] ||
|| 8\31 || 309.68 || [[Myna]] ||
|| 9\31 || 348.39 || [[Vicentino]]/[[Mohajira]]/[[Migration]] ||
|| 10\31 || 387.10 || [[Würschmidt]]/[[Worschmidt]] ||
|| 11\31 || 425.81 || [[Squares]]/[[Sentinel]] ||
|| 12\31 || 464.52 || [[Semisept]] ||
|| 13\31 || 503.23 || [[Meantone]]/[[Meanpop]] ||
|| 14\31 || 541.94 || [[Casablanca]]/[[Marrakesh]] ||
|| 15\31 || 580.65 || [[Tritonic]] ||
=Intervals= 
===1\31 octave - approx. 38.71¢ - Diesis=== 
A single step of 31-edo is about 38.71¢. Intervals around this size are called [[diesis|dieses]] (singular 'diesis'). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G#, to B#, but B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). In 11-limit tonal music, the single step stands in for just ratios 56:55 (31.19); 55:54 (31.77¢); 49:48 (39.70¢); 45:44 (38.91¢); 36:35 (48.77¢); 33:32 (53.27¢) and others. Demonstrated in [[SpiralProgressions]].

===2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second=== 
The difference between a major and minor third. The more 'expressive' of the 'half steps'. In 11-limit tonal music, 2\31 stands in for just ratios 28:27 (62.96¢); 25:24 (70.67¢); 22:21 (80.54¢); 21:20 (84.45¢) and others. Generates [[Starling temperaments#Valentine%20temperament|valentine temperament]] - aka [[Armodue theory#Semi-equalized%20Armodue|semi-equalized Armodue]].
====MOS Scales generated by 2\31:==== 
||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||
|| 15-tone ([[Maximal evenness|ME]] or quasi-equal) || [[1L 14s]] || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 3 ||   ||   ||
|| 16-tone || [[15L 1s]] || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 1 ||

===3\31 octave - approx. 116.13 - Major Semitone or Diatonic Semitone or Large Major Second=== 
The difference between a perfect fourth and a major third. The larger and clunkier of the 'half steps'. In 11-limit tonal music, 3\31 stands in for just ratios 16:15 (111.73¢); 15:14 (199.44¢) and others. Generates [[Gamelismic clan|miracle temperament]].
====MOS Scales generated by 3\31:==== 
||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||
|| nonatonic || [[1L 8s]] || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 7 ||   ||   ||   ||   ||   ||   ||
|| decatonic (quasi-equal) || [[9L 1s]] || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 4 ||   ||   ||   ||
|| 11-tone || [[10L 1s]] || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 1 ||
|| 21-tone (Blackjack) || [[11L 10s]] || 2 ||   || 1 || 2 ||   || 1 || 2 ||   || 1 || 2 ||   || 1 || 2 ||   || 1 || 2 ||   || 1 || 2 ||   || 1 || 2 ||   || 1 || 2 ||   || 1 || 2 ||   || 1 || 1 ||

===4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second=== 
Exactly one half of the minor third (and twice the minor semitone). In 11-limit tonal music, 4\31 stands in for 12:11 (150.64¢); 35:32 (155.14¢); 11:10 (165.00¢) and others. Generates [[Starling temperaments|nusecond temperament]].
====MOS Scales generated by 4\31:==== 
||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||
|| heptatonic || [[1L 6s]] || 4 ||   ||   ||   || 4 ||   ||   ||   || 4 ||   ||   ||   || 4 ||   ||   ||   || 4 ||   ||   ||   || 4 ||   ||   ||   || 7 ||   ||   ||   ||   ||   ||   ||
|| octatonic (quasi-equal) || [[7L 1s]] || 4 ||   ||   ||   || 4 ||   ||   ||   || 4 ||   ||   ||   || 4 ||   ||   ||   || 4 ||   ||   ||   || 4 ||   ||   ||   || 4 ||   ||   ||   || 3 ||   ||   ||
|| 15-tone || [[8L 7s]] || 1 || 3 ||   ||   || 1 || 3 ||   ||   || 1 || 3 ||   ||   || 1 || 3 ||   ||   || 1 || 3 ||   ||   || 1 || 3 ||   ||   || 1 || 3 ||   ||   || 3 ||   ||   ||
|| 23-tone || [[8L 15s]] || 1 || 1 || 2 ||   || 1 || 1 || 2 ||   || 1 || 1 || 2 ||   || 1 || 1 || 2 ||   || 1 || 1 || 2 ||   || 1 || 1 || 2 ||   || 1 || 1 || 2 ||   || 1 || 2 ||   ||

===5\31 octave - approx. 193.55¢ - Whole Tone or Major Second=== 
A rather smallish whole tone. Often called melodically dull. As it falls between (and functions as) just whole tones 9:8 and 10:9, 5\31 is considered a "meantone". Two meantones make a near-just major third. Generates [[Gamelismic clan|hemithirds temperament]] and [[Wuerschmidt family|hermiwuerschmidt temperament]].
====MOS Scales generated by 5\31:==== 
||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||
|| hexatonic (quasi-equal) || [[1L 5s]] || 5 ||   ||   ||   ||   || 5 ||   ||   ||   ||   || 5 ||   ||   ||   ||   || 5 ||   ||   ||   ||   || 5 ||   ||   ||   ||   || 6 ||   ||   ||   ||   ||   ||
|| heptatonic || [[6L 1s]] || 5 ||   ||   ||   ||   || 5 ||   ||   ||   ||   || 5 ||   ||   ||   ||   || 5 ||   ||   ||   ||   || 5 ||   ||   ||   ||   || 5 ||   ||   ||   ||   || 1 ||
|| 13-tone || [[6L 7s]] || 4 ||   ||   ||   || 1 || 4 ||   ||   ||   || 1 || 4 ||   ||   ||   || 1 || 4 ||   ||   ||   || 1 || 4 ||   ||   ||   || 1 || 4 ||   ||   ||   || 1 || 1 ||
|| 19-tone || [[6L 13s]] || 3 ||   ||   || 1 || 1 || 3 ||   ||   || 1 || 1 || 3 ||   ||   || 1 || 1 || 3 ||   ||   || 1 || 1 || 3 ||   ||   || 1 || 1 || 3 ||   ||   || 1 || 1 || 1 ||
|| 25-tone || [[6L 19s]] || 2 ||   || 1 || 1 || 1 || 2 ||   || 1 || 1 || 1 || 2 ||   || 1 || 1 || 1 || 2 ||   || 1 || 1 || 1 || 2 ||   || 1 || 1 || 1 || 2 ||   || 1 || 1 || 1 || 1 ||

===6\31 octave - approx. 232.26¢ - Supermajor Second=== 
Exactly one half of a narrow fourth, twice a major semitone, or thrice a minor semitone. In 7-limit tonal music, 6\31 stands in for 8:7 (231.17¢). Generates [[Meantone family|mothra temperament]].
====MOS Scales generated by 6\31:==== 
||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||
|| pentatonic (quasi-equal) || [[1L 4s]] || 6 ||   ||   ||   ||   ||   || 6 ||   ||   ||   ||   ||   || 6 ||   ||   ||   ||   ||   || 6 ||   ||   ||   ||   ||   || 7 ||   ||   ||   ||   ||   ||   ||
|| hexatonic || [[5L 1s]] || 6 ||   ||   ||   ||   ||   || 6 ||   ||   ||   ||   ||   || 6 ||   ||   ||   ||   ||   || 6 ||   ||   ||   ||   ||   || 6 ||   ||   ||   ||   ||   || 1 ||
|| 11-tone || [[5L 6s]] || 5 ||   ||   ||   ||   || 1 || 5 ||   ||   ||   ||   || 1 || 5 ||   ||   ||   ||   || 1 || 5 ||   ||   ||   ||   || 1 || 5 ||   ||   ||   ||   || 1 || 1 ||
|| 16-tone || [[5L 11s]] || 4 ||   ||   ||   || 1 || 1 || 4 ||   ||   ||   || 1 || 1 || 4 ||   ||   ||   || 1 || 1 || 4 ||   ||   ||   || 1 || 1 || 4 ||   ||   ||   || 1 || 1 || 1 ||
|| 21-tone || [[5L 16s]] || 3 ||   ||   || 1 || 1 || 1 || 3 ||   ||   || 1 || 1 || 1 || 3 ||   ||   || 1 || 1 || 1 || 3 ||   ||   || 1 || 1 || 1 || 3 ||   ||   || 1 || 1 || 1 || 1 ||
|| 26-tone || [[5L 21s]] || 2 ||   || 1 || 1 || 1 || 1 || 2 ||   || 1 || 1 || 1 || 1 || 2 ||   || 1 || 1 || 1 || 1 || 2 ||   || 1 || 1 || 1 || 1 || 2 ||   || 1 || 1 || 1 || 1 || 1 ||

===7\31 octave - approx. 270.97¢ - Subminor Third=== 
Exactly one half of a superfourth (11:8 approximation). In 7-limit tonal music, 7\31 stands in for 7:6 (266.87¢). A generator for Orwell temperament (but not as good as 12\53 or 19\84). Generates [[Semicomma family|orwell temperament]].
====MOS Scales generated by 7\31:==== 
||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||
|| pentatonic || [[4L 1s]] || 7 ||   ||   ||   ||   ||   ||   || 7 ||   ||   ||   ||   ||   ||   || 7 ||   ||   ||   ||   ||   ||   || 7 ||   ||   ||   ||   ||   ||   || 3 ||   ||   ||
|| nonatonic (quasi-equal; Orwell[9]) || [[4L 5s]] || 4 ||   ||   ||   || 3 ||   ||   || 4 ||   ||   ||   || 3 ||   ||   || 4 ||   ||   ||   || 3 ||   ||   || 4 ||   ||   ||   || 3 ||   ||   || 3 ||   ||   ||
|| 13-tone (Orwell[13]) || [[9L 4s]] || 1 || 3 ||   ||   || 3 ||   ||   || 1 || 3 ||   ||   || 3 ||   ||   || 1 || 3 ||   ||   || 3 ||   ||   || 1 || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   ||
|| 22-tone (Orwell[22]) || [[9L 13s]] || 1 || 1 || 2 ||   || 1 || 2 ||   || 1 || 1 || 2 ||   || 1 || 2 ||   || 1 || 1 || 2 ||   || 1 || 2 ||   || 1 || 1 || 2 ||   || 1 || 2 ||   || 1 || 2 ||   ||


===8\31 octave - approx. 309.68¢ - Minor Third=== 
A minor third, closer to the just 6:5 (315.64¢) than 12-edo. Exactly twice a neutral second, four times a minor semitone, and half of a large tritone. Generates [[Starling temperaments|myna temperament]].
====MOS Scales generated by 8\31:==== 
||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||
|| tetratonic (quasi-equal) || [[3L 1s]] || 8 ||   ||   ||   ||   ||   ||   ||   || 8 ||   ||   ||   ||   ||   ||   ||   || 8 ||   ||   ||   ||   ||   ||   ||   || 7 ||   ||   ||   ||   ||   ||   ||
|| heptatonic || [[4L 3s]] || 1 || 7 ||   ||   ||   ||   ||   ||   || 1 || 7 ||   ||   ||   ||   ||   ||   || 1 || 7 ||   ||   ||   ||   ||   ||   || 7 ||   ||   ||   ||   ||   ||   ||
|| 11-tone || [[4L 7s]] || 1 || 1 || 6 ||   ||   ||   ||   ||   || 1 || 1 || 6 ||   ||   ||   ||   ||   || 1 || 1 || 6 ||   ||   ||   ||   ||   || 1 || 6 ||   ||   ||   ||   ||   ||
|| 15-tone || [[4L 11s]] || 1 || 1 || 1 || 5 ||   ||   ||   ||   || 1 || 1 || 1 || 5 ||   ||   ||   ||   || 1 || 1 || 1 || 5 ||   ||   ||   ||   || 1 || 1 || 5 ||   ||   ||   ||   ||
|| 19-tone || [[4L 15s]] || 1 || 1 || 1 || 1 || 4 ||   ||   ||   || 1 || 1 || 1 || 1 || 4 ||   ||   ||   || 1 || 1 || 1 || 1 || 4 ||   ||   ||   || 1 || 1 || 1 || 4 ||   ||   ||   ||
|| 23-tone || [[4L 19s]] || 1 || 1 || 1 || 1 || 1 || 3 ||   ||   || 1 || 1 || 1 || 1 || 1 || 3 ||   ||   || 1 || 1 || 1 || 1 || 1 || 3 ||   ||   || 1 || 1 || 1 || 1 || 3 ||   ||   ||
|| 27-tone || [[4L 23s]] || 1 || 1 || 1 || 1 || 1 || 1 || 2 ||   || 1 || 1 || 1 || 1 || 1 || 1 || 2 ||   || 1 || 1 || 1 || 1 || 1 || 1 || 2 ||   || 1 || 1 || 1 || 1 || 1 || 2 ||   ||


===9\31 octave - approx. 348.39¢ - Neutral Third=== 
A neutral 3rd, practically equivalent to 11:9 (347.41¢). Exactly half a perfect fifth (making it a suitable generator for neutral third scales such as [[3L 4s]]). Is also thrice a major semitone. Generates [[Meantone family|mohajira temperament]].
====MOS Scales generated by 9\31:==== 
||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||
|| tetratonic || [[3L 1s]] || 9 ||   ||   ||   ||   ||   ||   ||   ||   || 9 ||   ||   ||   ||   ||   ||   ||   ||   || 9 ||   ||   ||   ||   ||   ||   ||   ||   || 4 ||   ||   ||   ||
|| heptatonic (quasi-equal) || [[3L 4s]] || 5 ||   ||   ||   ||   || 4 ||   ||   ||   || 5 ||   ||   ||   ||   || 4 ||   ||   ||   || 5 ||   ||   ||   ||   || 4 ||   ||   ||   || 4 ||   ||   ||   ||
|| 10-tone || [[7L 3s]] || 1 || 4 ||   ||   ||   || 4 ||   ||   ||   || 1 || 4 ||   ||   ||   || 4 ||   ||   ||   || 1 || 4 ||   ||   ||   || 4 ||   ||   ||   || 4 ||   ||   ||   ||
|| 17-tone || [[7L 10s]] || 1 || 1 || 3 ||   ||   || 1 || 3 ||   ||   || 1 || 1 || 3 ||   ||   || 1 || 3 ||   ||   || 1 || 1 || 3 ||   ||   || 1 || 3 ||   ||   || 1 || 3 ||   ||   ||
|| 24-tone || [[7L 17s]] || 1 || 1 || 1 || 2 ||   || 1 || 1 || 2 ||   || 1 || 1 || 1 || 2 ||   || 1 || 1 || 2 ||   || 1 || 1 || 1 || 2 ||   || 1 || 1 || 2 ||   || 1 || 1 || 2 ||   ||


