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 Zeta function integral tuning, http://www.research.att.com/~njas/sequences/A117538. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). For JI that uses primes 3 and 7, but no 5, try [[36edo]].

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

=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 temperament]].
====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 (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=== 
10¢ off from a just 11:8 (551.32¢); barely functional as such. 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.

=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.xs4all.nl/%7Ehuygensf/english/index.html|de Beer, Anton, ''The Development of 31-tone Music]]</span>
* <span class="wiki_link_ext">[[http://www.xs4all.nl/%7Ehuygensf/doc/fokkerorg.html|Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ]]</span>
* Fokker, A.D., "New Music with 31 Notes" translated by Leigh Gerdine
* <span class="wiki_link_ext">[[http://www.xs4all.nl/%7Ehuygensf/doc/rap31.html|Rapoport, Paul, ''About 31-tone Equal Temperament]]</span>
* <span class="wiki_link_ext">[[http://www.xs4all.nl/%7Ehuygensf/doc/terp31.html|Terpstra, Siemen, ''Toward a Theory of Meantone (and 31-et) Harmony'']]</span>
* <span class="wiki_link_ext">[[http://tonalsoft.com/enc/number/31edo.aspx|Tonalsoft Encyclopedia article]]</span>

Original HTML content:

<html><head><title>31edo</title></head><body><!-- ws:start:WikiTextTocRule:92:&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:92 --><!-- ws:start:WikiTextTocRule:93: --><a href="#Intervals">Intervals</a><!-- 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: --><!-- ws:end:WikiTextTocRule:113 --><!-- ws:start:WikiTextTocRule:114: --><!-- ws:end:WikiTextTocRule:114 --><!-- ws:start:WikiTextTocRule:115: --><!-- ws:end:WikiTextTocRule:115 --><!-- ws:start:WikiTextTocRule:116: --><!-- ws:end:WikiTextTocRule:116 --><!-- ws:start:WikiTextTocRule:117: --><!-- ws:end:WikiTextTocRule:117 --><!-- ws:start:WikiTextTocRule:118: --><!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --><!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --><!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: --><!-- ws:end:WikiTextTocRule:121 --><!-- ws:start:WikiTextTocRule:122: --><!-- ws:end:WikiTextTocRule:122 --><!-- ws:start:WikiTextTocRule:123: --><!-- ws:end:WikiTextTocRule:123 --><!-- ws:start:WikiTextTocRule:124: --><!-- ws:end:WikiTextTocRule:124 --><!-- ws:start:WikiTextTocRule:125: --><!-- ws:end:WikiTextTocRule:125 --><!-- ws:start:WikiTextTocRule:126: --><!-- ws:end:WikiTextTocRule:126 --><!-- ws:start:WikiTextTocRule:127: --><!-- ws:end:WikiTextTocRule:127 --><!-- ws:start:WikiTextTocRule:128: --><!-- ws:end:WikiTextTocRule:128 --><!-- ws:start:WikiTextTocRule:129: --><!-- ws:end:WikiTextTocRule:129 --><!-- ws:start:WikiTextTocRule:130: --><!-- ws:end:WikiTextTocRule:130 --><!-- ws:start:WikiTextTocRule:131: --><!-- ws:end:WikiTextTocRule:131 --><!-- ws:start:WikiTextTocRule:132: --><!-- ws:end:WikiTextTocRule:132 --><!-- ws:start:WikiTextTocRule:133: --> | <a href="#Modes">Modes</a><!-- ws:end:WikiTextTocRule:133 --><!-- ws:start:WikiTextTocRule:134: --><!-- ws:end:WikiTextTocRule:134 --><!-- ws:start:WikiTextTocRule:135: --> | <a href="#Music in 31-edo">Music in 31-edo</a><!-- ws:end:WikiTextTocRule:135 --><!-- ws:start:WikiTextTocRule:136: --><!-- ws:end:WikiTextTocRule:136 --><!-- ws:start:WikiTextTocRule:137: --> | <a href="#Practical Theory / Books">Practical Theory / Books</a><!-- ws:end:WikiTextTocRule:137 --><!-- ws:start:WikiTextTocRule:138: --> | <a href="#Other Articles">Other Articles</a><!-- ws:end:WikiTextTocRule:138 --><!-- ws:start:WikiTextTocRule:139: -->
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<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 Zeta function integral tuning, <!-- ws:start:WikiTextUrlRule:10864:http://www.research.att.com/~njas/sequences/A117538 --><a class="wiki_link_ext" href="http://www.research.att.com/~njas/sequences/A117538" rel="nofollow">http://www.research.att.com/~njas/sequences/A117538</a><!-- ws:end:WikiTextUrlRule:10864 -->. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course). For JI that uses primes 3 and 7, but no 5, try <a class="wiki_link" href="/36edo">36edo</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="Intervals"></a><!-- ws:end:WikiTextHeadingRule:0 -->Intervals</h1>