===10\31 octave - approx. 387.10¢ - Major Third=== 
A near-just major 3rd (compare with 5:4 = 386.31¢). Has led to the characterization of 31-edo as "smooth". Generates [[Wuerschmidt family|wurshmidt/worshmidt temperaments]].
====MOS Scales generated by 10\31:==== 
||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||
|| tritonic (quasi-equal) || [[1L 2s]] || 10 ||   ||   ||   ||   ||   ||   ||   ||   ||   || 10 ||   ||   ||   ||   ||   ||   ||   ||   ||   || 11 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
|| tetratonic || [[3L 1s]] || 10 ||   ||   ||   ||   ||   ||   ||   ||   ||   || 10 ||   ||   ||   ||   ||   ||   ||   ||   ||   || 10 ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 ||
|| heptatonic || [[3L 4s]] || 9 ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 9 ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 9 ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 ||
|| 10-tone || [[3L 7s]] || 8 ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 8 ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 8 ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 1 ||
|| 13-tone || [[3L 10s]] || 7 ||   ||   ||   ||   ||   ||   || 1 || 1 || 1 || 7 ||   ||   ||   ||   ||   ||   || 1 || 1 || 1 || 7 ||   ||   ||   ||   ||   ||   || 1 || 1 || 1 || 1 ||
|| 16-tone || [[3L 13s]] || 6 ||   ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 6 ||   ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 6 ||   ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 1 ||
|| 19-tone || [[3L 16s]] || 5 ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 5 ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 5 ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 ||
|| 22-tone || [[3L 19s]] || 4 ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 4 ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 4 ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 ||
|| 25-tone || [[3L 22s]] || 3 ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 3 ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 3 ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 ||
|| 28-tone || [[3L 25s]] || 2 ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 2 ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 2 ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 ||

===11\31 octave - approx. 425.806¢ - Supermajor Third=== 
In 11-limit tonal music, 11\31 functions as 14:11 (417.51¢), 32:25 (427.37¢), 9:7 (435.08¢) and others. Generates [[Meantone family|squares temperament]].
====MOS Scales generated by 11\31:==== 
||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||
|| tritonic || [[2L 1s]] || 11 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 11 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 9 ||   ||   ||   ||   ||   ||   ||   ||   ||
|| pentatonic || [[3L 2s]] || 2 ||   || 9 ||   ||   ||   ||   ||   ||   ||   ||   || 2 ||   || 9 ||   ||   ||   ||   ||   ||   ||   ||   || 9 ||   ||   ||   ||   ||   ||   ||   ||   ||
|| octatonic || [[3L 5s]] || 2 ||   || 2 ||   || 7 ||   ||   ||   ||   ||   ||   || 2 ||   || 2 ||   || 7 ||   ||   ||   ||   ||   ||   || 2 ||   || 7 ||   ||   ||   ||   ||   ||   ||
|| 11-tone || [[3L 8s]] || 2 ||   || 2 ||   || 2 ||   || 5 ||   ||   ||   ||   || 2 ||   || 2 ||   || 2 ||   || 5 ||   ||   ||   ||   || 2 ||   || 2 ||   || 5 ||   ||   ||   ||   ||
|| 14-tone (quasi-equal) || [[3L 11s]] || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 3 ||   ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 3 ||   ||   || 2 ||   || 2 ||   || 2 ||   || 3 ||   ||   ||
|| 17-tone || [[3L 14s]] || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 1 || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 ||   || 2 || 1 ||   || 2 ||   || 2 ||   || 2 ||   || 1 ||


===12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth=== 
Exactly twice a supermajor second, thrice a neutral second, or four times a major second. In 7-limit tonal music, 12\31 functions as 21:16 (470.78¢). Generates semisept temperament.
====MOS Scales generated by 12\31:==== 
||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||
|| tritonic || [[2L 1s]] || 12 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 12 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 7 ||   ||   ||   ||   ||   ||   ||
|| pentatonic || [[3L 2s]] || 5 ||   ||   ||   ||   || 7 ||   ||   ||   ||   ||   ||   || 5 ||   ||   ||   ||   || 7 ||   ||   ||   ||   ||   ||   || 7 ||   ||   ||   ||   ||   ||   ||
|| octatonic || [[5L 3s]] || 5 ||   ||   ||   ||   || 5 ||   ||   ||   ||   || 2 ||   || 5 ||   ||   ||   ||   || 5 ||   ||   ||   ||   || 2 ||   || 5 ||   ||   ||   ||   || 2 ||   ||
|| 13-tone (quasi-equal) || [[5L 8s]] || 3 ||   ||   || 2 ||   || 3 ||   ||   || 2 ||   || 2 ||   || 3 ||   ||   || 2 ||   || 3 ||   ||   || 2 ||   || 2 ||   || 3 ||   ||   || 2 ||   || 2 ||   ||
|| 18-tone || [[13L 5s]] || 1 || 2 ||   || 2 ||   || 1 || 2 ||   || 2 ||   || 2 ||   || 1 || 2 ||   || 2 ||   || 1 || 2 ||   || 2 ||   || 2 ||   || 1 || 2 ||   || 2 ||   || 2 ||   ||


===13\31 octave - approx. 503.23¢ - Perfect Fourth=== 
A sharp perfect fourth (compare to 4:3 = 498.04¢). As such, it functions marvelously as a generator for meantone temperament.
====MOS Scales generated by 13\31:==== 
||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||
|| tritonic || [[2L 1s]] || 13 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 13 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 5 ||   ||   ||   ||   ||
|| pentatonic || [[2L 3s]] || 8 ||   ||   ||   ||   ||   ||   ||   || 5 ||   ||   ||   ||   || 8 ||   ||   ||   ||   ||   ||   ||   || 5 ||   ||   ||   ||   || 5 ||   ||   ||   ||   ||
|| heptatonic || [[5L 2s]] || 3 ||   ||   || 5 ||   ||   ||   ||   || 5 ||   ||   ||   ||   || 3 ||   ||   || 5 ||   ||   ||   ||   || 5 ||   ||   ||   ||   || 5 ||   ||   ||   ||   ||
|| 12-tone (quasi-equal) || [[7L 5s]] || 3 ||   ||   || 3 ||   ||   || 2 ||   || 3 ||   ||   || 2 ||   || 3 ||   ||   || 3 ||   ||   || 2 ||   || 3 ||   ||   || 2 ||   || 3 ||   ||   || 2 ||   ||
|| 19-tone || [[12L 7s]] || 1 || 2 ||   || 1 || 2 ||   || 2 ||   || 1 || 2 ||   || 2 ||   || 1 || 2 ||   || 1 || 2 ||   || 2 ||   || 1 || 2 ||   || 2 ||   || 1 || 2 ||   || 2 ||   ||


===14\31 octave - approx. 541.94¢ - Superfourth=== 
9.4¢ off from a just 11:8 (551.32¢), which isn't bad relative to the size of a 31EDO step, as it's less than a quarter of a 31EDO step away. Exactly twice a subminor third. Generates [[Starling temperaments|casablanca temperament]].
====MOS Scales generated by 14\31:==== 
||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||
|| tritonic || [[2L 1s]] || 14 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 14 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 3 ||   ||   ||
|| pentatonic || [[2L 3s]] || 11 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 3 ||   ||   || 11 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 3 ||   ||   || 3 ||   ||   ||
|| heptatonic || [[2L 5s]] || 8 ||   ||   ||   ||   ||   ||   ||   || 3 ||   ||   || 3 ||   ||   || 8 ||   ||   ||   ||   ||   ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   ||
|| nonatonic || [[2L 7s]] || 5 ||   ||   ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 5 ||   ||   ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   ||
|| 11-tone (quasi-equal) || [[9L 2s]] || 2 ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 2 ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   || 3 ||   ||   ||
|| 20-tone || [[11L 9s]] || 2 ||   || 2 ||   || 1 || 2 ||   || 1 || 2 ||   || 1 || 2 ||   || 1 || 2 ||   || 2 ||   || 1 || 2 ||   || 1 || 2 ||   || 1 || 2 ||   || 1 || 2 ||   || 1 ||


===15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth=== 
In 7-limit tonal music, functions as 7:5 (582.51¢). Exactly thrice a whole tone. Generates tritonic temperament.
====MOS Scales generated by 15\31:==== 
||~ number of tones ||~ MOS class ||~ 0 ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||~ 16 ||~ 17 ||~ 18 ||~ 19 ||~ 20 ||~ 21 ||~ 22 ||~ 23 ||~ 24 ||~ 25 ||~ 26 ||~ 27 ||~ 28 ||~ 29 ||~ 30 ||
|| tritonic || [[2L 1s]] || 15 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 15 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 ||
|| pentatonic || [[2L 3s]] || 14 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 14 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 ||
|| heptatonic || [[2L 5s]] || 13 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 13 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 1 ||
|| nonatonic || [[2L 7s]] || 12 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 1 || 12 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 1 || 1 ||
|| 11-tone || [[2L 9s]] || 11 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 11 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 1 ||
|| 13-tone || [[2L 11s]] || 10 ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 10 ||   ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 ||
|| 15-tone || [[2L 13s]] || 9 ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 9 ||   ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 ||
|| 17-tone || [[2L 15s]] || 8 ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 8 ||   ||   ||   ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 ||
|| 19-tone || [[2L 17s]] || 7 ||   ||   ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 7 ||   ||   ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 ||
|| 21-tone || [[2L 19s]] || 6 ||   ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 6 ||   ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 ||
|| 23-tone || [[2L 21s]] || 5 ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 5 ||   ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 ||
|| 25-tone || [[2L 23s]] || 4 ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 4 ||   ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 ||
|| 27-tone || [[2L 25s]] || 3 ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 3 ||   ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 ||
|| 29-tone || [[2L 27s]] || 2 ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 2 ||   || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 ||


===16\31 octave=== 
The large tritone.

=Harmonic Scale= 
31edo approximates Mode 8 of the [[OverToneSeries|harmonic series]] O.K., but many intervals between the harmonics aren't distinguished, most importantly 9/8 (major tone) and 10/9 (minor tone), as 31EDO is a meantone temperament. The interval between the 8th and 11th harmonics is approximated O.K., but the intervals between the 11th harmonic and closer harmonics such as the 12th and 9th harmonics are approximated even better. 13/8 isn't distinguished from the 11-limit 18/11, as 144/143 is tempered out, so 13-limit intervals can't be distinguished from 11-limit ones.

|| Overtones in "Mode 8": || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 ||
|| ...as JI Ratio from 1/1: || 1/1 || 9/8 || 5/4 || 11/8 || 3/2 || 13/8 || 7/4 || 15/8 || 2/1 ||
|| ...in cents: || 0 || 203.9 || 386.3 || 551.3 || 702.0 || 840.5 || 968.8 || 1088.3 || 1200.0 ||
|| Nearest degree of 41edo: || 0 || 5 || 10 || 14 || 18 || 22 || 25 || 28 || 31 ||
|| ...in cents: || 0 || 193.5 || 387.1 || 541.9 || 696.8 || 851.6 || 967.7 || 1083.9 || 1200.0 ||
=Commas= 
31 EDO tempers out the following commas. (Note: This assumes the val < 31 49 72 87 107 115 |, comma values roundet to 5 significant digits.)
||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 9931568/9752117 ||< | -25 7 6 > ||> 31.567 ||= Ampersand's Comma ||=   ||=   ||
||= 81/80 ||< | -4 4 -1 > ||> 21.506 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 393216/390625 ||< | 17 1 -8 > ||> 11.445 ||= Wuerschmidt Comma ||=   ||=   ||
||= 2109375/2097152 ||< | -21 3 7 > ||> 10.061 ||= Semicomma ||= Fokker Comma ||=   ||
||= 6719816/6714445 ||< | 38 -2 -15 > ||> 1.3843 ||= Hemithirds Comma ||=   ||=   ||
||= 9859966/9733137 ||< | -10 7 8 -7 > ||> 22.413 ||= Blackjackisma ||=   ||=   ||
||= 64827/64000 ||< | -9 3 -3 4 > ||> 22.227 ||= Squalentine ||=   ||=   ||
||= 2430/2401 ||< | 1 5 1 -4 > ||> 20.785 ||= Nuwell ||=   ||=   ||
||= 50421/50000 ||< | -4 1 -5 5 > ||> 14.516 ||= Trimyna ||=   ||=   ||
||= 126/125 ||< | 1 2 -3 1 > ||> 13.795 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 1728/1715 ||< | 6 3 -1 -3 > ||> 13.074 ||= Orwellisma ||= Orwell Comma ||=   ||
||= 1029/1024 ||< | -10 1 0 3 > ||> 8.4327 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||< | -5 2 2 -1 > ||> 7.7115 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||< | 0 3 4 -5 > ||> 6.9903 ||= Mirkwai ||=   ||=   ||
||= 3136/3125 ||< | 6 0 -5 2 > ||> 6.0832 ||= Hemimean ||=   ||=   ||
||= 6144/6125 ||< | 11 1 -3 -2 > ||> 5.3621 ||= Porwell ||=   ||=   ||
||= 1065875/1063543 ||< | -26 -1 1 9 > ||> 3.7919 ||= Wadisma ||=   ||=   ||
||= 65625/65536 ||< | -16 1 5 1 > ||> 2.3495 ||= Horwell ||=   ||=   ||
||= 703125/702464 ||< | -11 2 7 -3 > ||> 1.6283 ||= Meter ||=   ||=   ||
||= 2401/2400 ||< | -5 -1 -2 4 > ||> 0.72120 ||= Breedsma ||=   ||=   ||
||= 99/98 ||< | -1 2 0 -2 1 > ||> 17.576 ||= Mothwellsma ||=   ||=   ||
||= 121/120 ||< | -3 -1 -1 0 2 > ||> 14.367 ||= Biyatisma ||=   ||=   ||
||= 176/175 ||< | 4 0 -2 -1 1 > ||> 9.8646 ||= Valinorsma ||=   ||=   ||
||= 243/242 ||< | -1 5 0 0 -2 > ||> 7.1391 ||= Rastma ||=   ||=   ||
||= 385/384 ||< | -7 -1 1 1 1 > ||> 4.5026 ||= Keenanisma ||=   ||=   ||
||= 441/440 ||< | -3 2 -1 2 -1 > ||> 3.9302 ||= Werckisma ||=   ||=   ||
||= 540/539 ||< | 2 3 1 -2 -1 > ||> 3.2090 ||= Swetisma ||=   ||=   ||
||= 3025/3024 ||< | -4 -3 2 -1 2 > ||> 0.57240 ||= Lehmerisma ||=   ||=   ||

=Modes= 

A large open list of modes (subsets) from 31edo that people have named: [[31edo modes]]. [[http://en.wikipedia.org/wiki/Rothenberg_propriety|Strictly proper]] [[Strictly proper 7-note 31edo scales|7-note 31edo scales]] in the sense of [[David Rothenberg]]. See also [[31edo MOS scales]]. Some of the popular ones:

* 31-tone major: 5 5 3 5 5 5 3
* Meantone[12] (Eb-G#): 2 3 3 2 3 2 3 2 3 3 2 3
* Harmonic scale 8: 5 5 4 4 4 4 3 3
* the [[Euler-Fokker genera]] (technically [[JI]] but representable in 31)

|||| ====Some 31 tone equal modes:==== ||
|| {{**2 3 3 2 3 2 3 2 3 3 2 3**}} || Meantone Chromatic (53/220-comma) ||
|| {{**5 5 3 5 5 5 3**}} || Thirty-one tone Major, Intense Diatonic Lydian, M.Ionian ||
|| {{**5 3 5 5 3 5 5**}} || Thirty-one tone Natural Minor, Intense Diatonic Hypodorian, Aeolian ||
|| {{**5 3 5 5 5 5 3**}} || Thirty-one tone Melodic Minor ||
|| {{**5 3 5 5 3 7 3**}} || Thirty-one tone Harmonic Minor ||
|| {{**5 5 3 5 3 7 3**}} || Thirty-one tone Harmonic Major ||
|| {{**5 5 3 5 3 5 5**}} || Thirty-one tone Major-Minor ||
|| {{**5 8 5 13**}} || Genus primum ||
|| {{**10 3 5 5 5 3**}} || Genus secundum ||
|| {{**8 2 8 3 7 3**}} || Genus tertium ||
|| {{**10 10 10 1**}} || Genus quartum ||
|| {{**5 7 6 7 5 1**}} || Genus quintum ||
|| {{**4 6 2 6 4 3 3 3**}} || Genus sextum ||
|| {{**4 6 5 6 4 6**}} || Genus septimum ||
|| {{**6 6 6 1 6 6**}} || Genus octavum ||
|| {{**4 6 9 6 4 2**}} || Genus nonum ||
|| {{**13 6 6 6**}} || Genus decimum ||
|| {{**5 5 3 5 5 3 2 3**}} || Genus diatonicum ||
|| {{**3 5 2 3 5 3 2 5 3**}} || Genus chromaticum ||
|| {{**5 5 2 1 5 5 2 3 3**}} || Genus diatonicum cum septimis ||
|| {{**3 4 3 3 2 1 4 1 4 1 2 3**}} || Genus enharmonicum vocale ||
|| {{**2 2 4 2 2 3 3 3 1 3 3 3**}} || Genus enharmonicum instrumentale ||
|| {{**3 2 3 2 3 2 3 3 2 3 2 3**}} || Genus diatonico-chromaticum ||
|| {{**5 2 1 2 5 3 2 1 4 1 2 3**}} || Genus bichromaticum ||
|| {{**4 4 5 4 4 5 5**}} || Neutral Diatonic Mixolydian ||
|| {{**4 5 4 4 5 5 4**}} || Neutral Diatonic Lydian ||
|| {{**5 4 4 5 5 4 4**}} || Neutral Diatonic Phrygian ||
|| {{**4 4 5 5 4 4 5**}} || Neutral Diatonic Dorian ||
|| {{**4 5 5 4 4 5 4**}} || Neutral Diatonic Hypolydian ||
|| {{**5 5 4 4 5 4 4**}} || Neutral Diatonic Hypophrygian ||
|| {{**5 4 4 5 4 4 5**}} || Neutral Diatonic Hypodorian ||
|| {{**4 5 4 4 5 4 5**}} || Neutral Mixolydian ||
|| {{**5 4 4 5 4 5 4**}} || Neutral Lydian ||
|| {{**4 4 5 4 5 4 5**}} || Neutral Phrygian ||
|| {{**4 5 4 5 4 5 4**}} || Neutral Dorian ||
|| {{**5 4 5 4 5 4 4**}} || Neutral Hypolydian ||
|| {{**4 5 4 5 4 4 5**}} || Neutral Hypophrygian ||
|| {{**5 4 5 4 4 5 4**}} || Neutral Hypodorian ||
|| {{**2 2 9 2 2 9 5**}} || Hemiolic Chromatic Mixolydian ||
|| {{**2 9 2 2 9 5 2**}} || Hemiolic Chromatic Lydian ||
|| {{**9 2 2 9 5 2 2**}} || Hemiolic Chromatic Phrygian ||
|| {{**2 2 9 5 2 2 9**}} || Hemiolic Chromatic Dorian ||
|| {{**2 9 5 2 2 9 2**}} || Hemiolic Chromatic Hypolydian ||
|| {{**9 5 2 2 9 2 2**}} || Hemiolic Chromatic Hypophrygian ||
|| {{**5 2 2 9 2 2 9**}} || Hemiolic Chromatic Hypodorian ||
|| {{**2 3 8 2 3 8 5**}} || Ratio 2:3 Chromatic Mixolydian ||
|| {{**3 8 2 3 8 5 2**}} || Ratio 2:3 Chromatic Lydian ||
|| {{**8 2 3 8 5 2 3**}} || Ratio 2:3 Chromatic Phrygian ||
|| {{**2 3 8 5 2 3 8**}} || Ratio 2:3 Chromatic Dorian ||
|| {{**3 8 5 2 3 8 2**}} || Ratio 2:3 Chromatic Hypolydian ||
|| {{**8 5 2 3 8 2 3**}} || Ratio 2:3 Chromatic Hypophrygian ||
|| {{**5 2 3 8 2 3 8**}} || Ratio 2:3 Chromatic Hypodorian ||
|| {{**3 5 5 3 5 5 5**}} || Intense Diatonic Mixolydian, M.Locrian ||
|| {{**5 3 5 5 5 3 5**}} || Intense Diatonic Phrygian, M.Dorian ||
|| {{**3 5 5 5 3 5 5**}} || Intense Diatonic Dorian, M.Phrygian ||
|| {{**5 5 5 3 5 5 3**}} || Intense Diatonic Hypolydian, M.Lydian ||
|| {{**5 5 3 5 5 3 5**}} || Intense Diatonic Hypophrygian, M.Mixolydian ||
|| {{**2 5 6 2 5 6 5**}} || Soft Diatonic Mixolydian ||
|| {{**5 6 2 5 6 5 2**}} || Soft Diatonic Lydian ||
|| {{**6 2 5 6 5 2 5**}} || Soft Diatonic Phrygian ||
|| {{**2 5 6 5 2 5 6**}} || Soft Diatonic Dorian ||
|| {{**5 6 5 2 5 6 2**}} || Soft Diatonic Hypolydian ||
|| {{**6 5 2 5 6 2 5**}} || Soft Diatonic Hypophrygian ||
|| {{**5 2 5 6 2 5 6**}} || Soft Diatonic Hypodorian ||
|| {{**1 2 10 1 2 10 5**}} || Enharmonic Mixolydian ||
|| {{**2 10 1 2 10 5 1**}} || Enharmonic Lydian ||
|| {{**10 1 2 10 5 1 2**}} || Enharmonic Phrygian ||
|| {{**1 2 10 5 1 2 10**}} || Enharmonic Dorian ||
|| {{**2 10 5 1 2 10 1**}} || Enharmonic Hypolydian ||
|| {{**10 5 1 2 10 1 2**}} || Enharmonic Hypophrygian ||
|| {{**5 1 2 10 1 2 10**}} || Enharmonic Hypodorian ||
|| {{**6 6 7 6 6**}} || Quasi-equal Pentatonic ||
|| {{**3 2 2 3 3 2 3 3 2 2 3 3**}} || Fokker 12-tone ||
|| {{**5 3 5 3 5 2 5 3**}} || Modus conjunctus ||
|| {{**3 5 2 5 3 5 3 5**}} || Octatonic ||
|| {{**3 3 4 3 5 3 4 3 3**}} || Hahn symmetric pentachordal ||
|| {{**3 4 3 3 5 3 4 3 3**}} || Hahn pentachordal ||
|| {{**4 4 2 5 3 3 4 3 3**}} || Hahn Nonatonic ||
|| {{**5 1 5 1 5 1 5 1 5 1 1**}} || de Vries 11-tone ||
|| {{**4 1 4 4 4 1 4 4 1 4**}} || Breed 10-tone ||
|| {{**4 2 4 2 4 2 4 3 3 3**}} || Lumma Decatonic ||
|| {{**5 3 3 3 3 5 3 3 3**}} || Rothenberg Generalized Diatonic ||
|| {{**5 2 6 5 2 5 6**}} || "Septimal" Natural Minor ||
|| {{**4 3 4 3 4 3 4 3 3**}} || Thirty-one tone Orwell ||
|| {{**2 5 2 2 5 2 2 2 5 2 2**}} || Secor Sentinel ||

=Music in 31-edo= 
[[31-edo compositions|An alphabetical list of Tricesimoprimal Compositions]].

==Thirty-one tone pedagogy== 
The [[MicroPedagogyCollective]] is currently at work producing demonstrative material which will encourage and enable more people to learn this system. There have been two [[ThirtyOneToneSinginCamp]]s as well.

=Practical Theory / Books= 

[[image:http://ronsword.com/images/TSG_sm.jpg width="87" height="116" link="@http://www.ronsword.com/books.html"]][[@http://www.ronsword.com/books.html|Sword, Ronald. "Tricesimoprimal Scales for Guitar." IAAA Press, UK-USA. First Ed: March 2009.]] - A comprehensive approach to 31-EDO and all the families associated for Guitar. Features over 300 scale charts / scale examples.

=Other Articles= 
* <span class="wiki_link_ext">[[http://www.huygens-fokker.org/docs/beerart.html|de Beer, Anton, ''The Development of 31-tone Music]]</span> [[http://www.webcitation.org/5xeFzBM9b|Permalink]]
* <span class="wiki_link_ext">[[http://www.huygens-fokker.org/docs/fokkerorg.html|Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ]]</span> [[http://www.webcitation.org/5xeG6Tmli|Permalink]]
* Fokker, A.D., "New Music with 31 Notes" translated by Leigh Gerdine
* <span class="wiki_link_ext">[[http://www.huygens-fokker.org/docs/rap31.html|Rapoport, Paul, ''About 31-tone Equal Temperament]]</span> [[http://www.webcitation.org/5xeGH4uBH|Permalink]]
* <span class="wiki_link_ext">[[http://www.huygens-fokker.org/docs/terp31.html|Terpstra, Siemen, ''Toward a Theory of Meantone (and 31-et) Harmony'']]</span> [[http://www.webcitation.org/5xeGMeCMd|Permalink]]
* <span class="wiki_link_ext">[[http://tonalsoft.com/enc/number/31edo.aspx|Tonalsoft Encyclopedia article]]</span> [[http://www.webcitation.org/5xeGYj7QU|Permalink]]

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<html><head><title>31edo</title></head><body><!-- ws:start:WikiTextTocRule:80:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:80 --><!-- ws:start:WikiTextTocRule:81: --><a href="#Quick index to intervals and linear temperaments">Quick index to intervals and linear temperaments</a><!-- ws:end:WikiTextTocRule:81 --><!-- ws:start:WikiTextTocRule:82: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:82 --><!-- ws:start:WikiTextTocRule:83: --><!-- ws:end:WikiTextTocRule:83 --><!-- ws:start:WikiTextTocRule:84: --><!-- ws:end:WikiTextTocRule:84 --><!-- ws:start:WikiTextTocRule:85: --><!-- ws:end:WikiTextTocRule:85 --><!-- ws:start:WikiTextTocRule:86: --><!-- ws:end:WikiTextTocRule:86 --><!-- ws:start:WikiTextTocRule:87: --><!-- ws:end:WikiTextTocRule:87 --><!-- ws:start:WikiTextTocRule:88: --><!-- ws:end:WikiTextTocRule:88 --><!-- ws:start:WikiTextTocRule:89: --><!-- ws:end:WikiTextTocRule:89 --><!-- ws:start:WikiTextTocRule:90: --><!-- ws:end:WikiTextTocRule:90 --><!-- ws:start:WikiTextTocRule:91: --><!-- ws:end:WikiTextTocRule:91 --><!-- ws:start:WikiTextTocRule:92: --><!-- ws:end:WikiTextTocRule:92 --><!-- ws:start:WikiTextTocRule:93: --><!-- ws:end:WikiTextTocRule:93 --><!-- ws:start:WikiTextTocRule:94: --><!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --><!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --><!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --><!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --><!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextTocRule:108: --><!-- ws:end:WikiTextTocRule:108 --><!-- ws:start:WikiTextTocRule:109: --><!-- ws:end:WikiTextTocRule:109 --><!-- ws:start:WikiTextTocRule:110: --><!-- ws:end:WikiTextTocRule:110 --><!-- ws:start:WikiTextTocRule:111: --><!-- ws:end:WikiTextTocRule:111 --><!-- ws:start:WikiTextTocRule:112: --><!-- ws:end:WikiTextTocRule:112 --><!-- ws:start:WikiTextTocRule:113: --> | <a href="#Harmonic Scale">Harmonic Scale</a><!-- ws:end:WikiTextTocRule:113 --><!-- ws:start:WikiTextTocRule:114: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:114 --><!-- ws:start:WikiTextTocRule:115: --> | <a href="#Modes">Modes</a><!-- ws:end:WikiTextTocRule:115 --><!-- ws:start:WikiTextTocRule:116: --><!-- ws:end:WikiTextTocRule:116 --><!-- ws:start:WikiTextTocRule:117: --> | <a href="#Music in 31-edo">Music in 31-edo</a><!-- ws:end:WikiTextTocRule:117 --><!-- ws:start:WikiTextTocRule:118: --><!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --> | <a href="#Practical Theory / Books">Practical Theory / Books</a><!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --> | <a href="#Other Articles">Other Articles</a><!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: -->
<!-- ws:end:WikiTextTocRule:121 --><hr />
<em>Thirty-one tone equal temperament</em>, also called <em>31-tET</em>, <em>31-EDO</em>, <em>31-et</em>, or <em>tricesimoprimal meantone temperament</em>, is the scale derived by dividing the octave into 31 <a class="wiki_link" href="/equal">equally</a> large steps. The term 'Tricesimoprimal' was first used by <a class="wiki_link" href="/Adriaan%20Fokker">Adriaan Fokker</a>. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 <a class="wiki_link" href="/cents">cents</a>. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). It's not bad for an 11-limit approximation, as it is consistent through it, but is the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablance in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.<br />
<br />
31edo is the 11th <a class="wiki_link" href="/prime%20numbers">prime</a> edo, following <a class="wiki_link" href="/29edo">29edo</a> and coming before <a class="wiki_link" href="/37edo">37edo</a>.<br />
<br />
For more encyclopedic info, see <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/31_equal_temperament" rel="nofollow">Wikipedia's article</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Quick index to intervals and linear temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Quick index to intervals and linear temperaments</h1>
 

<table class="wiki_table">
    <tr>
        <th>Generator<br />
</th>
        <th>Cents<br />
</th>
        <th>Temperaments<br />
</th>
    </tr>
    <tr>
        <td>1\31<br />
</td>
        <td>38.71<br />
</td>
        <td><a class="wiki_link" href="/Slender">Slender</a><br />
</td>
    </tr>
    <tr>
        <td>2\31<br />
</td>
        <td>77.42<br />
</td>
        <td><a class="wiki_link" href="/Valentine">Valentine</a>/<a class="wiki_link" href="/Lupercalia">Lupercalia</a><br />
</td>
    </tr>
    <tr>
        <td>3\31<br />
</td>
        <td>116.13<br />
</td>
        <td><a class="wiki_link" href="/Miracle">Miracle</a><br />
</td>
    </tr>
    <tr>
        <td>4\31<br />
</td>
        <td>154.84<br />
</td>
        <td><a class="wiki_link" href="/Nusecond">Nusecond</a><br />
</td>
    </tr>
    <tr>
        <td>5\31<br />
</td>
        <td>193.55<br />
</td>
        <td><a class="wiki_link" href="/Luna">Luna</a>/<a class="wiki_link" href="/Hemithird">Hemithird</a>/<a class="wiki_link" href="/Hemiw%C3%BCrschmidt">Hemiwürschmidt</a><br />
</td>
    </tr>
    <tr>
        <td>6\31<br />
</td>
        <td>232.26<br />
</td>
        <td><a class="wiki_link" href="/Cynder">Cynder</a>/<a class="wiki_link" href="/Mothra">Mothra</a><br />
</td>
    </tr>
    <tr>
        <td>7\31<br />
</td>
        <td>270.97<br />
</td>
        <td><a class="wiki_link" href="/Orson">Orson</a>/<a class="wiki_link" href="/Orwell">Orwell</a><br />
</td>
    </tr>
    <tr>
        <td>8\31<br />
</td>
        <td>309.68<br />
</td>
        <td><a class="wiki_link" href="/Myna">Myna</a><br />
</td>
    </tr>
    <tr>
        <td>9\31<br />
</td>
        <td>348.39<br />
</td>
        <td><a class="wiki_link" href="/Vicentino">Vicentino</a>/<a class="wiki_link" href="/Mohajira">Mohajira</a>/<a class="wiki_link" href="/Migration">Migration</a><br />
</td>
    </tr>
    <tr>
        <td>10\31<br />
</td>
        <td>387.10<br />
</td>
        <td><a class="wiki_link" href="/W%C3%BCrschmidt">Würschmidt</a>/<a class="wiki_link" href="/Worschmidt">Worschmidt</a><br />
</td>
    </tr>
    <tr>
        <td>11\31<br />
</td>
        <td>425.81<br />
</td>
        <td><a class="wiki_link" href="/Squares">Squares</a>/<a class="wiki_link" href="/Sentinel">Sentinel</a><br />
</td>
    </tr>
    <tr>
        <td>12\31<br />
</td>
        <td>464.52<br />
</td>
        <td><a class="wiki_link" href="/Semisept">Semisept</a><br />
</td>
    </tr>
    <tr>
        <td>13\31<br />
</td>
        <td>503.23<br />
</td>
        <td><a class="wiki_link" href="/Meantone">Meantone</a>/<a class="wiki_link" href="/Meanpop">Meanpop</a><br />
</td>
    </tr>
    <tr>
        <td>14\31<br />
</td>
        <td>541.94<br />
</td>
        <td><a class="wiki_link" href="/Casablanca">Casablanca</a>/<a class="wiki_link" href="/Marrakesh">Marrakesh</a><br />
</td>
    </tr>
    <tr>
        <td>15\31<br />
</td>
        <td>580.65<br />
</td>
        <td><a class="wiki_link" href="/Tritonic">Tritonic</a><br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:2 -->Intervals</h1>
 <!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="Intervals--1\31 octave - approx. 38.71¢ - Diesis"></a><!-- ws:end:WikiTextHeadingRule:4 -->1\31 octave - approx. 38.71¢ - Diesis</h3>
 A single step of 31-edo is about 38.71¢. Intervals around this size are called <a class="wiki_link" href="/diesis">dieses</a> (singular 'diesis'). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G#, to B#, but B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). In 11-limit tonal music, the single step stands in for just ratios 56:55 (31.19); 55:54 (31.77¢); 49:48 (39.70¢); 45:44 (38.91¢); 36:35 (48.77¢); 33:32 (53.27¢) and others. Demonstrated in <a class="wiki_link" href="/SpiralProgressions">SpiralProgressions</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Intervals--2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second"></a><!-- ws:end:WikiTextHeadingRule:6 -->2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second</h3>
 The difference between a major and minor third. The more 'expressive' of the 'half steps'. In 11-limit tonal music, 2\31 stands in for just ratios 28:27 (62.96¢); 25:24 (70.67¢); 22:21 (80.54¢); 21:20 (84.45¢) and others. Generates <a class="wiki_link" href="/Starling%20temperaments#Valentine%20temperament">valentine temperament</a> - aka <a class="wiki_link" href="/Armodue%20theory#Semi-equalized%20Armodue">semi-equalized Armodue</a>.<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h4&gt; --><h4 id="toc4"><a name="Intervals--2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second-MOS Scales generated by 2\31:"></a><!-- ws:end:WikiTextHeadingRule:8 -->MOS Scales generated by 2\31:</h4>
 

<table class="wiki_table">
    <tr>
        <th>number of tones<br />
</th>
        <th>MOS class<br />
</th>
        <th>0<br />
</th>
        <th>1<br />
</th>
        <th>2<br />
</th>
        <th>3<br />
</th>
        <th>4<br />
</th>
        <th>5<br />
</th>
        <th>6<br />
</th>
        <th>7<br />
</th>
        <th>8<br />
</th>
        <th>9<br />
</th>
        <th>10<br />
</th>
        <th>11<br />
</th>
        <th>12<br />
</th>
        <th>13<br />
</th>
        <th>14<br />
</th>
        <th>15<br />
</th>
        <th>16<br />
</th>
        <th>17<br />
</th>
        <th>18<br />
</th>
        <th>19<br />
</th>
        <th>20<br />
</th>
        <th>21<br />
</th>
        <th>22<br />
</th>
        <th>23<br />
</th>
        <th>24<br />
</th>
        <th>25<br />
</th>
        <th>26<br />
</th>
        <th>27<br />
</th>
        <th>28<br />
</th>
        <th>29<br />
</th>
        <th>30<br />
</th>
    </tr>
    <tr>
        <td>15-tone (<a class="wiki_link" href="/Maximal%20evenness">ME</a> or quasi-equal)<br />
</td>
        <td><a class="wiki_link" href="/1L%2014s">1L 14s</a><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>16-tone<br />
</td>
        <td><a class="wiki_link" href="/15L%201s">15L 1s</a><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Intervals--3\31 octave - approx. 116.13 - Major Semitone or Diatonic Semitone or Large Major Second"></a><!-- ws:end:WikiTextHeadingRule:10 -->3\31 octave - approx. 116.13 - Major Semitone or Diatonic Semitone or Large Major Second</h3>
 The difference between a perfect fourth and a major third. The larger and clunkier of the 'half steps'. In 11-limit tonal music, 3\31 stands in for just ratios 16:15 (111.73¢); 15:14 (199.44¢) and others. Generates <a class="wiki_link" href="/Gamelismic%20clan">miracle temperament</a>.<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h4&gt; --><h4 id="toc6"><a name="Intervals--3\31 octave - approx. 116.13 - Major Semitone or Diatonic Semitone or Large Major Second-MOS Scales generated by 3\31:"></a><!-- ws:end:WikiTextHeadingRule:12 -->MOS Scales generated by 3\31:</h4>
 

<table class="wiki_table">
    <tr>
        <th>number of tones<br />
</th>
        <th>MOS class<br />
</th>
        <th>0<br />
</th>
        <th>1<br />
</th>
        <th>2<br />
</th>
        <th>3<br />
</th>
        <th>4<br />
</th>
        <th>5<br />
</th>
        <th>6<br />
</th>
        <th>7<br />
</th>
        <th>8<br />
</th>
        <th>9<br />
</th>
        <th>10<br />
</th>
        <th>11<br />
</th>
        <th>12<br />
</th>
        <th>13<br />
</th>
        <th>14<br />
</th>
        <th>15<br />
</th>
        <th>16<br />
</th>
        <th>17<br />
</th>
        <th>18<br />
</th>
        <th>19<br />
</th>
        <th>20<br />
</th>
        <th>21<br />
</th>
        <th>22<br />
</th>
        <th>23<br />
</th>
        <th>24<br />
</th>
        <th>25<br />
</th>
        <th>26<br />
</th>
        <th>27<br />
</th>
        <th>28<br />
</th>
        <th>29<br />
</th>
        <th>30<br />
</th>
    </tr>
    <tr>
        <td>nonatonic<br />
</td>
        <td><a class="wiki_link" href="/1L%208s">1L 8s</a><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>decatonic (quasi-equal)<br />
</td>
        <td><a class="wiki_link" href="/9L%201s">9L 1s</a><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>11-tone<br />
</td>
        <td><a class="wiki_link" href="/10L%201s">10L 1s</a><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>21-tone (Blackjack)<br />
</td>
        <td><a class="wiki_link" href="/11L%2010s">11L 10s</a><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Intervals--4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second"></a><!-- ws:end:WikiTextHeadingRule:14 -->4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second</h3>
 Exactly one half of the minor third (and twice the minor semitone). In 11-limit tonal music, 4\31 stands in for 12:11 (150.64¢); 35:32 (155.14¢); 11:10 (165.00¢) and others. Generates <a class="wiki_link" href="/Starling%20temperaments">nusecond temperament</a>.<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h4&gt; --><h4 id="toc8"><a name="Intervals--4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second-MOS Scales generated by 4\31:"></a><!-- ws:end:WikiTextHeadingRule:16 -->MOS Scales generated by 4\31:</h4>
 

<table class="wiki_table">
    <tr>
        <th>number of tones<br />
</th>
        <th>MOS class<br />
</th>
        <th>0<br />
</th>
        <th>1<br />
</th>
        <th>2<br />
</th>
        <th>3<br />
</th>
        <th>4<br />
</th>
        <th>5<br />
</th>
        <th>6<br />
</th>
        <th>7<br />
</th>
        <th>8<br />
</th>
        <th>9<br />
</th>
        <th>10<br />
</th>
        <th>11<br />
</th>
        <th>12<br />
</th>
        <th>13<br />
</th>
        <th>14<br />
</th>
        <th>15<br />
</th>
        <th>16<br />
</th>
        <th>17<br />
</th>
        <th>18<br />
</th>
        <th>19<br />
</th>
        <th>20<br />
</th>
        <th>21<br />
</th>
        <th>22<br />
</th>
        <th>23<br />
</th>
        <th>24<br />
</th>
        <th>25<br />
</th>
        <th>26<br />
</th>
        <th>27<br />
</th>
        <th>28<br />
</th>
        <th>29<br />
</th>
        <th>30<br />
</th>
    </tr>
    <tr>
        <td>heptatonic<br />
</td>
        <td><a class="wiki_link" href="/1L%206s">1L 6s</a><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>octatonic (quasi-equal)<br />
</td>
        <td><a class="wiki_link" href="/7L%201s">7L 1s</a><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>15-tone<br />
</td>
        <td><a class="wiki_link" href="/8L%207s">8L 7s</a><br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>23-tone<br />
</td>
        <td><a class="wiki_link" href="/8L%2015s">8L 15s</a><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="Intervals--5\31 octave - approx. 193.55¢ - Whole Tone or Major Second"></a><!-- ws:end:WikiTextHeadingRule:18 -->5\31 octave - approx. 193.55¢ - Whole Tone or Major Second</h3>
 A rather smallish whole tone. Often called melodically dull. As it falls between (and functions as) just whole tones 9:8 and 10:9, 5\31 is considered a &quot;meantone&quot;. Two meantones make a near-just major third. Generates <a class="wiki_link" href="/Gamelismic%20clan">hemithirds temperament</a> and <a class="wiki_link" href="/Wuerschmidt%20family">hermiwuerschmidt temperament</a>.<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h4&gt; --><h4 id="toc10"><a name="Intervals--5\31 octave - approx. 193.55¢ - Whole Tone or Major Second-MOS Scales generated by 5\31:"></a><!-- ws:end:WikiTextHeadingRule:20 -->MOS Scales generated by 5\31:</h4>
 

<table class="wiki_table">
    <tr>
        <th>number of tones<br />
</th>
        <th>MOS class<br />
</th>
        <th>0<br />
</th>
        <th>1<br />
</th>
        <th>2<br />
</th>
        <th>3<br />
</th>
        <th>4<br />
</th>
        <th>5<br />
</th>
        <th>6<br />
</th>
        <th>7<br />
</th>
        <th>8<br />
</th>
        <th>9<br />
</th>
        <th>10<br />
</th>
        <th>11<br />
</th>
        <th>12<br />
</th>
        <th>13<br />
</th>
        <th>14<br />
</th>
        <th>15<br />
</th>
        <th>16<br />
</th>
        <th>17<br />
</th>
        <th>18<br />
</th>
        <th>19<br />
</th>
        <th>20<br />
</th>
        <th>21<br />
</th>
        <th>22<br />
</th>
        <th>23<br />
</th>
        <th>24<br />
</th>
        <th>25<br />
</th>
        <th>26<br />
</th>
        <th>27<br />
</th>
        <th>28<br />
</th>
        <th>29<br />
</th>
        <th>30<br />
</th>
    </tr>
    <tr>
        <td>hexatonic (quasi-equal)<br />
</td>
        <td><a class="wiki_link" href="/1L%205s">1L 5s</a><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>heptatonic<br />
</td>
        <td><a class="wiki_link" href="/6L%201s">6L 1s</a><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>13-tone<br />
</td>
        <td><a class="wiki_link" href="/6L%207s">6L 7s</a><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>19-tone<br />
</td>
        <td><a class="wiki_link" href="/6L%2013s">6L 13s</a><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>25-tone<br />
</td>
        <td><a class="wiki_link" href="/6L%2019s">6L 19s</a><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Intervals--6\31 octave - approx. 232.26¢ - Supermajor Second"></a><!-- ws:end:WikiTextHeadingRule:22 -->6\31 octave - approx. 232.26¢ - Supermajor Second</h3>
 Exactly one half of a narrow fourth, twice a major semitone, or thrice a minor semitone. In 7-limit tonal music, 6\31 stands in for 8:7 (231.17¢). Generates <a class="wiki_link" href="/Meantone%20family">mothra temperament</a>.<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h4&gt; --><h4 id="toc12"><a name="Intervals--6\31 octave - approx. 232.26¢ - Supermajor Second-MOS Scales generated by 6\31:"></a><!-- ws:end:WikiTextHeadingRule:24 -->MOS Scales generated by 6\31:</h4>
 

<table class="wiki_table">
    <tr>
        <th>number of tones<br />
</th>
        <th>MOS class<br />
</th>
        <th>0<br />
</th>
        <th>1<br />
</th>
        <th>2<br />
</th>
        <th>3<br />
</th>
        <th>4<br />
</th>
        <th>5<br />
</th>
        <th>6<br />
</th>
        <th>7<br />
</th>
        <th>8<br />
</th>
        <th>9<br />
</th>
        <th>10<br />
</th>
        <th>11<br />
</th>
        <th>12<br />
</th>
        <th>13<br />
</th>
        <th>14<br />
</th>
        <th>15<br />
</th>
        <th>16<br />
</th>
        <th>17<br />
</th>
        <th>18<br />
</th>
        <th>19<br />
</th>
        <th>20<br />
</th>
        <th>21<br />
</th>
        <th>22<br />
</th>
        <th>23<br />
</th>
        <th>24<br />
</th>
        <th>25<br />
</th>
        <th>26<br />
</th>
        <th>27<br />
</th>
        <th>28<br />
</th>
        <th>29<br />
</th>
        <th>30<br />
</th>
    </tr>
    <tr>
        <td>pentatonic (quasi-equal)<br />
</td>
        <td><a class="wiki_link" href="/1L%204s">1L 4s</a><br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>hexatonic<br />
</td>
        <td><a class="wiki_link" href="/5L%201s">5L 1s</a><br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>11-tone<br />
</td>
        <td><a class="wiki_link" href="/5L%206s">5L 6s</a><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>16-tone<br />
</td>
        <td><a class="wiki_link" href="/5L%2011s">5L 11s</a><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>21-tone<br />
</td>
        <td><a class="wiki_link" href="/5L%2016s">5L 16s</a><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>26-tone<br />
</td>
        <td><a class="wiki_link" href="/5L%2021s">5L 21s</a><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Intervals--7\31 octave - approx. 270.97¢ - Subminor Third"></a><!-- ws:end:WikiTextHeadingRule:26 -->7\31 octave - approx. 270.97¢ - Subminor Third</h3>
 Exactly one half of a superfourth (11:8 approximation). In 7-limit tonal music, 7\31 stands in for 7:6 (266.87¢). A generator for Orwell temperament (but not as good as 12\53 or 19\84). Generates <a class="wiki_link" href="/Semicomma%20family">orwell temperament</a>.<br />
<!-- ws:start:WikiTextHeadingRule:28:&lt;h4&gt; --><h4 id="toc14"><a name="Intervals--7\31 octave - approx. 270.97¢ - Subminor Third-MOS Scales generated by 7\31:"></a><!-- ws:end:WikiTextHeadingRule:28 -->MOS Scales generated by 7\31:</h4>
 

<table class="wiki_table">
    <tr>
        <th>number of tones<br />
</th>
        <th>MOS class<br />
</th>
        <th>0<br />
</th>
        <th>1<br />
</th>
        <th>2<br />
</th>
        <th>3<br />
</th>
        <th>4<br />
</th>
        <th>5<br />
</th>
        <th>6<br />
</th>
        <th>7<br />
</th>
        <th>8<br />
</th>
        <th>9<br />
</th>
        <th>10<br />
</th>
        <th>11<br />
</th>
        <th>12<br />
</th>
        <th>13<br />
</th>
        <th>14<br />
</th>
        <th>15<br />
</th>
        <th>16<br />
</th>
        <th>17<br />
</th>
        <th>18<br />
</th>
        <th>19<br />
</th>
        <th>20<br />
</th>
        <th>21<br />
</th>
        <th>22<br />
</th>
        <th>23<br />
</th>
        <th>24<br />
</th>
        <th>25<br />
</th>
        <th>26<br />
</th>
        <th>27<br />
</th>
        <th>28<br />
</th>
        <th>29<br />
</th>
        <th>30<br />
</th>
    </tr>
    <tr>
        <td>pentatonic<br />
</td>
        <td><a class="wiki_link" href="/4L%201s">4L 1s</a><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>nonatonic (quasi-equal; Orwell[9])<br />
</td>
        <td><a class="wiki_link" href="/4L%205s">4L 5s</a><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>13-tone (Orwell[13])<br />
</td>
        <td><a class="wiki_link" href="/9L%204s">9L 4s</a><br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>22-tone (Orwell[22])<br />
</td>
        <td><a class="wiki_link" href="/9L%2013s">9L 13s</a><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="Intervals--8\31 octave - approx. 309.68¢ - Minor Third"></a><!-- ws:end:WikiTextHeadingRule:30 -->8\31 octave - approx. 309.68¢ - Minor Third</h3>
 A minor third, closer to the just 6:5 (315.64¢) than 12-edo. Exactly twice a neutral second, four times a minor semitone, and half of a large tritone. Generates <a class="wiki_link" href="/Starling%20temperaments">myna temperament</a>.<br />
<!-- ws:start:WikiTextHeadingRule:32:&lt;h4&gt; --><h4 id="toc16"><a name="Intervals--8\31 octave - approx. 309.68¢ - Minor Third-MOS Scales generated by 8\31:"></a><!-- ws:end:WikiTextHeadingRule:32 -->MOS Scales generated by 8\31:</h4>
 

<table class="wiki_table">
    <tr>
        <th>number of tones<br />
</th>
        <th>MOS class<br />
</th>
        <th>0<br />
</th>
        <th>1<br />
</th>
        <th>2<br />
</th>
        <th>3<br />
</th>
        <th>4<br />
</th>
        <th>5<br />
</th>
        <th>6<br />
</th>
        <th>7<br />
</th>
        <th>8<br />
</th>
        <th>9<br />
</th>
        <th>10<br />
</th>
        <th>11<br />
</th>
        <th>12<br />
</th>
        <th>13<br />
</th>
        <th>14<br />
</th>
        <th>15<br />
</th>
        <th>16<br />
</th>
        <th>17<br />
</th>
        <th>18<br />
</th>
        <th>19<br />
</th>
        <th>20<br />
</th>
        <th>21<br />
</th>
        <th>22<br />
</th>
        <th>23<br />
</th>
        <th>24<br />
</th>
        <th>25<br />
</th>
        <th>26<br />
</th>
        <th>27<br />
</th>
        <th>28<br />
</th>
        <th>29<br />
</th>
        <th>30<br />
</th>
    </tr>
    <tr>
        <td>tetratonic (quasi-equal)<br />
</td>
        <td><a class="wiki_link" href="/3L%201s">3L 1s</a><br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>heptatonic<br />
</td>
        <td><a class="wiki_link" href="/4L%203s">4L 3s</a><br />
</td>
        <td>1<br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>11-tone<br />
</td>
        <td><a class="wiki_link" href="/4L%207s">4L 7s</a><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>15-tone<br />
</td>
        <td><a class="wiki_link" href="/4L%2011s">4L 11s</a><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>19-tone<br />
</td>
        <td><a class="wiki_link" href="/4L%2015s">4L 15s</a><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>23-tone<br />
</td>
        <td><a class="wiki_link" href="/4L%2019s">4L 19s</a><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>27-tone<br />
</td>
        <td><a class="wiki_link" href="/4L%2023s">4L 23s</a><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:34:&lt;h3&gt; --><h3 id="toc17"><a name="Intervals--9\31 octave - approx. 348.39¢ - Neutral Third"></a><!-- ws:end:WikiTextHeadingRule:34 -->9\31 octave - approx. 348.39¢ - Neutral Third</h3>
 A neutral 3rd, practically equivalent to 11:9 (347.41¢). Exactly half a perfect fifth (making it a suitable generator for neutral third scales such as <a class="wiki_link" href="/3L%204s">3L 4s</a>). Is also thrice a major semitone. Generates <a class="wiki_link" href="/Meantone%20family">mohajira temperament</a>.<br />
<!-- ws:start:WikiTextHeadingRule:36:&lt;h4&gt; --><h4 id="toc18"><a name="Intervals--9\31 octave - approx. 348.39¢ - Neutral Third-MOS Scales generated by 9\31:"></a><!-- ws:end:WikiTextHeadingRule:36 -->MOS Scales generated by 9\31:</h4>
 

<table class="wiki_table">
    <tr>
        <th>number of tones<br />
</th>
        <th>MOS class<br />
</th>
        <th>0<br />
</th>
        <th>1<br />
</th>
        <th>2<br />
</th>
        <th>3<br />
</th>
        <th>4<br />
</th>
        <th>5<br />
</th>
        <th>6<br />
</th>
        <th>7<br />
</th>
        <th>8<br />
</th>
        <th>9<br />
</th>
        <th>10<br />
</th>
        <th>11<br />
</th>
        <th>12<br />
</th>
        <th>13<br />
</th>
        <th>14<br />
</th>
        <th>15<br />
</th>
        <th>16<br />
</th>
        <th>17<br />
</th>
        <th>18<br />
</th>
        <th>19<br />
</th>
        <th>20<br />
</th>
        <th>21<br />
</th>
        <th>22<br />
</th>
        <th>23<br />
</th>
        <th>24<br />
</th>
        <th>25<br />
</th>
        <th>26<br />
</th>
        <th>27<br />
</th>
        <th>28<br />
</th>
        <th>29<br />
</th>
        <th>30<br />
</th>
    </tr>
    <tr>
        <td>tetratonic<br />
</td>
        <td><a class="wiki_link" href="/3L%201s">3L 1s</a><br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>heptatonic (quasi-equal)<br />
</td>
        <td><a class="wiki_link" href="/3L%204s">3L 4s</a><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>10-tone<br />
</td>
        <td><a class="wiki_link" href="/7L%203s">7L 3s</a><br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>17-tone<br />
</td>
        <td><a class="wiki_link" href="/7L%2010s">7L 10s</a><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>24-tone<br />
</td>
        <td><a class="wiki_link" href="/7L%2017s">7L 17s</a><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:38:&lt;h3&gt; --><h3 id="toc19"><a name="Intervals--10\31 octave - approx. 387.10¢ - Major Third"></a><!-- ws:end:WikiTextHeadingRule:38 -->10\31 octave - approx. 387.10¢ - Major Third</h3>
 A near-just major 3rd (compare with 5:4 = 386.31¢). Has led to the characterization of 31-edo as &quot;smooth&quot;. Generates <a class="wiki_link" href="/Wuerschmidt%20family">wurshmidt/worshmidt temperaments</a>.<br />
<!-- ws:start:WikiTextHeadingRule:40:&lt;h4&gt; --><h4 id="toc20"><a name="Intervals--10\31 octave - approx. 387.10¢ - Major Third-MOS Scales generated by 10\31:"></a><!-- ws:end:WikiTextHeadingRule:40 -->MOS Scales generated by 10\31:</h4>
 

<table class="wiki_table">
    <tr>
        <th>number of tones<br />
</th>
        <th>MOS class<br />
</th>
        <th>0<br />
</th>
        <th>1<br />
</th>
        <th>2<br />
</th>
        <th>3<br />
</th>
        <th>4<br />
</th>
        <th>5<br />
</th>
        <th>6<br />
</th>
        <th>7<br />
</th>
        <th>8<br />
</th>
        <th>9<br />
</th>
        <th>10<br />
</th>
        <th>11<br />
</th>
        <th>12<br />
</th>
        <th>13<br />
</th>
        <th>14<br />
</th>
        <th>15<br />
</th>
        <th>16<br />
</th>
        <th>17<br />
</th>
        <th>18<br />
</th>
        <th>19<br />
</th>
        <th>20<br />
</th>
        <th>21<br />
</th>
        <th>22<br />
</th>
        <th>23<br />
</th>
        <th>24<br />
</th>
        <th>25<br />
</th>
        <th>26<br />
</th>
        <th>27<br />
</th>
        <th>28<br />
</th>
        <th>29<br />
</th>
        <th>30<br />
</th>
    </tr>
    <tr>
        <td>tritonic (quasi-equal)<br />
</td>
        <td><a class="wiki_link" href="/1L%202s">1L 2s</a><br />
</td>
        <td>10<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>10<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>11<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>tetratonic<br />
</td>
        <td><a class="wiki_link" href="/3L%201s">3L 1s</a><br />
</td>
        <td>10<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>10<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>10<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>heptatonic<br />
</td>
        <td><a class="wiki_link" href="/3L%204s">3L 4s</a><br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>10-tone<br />
</td>
        <td><a class="wiki_link" href="/3L%207s">3L 7s</a><br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>13-tone<br />
</td>
        <td><a class="wiki_link" href="/3L%2010s">3L 10s</a><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>16-tone<br />
</td>
        <td><a class="wiki_link" href="/3L%2013s">3L 13s</a><br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>19-tone<br />
</td>
        <td><a class="wiki_link" href="/3L%2016s">3L 16s</a><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>22-tone<br />
</td>
        <td><a class="wiki_link" href="/3L%2019s">3L 19s</a><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>25-tone<br />
</td>
        <td><a class="wiki_link" href="/3L%2022s">3L 22s</a><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>28-tone<br />
</td>
        <td><a class="wiki_link" href="/3L%2025s">3L 25s</a><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:42:&lt;h3&gt; --><h3 id="toc21"><a name="Intervals--11\31 octave - approx. 425.806¢ - Supermajor Third"></a><!-- ws:end:WikiTextHeadingRule:42 -->11\31 octave - approx. 425.806¢ - Supermajor Third</h3>
 In 11-limit tonal music, 11\31 functions as 14:11 (417.51¢), 32:25 (427.37¢), 9:7 (435.08¢) and others. Generates <a class="wiki_link" href="/Meantone%20family">squares temperament</a>.<br />
<!-- ws:start:WikiTextHeadingRule:44:&lt;h4&gt; --><h4 id="toc22"><a name="Intervals--11\31 octave - approx. 425.806¢ - Supermajor Third-MOS Scales generated by 11\31:"></a><!-- ws:end:WikiTextHeadingRule:44 -->MOS Scales generated by 11\31:</h4>
 

<table class="wiki_table">
    <tr>
        <th>number of tones<br />
</th>
        <th>MOS class<br />
</th>
        <th>0<br />
</th>
        <th>1<br />
</th>
        <th>2<br />
</th>
        <th>3<br />
</th>
        <th>4<br />
</th>
        <th>5<br />
</th>
        <th>6<br />
</th>
        <th>7<br />
</th>
        <th>8<br />
</th>
        <th>9<br />
</th>
        <th>10<br />
</th>
        <th>11<br />
</th>
        <th>12<br />
</th>
        <th>13<br />
</th>
        <th>14<br />
</th>
        <th>15<br />
</th>
        <th>16<br />
</th>
        <th>17<br />
</th>
        <th>18<br />
</th>
        <th>19<br />
</th>
        <th>20<br />
</th>
        <th>21<br />
</th>
        <th>22<br />
</th>
        <th>23<br />
</th>
        <th>24<br />
</th>
        <th>25<br />
</th>
        <th>26<br />
</th>
        <th>27<br />
</th>
        <th>28<br />
</th>
        <th>29<br />
</th>
        <th>30<br />
</th>
    </tr>
    <tr>
        <td>tritonic<br />
</td>
        <td><a class="wiki_link" href="/2L%201s">2L 1s</a><br />
</td>
        <td>11<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>11<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>pentatonic<br />
</td>
        <td><a class="wiki_link" href="/3L%202s">3L 2s</a><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>octatonic<br />
</td>
        <td><a class="wiki_link" href="/3L%205s">3L 5s</a><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>11-tone<br />
</td>
        <td><a class="wiki_link" href="/3L%208s">3L 8s</a><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>14-tone (quasi-equal)<br />
</td>
        <td><a class="wiki_link" href="/3L%2011s">3L 11s</a><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>17-tone<br />
</td>
        <td><a class="wiki_link" href="/3L%2014s">3L 14s</a><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td>1<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:46:&lt;h3&gt; --><h3 id="toc23"><a name="Intervals--12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth"></a><!-- ws:end:WikiTextHeadingRule:46 -->12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth</h3>
 Exactly twice a supermajor second, thrice a neutral second, or four times a major second. In 7-limit tonal music, 12\31 functions as 21:16 (470.78¢). Generates semisept temperament.<br />
<!-- ws:start:WikiTextHeadingRule:48:&lt;h4&gt; --><h4 id="toc24"><a name="Intervals--12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth-MOS Scales generated by 12\31:"></a><!-- ws:end:WikiTextHeadingRule:48 -->MOS Scales generated by 12\31:</h4>
 

<table class="wiki_table">
    <tr>
        <th>number of tones<br />
</th>
        <th>MOS class<br />
</th>
        <th>0<br />
</th>
        <th>1<br />
</th>
        <th>2<br />
</th>
        <th>3<br />
</th>
        <th>4<br />
</th>
        <th>5<br />
</th>
        <th>6<br />
</th>
        <th>7<br />
</th>
        <th>8<br />
</th>
        <th>9<br />
</th>
        <th>10<br />
</th>
        <th>11<br />
</th>
        <th>12<br />
</th>
        <th>13<br />
</th>
        <th>14<br />
</th>
        <th>15<br />
</th>
        <th>16<br />
</th>
        <th>17<br />
</th>
        <th>18<br />
</th>
        <th>19<br />
</th>
        <th>20<br />
</th>
        <th>21<br />
</th>
        <th>22<br />
</th>
        <th>23<br />
</th>
        <th>24<br />
</th>
        <th>25<br />
</th>
        <th>26<br />
</th>
        <th>27<br />
</th>
        <th>28<br />
</th>
        <th>29<br />
</th>
        <th>30<br />
</th>
    </tr>
    <tr>
        <td>tritonic<br />
</td>
        <td><a class="wiki_link" href="/2L%201s">2L 1s</a><br />
</td>
        <td>12<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>12<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>pentatonic<br />
</td>
        <td><a class="wiki_link" href="/3L%202s">3L 2s</a><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>octatonic<br />
</td>
        <td><a class="wiki_link" href="/5L%203s">5L 3s</a><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>13-tone (quasi-equal)<br />
</td>
        <td><a class="wiki_link" href="/5L%208s">5L 8s</a><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>18-tone<br />
</td>
        <td><a class="wiki_link" href="/13L%205s">13L 5s</a><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:50:&lt;h3&gt; --><h3 id="toc25"><a name="Intervals--13\31 octave - approx. 503.23¢ - Perfect Fourth"></a><!-- ws:end:WikiTextHeadingRule:50 -->13\31 octave - approx. 503.23¢ - Perfect Fourth</h3>
 A sharp perfect fourth (compare to 4:3 = 498.04¢). As such, it functions marvelously as a generator for meantone temperament.<br />
<!-- ws:start:WikiTextHeadingRule:52:&lt;h4&gt; --><h4 id="toc26"><a name="Intervals--13\31 octave - approx. 503.23¢ - Perfect Fourth-MOS Scales generated by 13\31:"></a><!-- ws:end:WikiTextHeadingRule:52 -->MOS Scales generated by 13\31:</h4>
 

<table class="wiki_table">
    <tr>
        <th>number of tones<br />
</th>
        <th>MOS class<br />
</th>
        <th>0<br />
</th>
        <th>1<br />
</th>
        <th>2<br />
</th>
        <th>3<br />
</th>
        <th>4<br />
</th>
        <th>5<br />
</th>
        <th>6<br />
</th>
        <th>7<br />
</th>
        <th>8<br />
</th>
        <th>9<br />
</th>
        <th>10<br />
</th>
        <th>11<br />
</th>
        <th>12<br />
</th>
        <th>13<br />
</th>
        <th>14<br />
</th>
        <th>15<br />
</th>
        <th>16<br />
</th>
        <th>17<br />
</th>
        <th>18<br />
</th>
        <th>19<br />
</th>
        <th>20<br />
</th>
        <th>21<br />
</th>
        <th>22<br />
</th>
        <th>23<br />
</th>
        <th>24<br />
</th>
        <th>25<br />
</th>
        <th>26<br />
</th>
        <th>27<br />
</th>
        <th>28<br />
</th>
        <th>29<br />
</th>
        <th>30<br />
</th>
    </tr>
    <tr>
        <td>tritonic<br />
</td>
        <td><a class="wiki_link" href="/2L%201s">2L 1s</a><br />
</td>
        <td>13<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>13<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>pentatonic<br />
</td>
        <td><a class="wiki_link" href="/2L%203s">2L 3s</a><br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>heptatonic<br />
</td>
        <td><a class="wiki_link" href="/5L%202s">5L 2s</a><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>12-tone (quasi-equal)<br />
</td>
        <td><a class="wiki_link" href="/7L%205s">7L 5s</a><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>19-tone<br />
</td>
        <td><a class="wiki_link" href="/12L%207s">12L 7s</a><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:54:&lt;h3&gt; --><h3 id="toc27"><a name="Intervals--14\31 octave - approx. 541.94¢ - Superfourth"></a><!-- ws:end:WikiTextHeadingRule:54 -->14\31 octave - approx. 541.94¢ - Superfourth</h3>
 9.4¢ off from a just 11:8 (551.32¢), which isn't bad relative to the size of a 31EDO step, as it's less than a quarter of a 31EDO step away. Exactly twice a subminor third. Generates <a class="wiki_link" href="/Starling%20temperaments">casablanca temperament</a>.<br />
<!-- ws:start:WikiTextHeadingRule:56:&lt;h4&gt; --><h4 id="toc28"><a name="Intervals--14\31 octave - approx. 541.94¢ - Superfourth-MOS Scales generated by 14\31:"></a><!-- ws:end:WikiTextHeadingRule:56 -->MOS Scales generated by 14\31:</h4>
 

<table class="wiki_table">
    <tr>
        <th>number of tones<br />
</th>
        <th>MOS class<br />
</th>
        <th>0<br />
</th>
        <th>1<br />
</th>
        <th>2<br />
</th>
        <th>3<br />
</th>
        <th>4<br />
</th>
        <th>5<br />
</th>
        <th>6<br />
</th>
        <th>7<br />
</th>
        <th>8<br />
</th>
        <th>9<br />
</th>
        <th>10<br />
</th>
        <th>11<br />
</th>
        <th>12<br />
</th>
        <th>13<br />
</th>
        <th>14<br />
</th>
        <th>15<br />
</th>
        <th>16<br />
</th>
        <th>17<br />
</th>
        <th>18<br />
</th>
        <th>19<br />
</th>
        <th>20<br />
</th>
        <th>21<br />
</th>
        <th>22<br />
</th>
        <th>23<br />
</th>
        <th>24<br />
</th>
        <th>25<br />
</th>
        <th>26<br />
</th>
        <th>27<br />
</th>
        <th>28<br />
</th>
        <th>29<br />
</th>
        <th>30<br />
</th>
    </tr>
    <tr>
        <td>tritonic<br />
</td>
        <td><a class="wiki_link" href="/2L%201s">2L 1s</a><br />
</td>
        <td>14<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>14<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>pentatonic<br />
</td>
        <td><a class="wiki_link" href="/2L%203s">2L 3s</a><br />
</td>
        <td>11<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>11<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>heptatonic<br />
</td>
        <td><a class="wiki_link" href="/2L%205s">2L 5s</a><br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>nonatonic<br />
</td>
        <td><a class="wiki_link" href="/2L%207s">2L 7s</a><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>11-tone (quasi-equal)<br />
</td>
        <td><a class="wiki_link" href="/9L%202s">9L 2s</a><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>20-tone<br />
</td>
        <td><a class="wiki_link" href="/11L%209s">11L 9s</a><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:58:&lt;h3&gt; --><h3 id="toc29"><a name="Intervals--15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth"></a><!-- ws:end:WikiTextHeadingRule:58 -->15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth</h3>
 In 7-limit tonal music, functions as 7:5 (582.51¢). Exactly thrice a whole tone. Generates tritonic temperament.<br />
<!-- ws:start:WikiTextHeadingRule:60:&lt;h4&gt; --><h4 id="toc30"><a name="Intervals--15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth-MOS Scales generated by 15\31:"></a><!-- ws:end:WikiTextHeadingRule:60 -->MOS Scales generated by 15\31:</h4>
 

<table class="wiki_table">
    <tr>
        <th>number of tones<br />
</th>
        <th>MOS class<br />
</th>
        <th>0<br />
</th>
        <th>1<br />
</th>
        <th>2<br />
</th>
        <th>3<br />
</th>
        <th>4<br />
</th>
        <th>5<br />
</th>
        <th>6<br />
</th>
        <th>7<br />
</th>
        <th>8<br />
</th>
        <th>9<br />
</th>
        <th>10<br />
</th>
        <th>11<br />
</th>
        <th>12<br />
</th>
        <th>13<br />
</th>
        <th>14<br />
</th>
        <th>15<br />
</th>
        <th>16<br />
</th>
        <th>17<br />
</th>
        <th>18<br />
</th>
        <th>19<br />
</th>
        <th>20<br />
</th>
        <th>21<br />
</th>
        <th>22<br />
</th>
        <th>23<br />
</th>
        <th>24<br />
</th>
        <th>25<br />
</th>
        <th>26<br />
</th>
        <th>27<br />
</th>
        <th>28<br />
</th>
        <th>29<br />
</th>
        <th>30<br />
</th>
    </tr>
    <tr>
        <td>tritonic<br />
</td>
        <td><a class="wiki_link" href="/2L%201s">2L 1s</a><br />
</td>
        <td>15<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>15<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>pentatonic<br />
</td>
        <td><a class="wiki_link" href="/2L%203s">2L 3s</a><br />
</td>
        <td>14<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>14<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>heptatonic<br />
</td>
        <td><a class="wiki_link" href="/2L%205s">2L 5s</a><br />
</td>
        <td>13<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>13<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>nonatonic<br />
</td>
        <td><a class="wiki_link" href="/2L%207s">2L 7s</a><br />
</td>
        <td>12<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>12<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>11-tone<br />
</td>
        <td><a class="wiki_link" href="/2L%209s">2L 9s</a><br />
</td>
        <td>11<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>11<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>13-tone<br />
</td>
        <td><a class="wiki_link" href="/2L%2011s">2L 11s</a><br />
</td>
        <td>10<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>10<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>15-tone<br />
</td>
        <td><a class="wiki_link" href="/2L%2013s">2L 13s</a><br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>17-tone<br />
</td>
        <td><a class="wiki_link" href="/2L%2015s">2L 15s</a><br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>19-tone<br />
</td>
        <td><a class="wiki_link" href="/2L%2017s">2L 17s</a><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>21-tone<br />
</td>
        <td><a class="wiki_link" href="/2L%2019s">2L 19s</a><br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>6<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>23-tone<br />
</td>
        <td><a class="wiki_link" href="/2L%2021s">2L 21s</a><br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>5<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>25-tone<br />
</td>
        <td><a class="wiki_link" href="/2L%2023s">2L 23s</a><br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>4<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>27-tone<br />
</td>
        <td><a class="wiki_link" href="/2L%2025s">2L 25s</a><br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
    <tr>
        <td>29-tone<br />
</td>
        <td><a class="wiki_link" href="/2L%2027s">2L 27s</a><br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
        <td>1<br />
</td>
    </tr>
</table>

<br />
<br />
<!-- ws:start:WikiTextHeadingRule:62:&lt;h3&gt; --><h3 id="toc31"><a name="Intervals--16\31 octave"></a><!-- ws:end:WikiTextHeadingRule:62 -->16\31 octave</h3>
 The large tritone.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:64:&lt;h1&gt; --><h1 id="toc32"><a name="Harmonic Scale"></a><!-- ws:end:WikiTextHeadingRule:64 -->Harmonic Scale</h1>
 31edo approximates Mode 8 of the <a class="wiki_link" href="/OverToneSeries">harmonic series</a> O.K., but many intervals between the harmonics aren't distinguished, most importantly 9/8 (major tone) and 10/9 (minor tone), as 31EDO is a meantone temperament. The interval between the 8th and 11th harmonics is approximated O.K., but the intervals between the 11th harmonic and closer harmonics such as the 12th and 9th harmonics are approximated even better. 13/8 isn't distinguished from the 11-limit 18/11, as 144/143 is tempered out, so 13-limit intervals can't be distinguished from 11-limit ones.<br />
<br />


<table class="wiki_table">
    <tr>
        <td>Overtones in &quot;Mode 8&quot;:<br />
</td>
        <td>8<br />
</td>
        <td>9<br />
</td>
        <td>10<br />
</td>
        <td>11<br />
</td>
        <td>12<br />
</td>
        <td>13<br />
</td>
        <td>14<br />
</td>
        <td>15<br />
</td>
        <td>16<br />
</td>
    </tr>
    <tr>
        <td>...as JI Ratio from 1/1:<br />
</td>
        <td>1/1<br />
</td>
        <td>9/8<br />
</td>
        <td>5/4<br />
</td>
        <td>11/8<br />
</td>
        <td>3/2<br />
</td>
        <td>13/8<br />
</td>
        <td>7/4<br />
</td>
        <td>15/8<br />
</td>
        <td>2/1<br />
</td>
    </tr>
    <tr>
        <td>...in cents:<br />
</td>
        <td>0<br />
</td>
        <td>203.9<br />
</td>
        <td>386.3<br />
</td>
        <td>551.3<br />
</td>
        <td>702.0<br />
</td>
        <td>840.5<br />
</td>
        <td>968.8<br />
</td>
        <td>1088.3<br />
</td>
        <td>1200.0<br />
</td>
    </tr>
    <tr>
        <td>Nearest degree of 41edo:<br />
</td>
        <td>0<br />
</td>
        <td>5<br />
</td>
        <td>10<br />
</td>
        <td>14<br />
</td>
        <td>18<br />
</td>
        <td>22<br />
</td>
        <td>25<br />
</td>
        <td>28<br />
</td>
        <td>31<br />
</td>
    </tr>
    <tr>
        <td>...in cents:<br />
</td>
        <td>0<br />
</td>
        <td>193.5<br />
</td>
        <td>387.1<br />
</td>
        <td>541.9<br />
</td>
        <td>696.8<br />
</td>
        <td>851.6<br />
</td>
        <td>967.7<br />
</td>
        <td>1083.9<br />
</td>
        <td>1200.0<br />
</td>
    </tr>
</table>

<!-- ws:start:WikiTextHeadingRule:66:&lt;h1&gt; --><h1 id="toc33"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:66 -->Commas</h1>
 31 EDO tempers out the following commas. (Note: This assumes the val &lt; 31 49 72 87 107 115 |, comma values roundet to 5 significant digits.)<br />


<table class="wiki_table">
    <tr>
        <th>Comma<br />
</th>
        <th>Monzo<br />
</th>
        <th>Value (Cents)<br />
</th>
        <th>Name 1<br />
</th>
        <th>Name 2<br />
</th>
        <th>Name 3<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">9931568/9752117<br />
</td>
        <td style="text-align: left;">| -25 7 6 &gt;<br />
</td>
        <td style="text-align: right;">31.567<br />
</td>
        <td style="text-align: center;">Ampersand's Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">81/80<br />
</td>
        <td style="text-align: left;">| -4 4 -1 &gt;<br />
</td>
        <td style="text-align: right;">21.506<br />
</td>
        <td style="text-align: center;">Syntonic Comma<br />
</td>
        <td style="text-align: center;">Didymos Comma<br />
</td>
        <td style="text-align: center;">Meantone Comma<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">393216/390625<br />
</td>
        <td style="text-align: left;">| 17 1 -8 &gt;<br />
</td>
        <td style="text-align: right;">11.445<br />
</td>
        <td style="text-align: center;">Wuerschmidt Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">2109375/2097152<br />
</td>
        <td style="text-align: left;">| -21 3 7 &gt;<br />
</td>
        <td style="text-align: right;">10.061<br />
</td>
        <td style="text-align: center;">Semicomma<br />
</td>
        <td style="text-align: center;">Fokker Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">6719816/6714445<br />
</td>
        <td style="text-align: left;">| 38 -2 -15 &gt;<br />
</td>
        <td style="text-align: right;">1.3843<br />
</td>
        <td style="text-align: center;">Hemithirds Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">9859966/9733137<br />
</td>
        <td style="text-align: left;">| -10 7 8 -7 &gt;<br />
</td>
        <td style="text-align: right;">22.413<br />
</td>
        <td style="text-align: center;">Blackjackisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">64827/64000<br />
</td>
        <td style="text-align: left;">| -9 3 -3 4 &gt;<br />
</td>
        <td style="text-align: right;">22.227<br />
</td>
        <td style="text-align: center;">Squalentine<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">2430/2401<br />
</td>
        <td style="text-align: left;">| 1 5 1 -4 &gt;<br />
</td>
        <td style="text-align: right;">20.785<br />
</td>
        <td style="text-align: center;">Nuwell<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">50421/50000<br />
</td>
        <td style="text-align: left;">| -4 1 -5 5 &gt;<br />
</td>
        <td style="text-align: right;">14.516<br />
</td>
        <td style="text-align: center;">Trimyna<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">126/125<br />
</td>
        <td style="text-align: left;">| 1 2 -3 1 &gt;<br />
</td>
        <td style="text-align: right;">13.795<br />
</td>
        <td style="text-align: center;">Septimal Semicomma<br />
</td>
        <td style="text-align: center;">Starling Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1728/1715<br />
</td>
        <td style="text-align: left;">| 6 3 -1 -3 &gt;<br />
</td>
        <td style="text-align: right;">13.074<br />
</td>
        <td style="text-align: center;">Orwellisma<br />
</td>
        <td style="text-align: center;">Orwell Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1029/1024<br />
</td>
        <td style="text-align: left;">| -10 1 0 3 &gt;<br />
</td>
        <td style="text-align: right;">8.4327<br />
</td>
        <td style="text-align: center;">Gamelisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">225/224<br />
</td>
        <td style="text-align: left;">| -5 2 2 -1 &gt;<br />
</td>
        <td style="text-align: right;">7.7115<br />
</td>
        <td style="text-align: center;">Septimal Kleisma<br />
</td>
        <td style="text-align: center;">Marvel Comma<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">16875/16807<br />
</td>
        <td style="text-align: left;">| 0 3 4 -5 &gt;<br />
</td>
        <td style="text-align: right;">6.9903<br />
</td>
        <td style="text-align: center;">Mirkwai<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3136/3125<br />
</td>
        <td style="text-align: left;">| 6 0 -5 2 &gt;<br />
</td>
        <td style="text-align: right;">6.0832<br />
</td>
        <td style="text-align: center;">Hemimean<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">6144/6125<br />
</td>
        <td style="text-align: left;">| 11 1 -3 -2 &gt;<br />
</td>
        <td style="text-align: right;">5.3621<br />
</td>
        <td style="text-align: center;">Porwell<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1065875/1063543<br />
</td>
        <td style="text-align: left;">| -26 -1 1 9 &gt;<br />
</td>
        <td style="text-align: right;">3.7919<br />
</td>
        <td style="text-align: center;">Wadisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">65625/65536<br />
</td>
        <td style="text-align: left;">| -16 1 5 1 &gt;<br />
</td>
        <td style="text-align: right;">2.3495<br />
</td>
        <td style="text-align: center;">Horwell<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">703125/702464<br />
</td>
        <td style="text-align: left;">| -11 2 7 -3 &gt;<br />
</td>
        <td style="text-align: right;">1.6283<br />
</td>
        <td style="text-align: center;">Meter<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">2401/2400<br />
</td>
        <td style="text-align: left;">| -5 -1 -2 4 &gt;<br />
</td>
        <td style="text-align: right;">0.72120<br />
</td>
        <td style="text-align: center;">Breedsma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">99/98<br />
</td>
        <td style="text-align: left;">| -1 2 0 -2 1 &gt;<br />
</td>
        <td style="text-align: right;">17.576<br />
</td>
        <td style="text-align: center;">Mothwellsma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">121/120<br />
</td>
        <td style="text-align: left;">| -3 -1 -1 0 2 &gt;<br />
</td>
        <td style="text-align: right;">14.367<br />
</td>
        <td style="text-align: center;">Biyatisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">176/175<br />
</td>
        <td style="text-align: left;">| 4 0 -2 -1 1 &gt;<br />
</td>
        <td style="text-align: right;">9.8646<br />
</td>
        <td style="text-align: center;">Valinorsma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">243/242<br />
</td>
        <td style="text-align: left;">| -1 5 0 0 -2 &gt;<br />
</td>
        <td style="text-align: right;">7.1391<br />
</td>
        <td style="text-align: center;">Rastma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">385/384<br />
</td>
        <td style="text-align: left;">| -7 -1 1 1 1 &gt;<br />
</td>
        <td style="text-align: right;">4.5026<br />
</td>
        <td style="text-align: center;">Keenanisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">441/440<br />
</td>
        <td style="text-align: left;">| -3 2 -1 2 -1 &gt;<br />
</td>
        <td style="text-align: right;">3.9302<br />
</td>
        <td style="text-align: center;">Werckisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">540/539<br />
</td>
        <td style="text-align: left;">| 2 3 1 -2 -1 &gt;<br />
</td>
        <td style="text-align: right;">3.2090<br />
</td>
        <td style="text-align: center;">Swetisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3025/3024<br />
</td>
        <td style="text-align: left;">| -4 -3 2 -1 2 &gt;<br />
</td>
        <td style="text-align: right;">0.57240<br />
</td>
        <td style="text-align: center;">Lehmerisma<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:68:&lt;h1&gt; --><h1 id="toc34"><a name="Modes"></a><!-- ws:end:WikiTextHeadingRule:68 -->Modes</h1>
 <br />
A large open list of modes (subsets) from 31edo that people have named: <a class="wiki_link" href="/31edo%20modes">31edo modes</a>. <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rothenberg_propriety" rel="nofollow">Strictly proper</a> <a class="wiki_link" href="/Strictly%20proper%207-note%2031edo%20scales">7-note 31edo scales</a> in the sense of <a class="wiki_link" href="/David%20Rothenberg">David Rothenberg</a>. See also <a class="wiki_link" href="/31edo%20MOS%20scales">31edo MOS scales</a>. Some of the popular ones:<br />
<br />
<ul><li>31-tone major: 5 5 3 5 5 5 3</li><li>Meantone[12] (Eb-G#): 2 3 3 2 3 2 3 2 3 3 2 3</li><li>Harmonic scale 8: 5 5 4 4 4 4 3 3</li><li>the <a class="wiki_link" href="/Euler-Fokker%20genera">Euler-Fokker genera</a> (technically <a class="wiki_link" href="/JI">JI</a> but representable in 31)</li></ul><br />


<table class="wiki_table">
    <tr>
        <td colspan="2"><!-- ws:start:WikiTextHeadingRule:70:&lt;h4&gt; --><h4 id="toc35"><a name="Modes---Some 31 tone equal modes:"></a><!-- ws:end:WikiTextHeadingRule:70 -->Some 31 tone equal modes:</h4>
</td>
    </tr>
    <tr>
        <td><tt><strong>2 3 3 2 3 2 3 2 3 3 2 3</strong></tt><br />
</td>
        <td>Meantone Chromatic (53/220-comma)<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 5 3 5 5 5 3</strong></tt><br />
</td>
        <td>Thirty-one tone Major, Intense Diatonic Lydian, M.Ionian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 3 5 5 3 5 5</strong></tt><br />
</td>
        <td>Thirty-one tone Natural Minor, Intense Diatonic Hypodorian, Aeolian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 3 5 5 5 5 3</strong></tt><br />
</td>
        <td>Thirty-one tone Melodic Minor<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 3 5 5 3 7 3</strong></tt><br />
</td>
        <td>Thirty-one tone Harmonic Minor<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 5 3 5 3 7 3</strong></tt><br />
</td>
        <td>Thirty-one tone Harmonic Major<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 5 3 5 3 5 5</strong></tt><br />
</td>
        <td>Thirty-one tone Major-Minor<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 8 5 13</strong></tt><br />
</td>
        <td>Genus primum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>10 3 5 5 5 3</strong></tt><br />
</td>
        <td>Genus secundum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>8 2 8 3 7 3</strong></tt><br />
</td>
        <td>Genus tertium<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>10 10 10 1</strong></tt><br />
</td>
        <td>Genus quartum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 7 6 7 5 1</strong></tt><br />
</td>
        <td>Genus quintum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 6 2 6 4 3 3 3</strong></tt><br />
</td>
        <td>Genus sextum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 6 5 6 4 6</strong></tt><br />
</td>
        <td>Genus septimum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>6 6 6 1 6 6</strong></tt><br />
</td>
        <td>Genus octavum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 6 9 6 4 2</strong></tt><br />
</td>
        <td>Genus nonum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>13 6 6 6</strong></tt><br />
</td>
        <td>Genus decimum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 5 3 5 5 3 2 3</strong></tt><br />
</td>
        <td>Genus diatonicum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 5 2 3 5 3 2 5 3</strong></tt><br />
</td>
        <td>Genus chromaticum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 5 2 1 5 5 2 3 3</strong></tt><br />
</td>
        <td>Genus diatonicum cum septimis<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 4 3 3 2 1 4 1 4 1 2 3</strong></tt><br />
</td>
        <td>Genus enharmonicum vocale<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 2 4 2 2 3 3 3 1 3 3 3</strong></tt><br />
</td>
        <td>Genus enharmonicum instrumentale<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 2 3 2 3 2 3 3 2 3 2 3</strong></tt><br />
</td>
        <td>Genus diatonico-chromaticum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 2 1 2 5 3 2 1 4 1 2 3</strong></tt><br />
</td>
        <td>Genus bichromaticum<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 4 5 4 4 5 5</strong></tt><br />
</td>
        <td>Neutral Diatonic Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 5 4 4 5 5 4</strong></tt><br />
</td>
        <td>Neutral Diatonic Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 4 4 5 5 4 4</strong></tt><br />
</td>
        <td>Neutral Diatonic Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 4 5 5 4 4 5</strong></tt><br />
</td>
        <td>Neutral Diatonic Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 5 5 4 4 5 4</strong></tt><br />
</td>
        <td>Neutral Diatonic Hypolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 5 4 4 5 4 4</strong></tt><br />
</td>
        <td>Neutral Diatonic Hypophrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 4 4 5 4 4 5</strong></tt><br />
</td>
        <td>Neutral Diatonic Hypodorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 5 4 4 5 4 5</strong></tt><br />
</td>
        <td>Neutral Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 4 4 5 4 5 4</strong></tt><br />
</td>
        <td>Neutral Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 4 5 4 5 4 5</strong></tt><br />
</td>
        <td>Neutral Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 5 4 5 4 5 4</strong></tt><br />
</td>
        <td>Neutral Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 4 5 4 5 4 4</strong></tt><br />
</td>
        <td>Neutral Hypolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 5 4 5 4 4 5</strong></tt><br />
</td>
        <td>Neutral Hypophrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 4 5 4 4 5 4</strong></tt><br />
</td>
        <td>Neutral Hypodorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 2 9 2 2 9 5</strong></tt><br />
</td>
        <td>Hemiolic Chromatic Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 9 2 2 9 5 2</strong></tt><br />
</td>
        <td>Hemiolic Chromatic Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>9 2 2 9 5 2 2</strong></tt><br />
</td>
        <td>Hemiolic Chromatic Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 2 9 5 2 2 9</strong></tt><br />
</td>
        <td>Hemiolic Chromatic Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 9 5 2 2 9 2</strong></tt><br />
</td>
        <td>Hemiolic Chromatic Hypolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>9 5 2 2 9 2 2</strong></tt><br />
</td>
        <td>Hemiolic Chromatic Hypophrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 2 2 9 2 2 9</strong></tt><br />
</td>
        <td>Hemiolic Chromatic Hypodorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 3 8 2 3 8 5</strong></tt><br />
</td>
        <td>Ratio 2:3 Chromatic Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 8 2 3 8 5 2</strong></tt><br />
</td>
        <td>Ratio 2:3 Chromatic Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>8 2 3 8 5 2 3</strong></tt><br />
</td>
        <td>Ratio 2:3 Chromatic Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 3 8 5 2 3 8</strong></tt><br />
</td>
        <td>Ratio 2:3 Chromatic Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 8 5 2 3 8 2</strong></tt><br />
</td>
        <td>Ratio 2:3 Chromatic Hypolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>8 5 2 3 8 2 3</strong></tt><br />
</td>
        <td>Ratio 2:3 Chromatic Hypophrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 2 3 8 2 3 8</strong></tt><br />
</td>
        <td>Ratio 2:3 Chromatic Hypodorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 5 5 3 5 5 5</strong></tt><br />
</td>
        <td>Intense Diatonic Mixolydian, M.Locrian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 3 5 5 5 3 5</strong></tt><br />
</td>
        <td>Intense Diatonic Phrygian, M.Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 5 5 5 3 5 5</strong></tt><br />
</td>
        <td>Intense Diatonic Dorian, M.Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 5 5 3 5 5 3</strong></tt><br />
</td>
        <td>Intense Diatonic Hypolydian, M.Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 5 3 5 5 3 5</strong></tt><br />
</td>
        <td>Intense Diatonic Hypophrygian, M.Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 5 6 2 5 6 5</strong></tt><br />
</td>
        <td>Soft Diatonic Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 6 2 5 6 5 2</strong></tt><br />
</td>
        <td>Soft Diatonic Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>6 2 5 6 5 2 5</strong></tt><br />
</td>
        <td>Soft Diatonic Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 5 6 5 2 5 6</strong></tt><br />
</td>
        <td>Soft Diatonic Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 6 5 2 5 6 2</strong></tt><br />
</td>
        <td>Soft Diatonic Hypolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>6 5 2 5 6 2 5</strong></tt><br />
</td>
        <td>Soft Diatonic Hypophrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 2 5 6 2 5 6</strong></tt><br />
</td>
        <td>Soft Diatonic Hypodorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>1 2 10 1 2 10 5</strong></tt><br />
</td>
        <td>Enharmonic Mixolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 10 1 2 10 5 1</strong></tt><br />
</td>
        <td>Enharmonic Lydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>10 1 2 10 5 1 2</strong></tt><br />
</td>
        <td>Enharmonic Phrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>1 2 10 5 1 2 10</strong></tt><br />
</td>
        <td>Enharmonic Dorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 10 5 1 2 10 1</strong></tt><br />
</td>
        <td>Enharmonic Hypolydian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>10 5 1 2 10 1 2</strong></tt><br />
</td>
        <td>Enharmonic Hypophrygian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 1 2 10 1 2 10</strong></tt><br />
</td>
        <td>Enharmonic Hypodorian<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>6 6 7 6 6</strong></tt><br />
</td>
        <td>Quasi-equal Pentatonic<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 2 2 3 3 2 3 3 2 2 3 3</strong></tt><br />
</td>
        <td>Fokker 12-tone<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 3 5 3 5 2 5 3</strong></tt><br />
</td>
        <td>Modus conjunctus<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 5 2 5 3 5 3 5</strong></tt><br />
</td>
        <td>Octatonic<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 3 4 3 5 3 4 3 3</strong></tt><br />
</td>
        <td>Hahn symmetric pentachordal<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>3 4 3 3 5 3 4 3 3</strong></tt><br />
</td>
        <td>Hahn pentachordal<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 4 2 5 3 3 4 3 3</strong></tt><br />
</td>
        <td>Hahn Nonatonic<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 1 5 1 5 1 5 1 5 1 1</strong></tt><br />
</td>
        <td>de Vries 11-tone<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 1 4 4 4 1 4 4 1 4</strong></tt><br />
</td>
        <td>Breed 10-tone<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 2 4 2 4 2 4 3 3 3</strong></tt><br />
</td>
        <td>Lumma Decatonic<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 3 3 3 3 5 3 3 3</strong></tt><br />
</td>
        <td>Rothenberg Generalized Diatonic<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>5 2 6 5 2 5 6</strong></tt><br />
</td>
        <td>&quot;Septimal&quot; Natural Minor<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>4 3 4 3 4 3 4 3 3</strong></tt><br />
</td>
        <td>Thirty-one tone Orwell<br />
</td>
    </tr>
    <tr>
        <td><tt><strong>2 5 2 2 5 2 2 2 5 2 2</strong></tt><br />
</td>
        <td>Secor Sentinel<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:72:&lt;h1&gt; --><h1 id="toc36"><a name="Music in 31-edo"></a><!-- ws:end:WikiTextHeadingRule:72 -->Music in 31-edo</h1>
 <a class="wiki_link" href="/31-edo%20compositions">An alphabetical list of Tricesimoprimal Compositions</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:74:&lt;h2&gt; --><h2 id="toc37"><a name="Music in 31-edo-Thirty-one tone pedagogy"></a><!-- ws:end:WikiTextHeadingRule:74 -->Thirty-one tone pedagogy</h2>
 The <a class="wiki_link" href="/MicroPedagogyCollective">MicroPedagogyCollective</a> is currently at work producing demonstrative material which will encourage and enable more people to learn this system. There have been two <a class="wiki_link" href="/ThirtyOneToneSinginCamp">ThirtyOneToneSinginCamp</a>s as well.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:76:&lt;h1&gt; --><h1 id="toc38"><a name="Practical Theory / Books"></a><!-- ws:end:WikiTextHeadingRule:76 -->Practical Theory / Books</h1>
 <br />
<!-- ws:start:WikiTextRemoteImageRule:7938:&lt;a href=&quot;http://www.ronsword.com/books.html&quot; target=&quot;_blank&quot; rel=&quot;nofollow&quot;&gt;&lt;img src=&quot;http://ronsword.com/images/TSG_sm.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 116px; width: 87px;&quot; /&gt;&lt;/a&gt; --><a href="http://www.ronsword.com/books.html" target="_blank" rel="nofollow"><img src="http://ronsword.com/images/TSG_sm.jpg" alt="external image TSG_sm.jpg" title="external image TSG_sm.jpg" style="height: 116px; width: 87px;" /></a><!-- ws:end:WikiTextRemoteImageRule:7938 --><a class="wiki_link_ext" href="http://www.ronsword.com/books.html" rel="nofollow" target="_blank">Sword, Ronald. &quot;Tricesimoprimal Scales for Guitar.&quot; IAAA Press, UK-USA. First Ed: March 2009.</a> - A comprehensive approach to 31-EDO and all the families associated for Guitar. Features over 300 scale charts / scale examples.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:78:&lt;h1&gt; --><h1 id="toc39"><a name="Other Articles"></a><!-- ws:end:WikiTextHeadingRule:78 -->Other Articles</h1>
 <ul><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/beerart.html" rel="nofollow">de Beer, Anton, ''The Development of 31-tone Music</a></span> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeFzBM9b" rel="nofollow">Permalink</a></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/fokkerorg.html" rel="nofollow">Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ</a></span> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeG6Tmli" rel="nofollow">Permalink</a></li><li>Fokker, A.D., &quot;New Music with 31 Notes&quot; translated by Leigh Gerdine</li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/rap31.html" rel="nofollow">Rapoport, Paul, ''About 31-tone Equal Temperament</a></span> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeGH4uBH" rel="nofollow">Permalink</a></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.huygens-fokker.org/docs/terp31.html" rel="nofollow">Terpstra, Siemen, ''Toward a Theory of Meantone (and 31-et) Harmony''</a></span> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeGMeCMd" rel="nofollow">Permalink</a></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://tonalsoft.com/enc/number/31edo.aspx" rel="nofollow">Tonalsoft Encyclopedia article</a></span> <a class="wiki_link_ext" href="http://www.webcitation.org/5xeGYj7QU" rel="nofollow">Permalink</a></li></ul></body></html>
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