 <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="Intervals--1\31 octave - approx. 38.71¢ - Diesis"></a><!-- ws:end:WikiTextHeadingRule:2 -->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:4:&lt;h3&gt; --><h3 id="toc2"><a name="Intervals--2\31 octave - approx. 77.42¢ - Minor Semitone or Chromatic Semitone or Small Minor Second"></a><!-- ws:end:WikiTextHeadingRule:4 -->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 temperament</a>.<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h4&gt; --><h4 id="toc3"><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:6 -->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 (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:8:&lt;h3&gt; --><h3 id="toc4"><a name="Intervals--3\31 octave - approx. 116.13 - Major Semitone or Diatonic Semitone or Large Major Second"></a><!-- ws:end:WikiTextHeadingRule:8 -->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:10:&lt;h4&gt; --><h4 id="toc5"><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:10 -->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 (<a class="wiki_link" href="/quasi-equal">quasi-equal</a>)<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:12:&lt;h3&gt; --><h3 id="toc6"><a name="Intervals--4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second"></a><!-- ws:end:WikiTextHeadingRule:12 -->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:14:&lt;h4&gt; --><h4 id="toc7"><a name="Intervals--4\31 octave - approx. 154.84¢ - Neutral Tone or Neutral Second-MOS Scales generated by 4\31:"></a><!-- ws:end:WikiTextHeadingRule:14 -->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:16:&lt;h3&gt; --><h3 id="toc8"><a name="Intervals--5\31 octave - approx. 193.55¢ - Whole Tone or Major Second"></a><!-- ws:end:WikiTextHeadingRule:16 -->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:18:&lt;h4&gt; --><h4 id="toc9"><a name="Intervals--5\31 octave - approx. 193.55¢ - Whole Tone or Major Second-MOS Scales generated by 5\31:"></a><!-- ws:end:WikiTextHeadingRule:18 -->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:20:&lt;h3&gt; --><h3 id="toc10"><a name="Intervals--6\31 octave - approx. 232.26¢ - Supermajor Second"></a><!-- ws:end:WikiTextHeadingRule:20 -->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:22:&lt;h4&gt; --><h4 id="toc11"><a name="Intervals--6\31 octave - approx. 232.26¢ - Supermajor Second-MOS Scales generated by 6\31:"></a><!-- ws:end:WikiTextHeadingRule:22 -->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:24:&lt;h3&gt; --><h3 id="toc12"><a name="Intervals--7\31 octave - approx. 270.97¢ - Subminor Third"></a><!-- ws:end:WikiTextHeadingRule:24 -->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:26:&lt;h4&gt; --><h4 id="toc13"><!-- ws:end:WikiTextHeadingRule:26 --> </h4>
 <!-- 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"><!-- ws:end:WikiTextHeadingRule:32 --> </h4>
 <!-- ws:start:WikiTextHeadingRule:34:&lt;h4&gt; --><h4 id="toc17"><a name="Intervals--8\31 octave - approx. 309.68¢ - Minor Third-MOS Scales generated by 8\31:"></a><!-- ws:end:WikiTextHeadingRule:34 -->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:36:&lt;h3&gt; --><h3 id="toc18"><a name="Intervals--9\31 octave - approx. 348.39¢ - Neutral Third"></a><!-- ws:end:WikiTextHeadingRule:36 -->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:38:&lt;h4&gt; --><h4 id="toc19"><!-- ws:end:WikiTextHeadingRule:38 --> </h4>
 <!-- ws:start:WikiTextHeadingRule:40:&lt;h4&gt; --><h4 id="toc20"><a name="Intervals--9\31 octave - approx. 348.39¢ - Neutral Third-MOS Scales generated by 9\31:"></a><!-- ws:end:WikiTextHeadingRule:40 -->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:42:&lt;h3&gt; --><h3 id="toc21"><a name="Intervals--10\31 octave - approx. 387.10¢ - Major Third"></a><!-- ws:end:WikiTextHeadingRule:42 -->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:44:&lt;h4&gt; --><h4 id="toc22"><!-- ws:end:WikiTextHeadingRule:44 --> </h4>
 <!-- ws:start:WikiTextHeadingRule:46:&lt;h4&gt; --><h4 id="toc23"><a name="Intervals--10\31 octave - approx. 387.10¢ - Major Third-MOS Scales generated by 10\31:"></a><!-- ws:end:WikiTextHeadingRule:46 -->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:48:&lt;h3&gt; --><h3 id="toc24"><a name="Intervals--11\31 octave - approx. 425.806¢ - Supermajor Third"></a><!-- ws:end:WikiTextHeadingRule:48 -->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:50:&lt;h4&gt; --><h4 id="toc25"><!-- ws:end:WikiTextHeadingRule:50 --> </h4>
 <!-- ws:start:WikiTextHeadingRule:52:&lt;h4&gt; --><h4 id="toc26"><a name="Intervals--11\31 octave - approx. 425.806¢ - Supermajor Third-MOS Scales generated by 11\31:"></a><!-- ws:end:WikiTextHeadingRule:52 -->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:54:&lt;h3&gt; --><h3 id="toc27"><a name="Intervals--12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth"></a><!-- ws:end:WikiTextHeadingRule:54 -->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:56:&lt;h4&gt; --><h4 id="toc28"><!-- ws:end:WikiTextHeadingRule:56 --> </h4>
 <!-- ws:start:WikiTextHeadingRule:58:&lt;h4&gt; --><h4 id="toc29"><a name="Intervals--12\31 octave - approx. 464.52¢ - Narrow Fourth or Subfourth-MOS Scales generated by 12\31:"></a><!-- ws:end:WikiTextHeadingRule:58 -->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:60:&lt;h3&gt; --><h3 id="toc30"><a name="Intervals--13\31 octave - approx. 503.23¢ - Perfect Fourth"></a><!-- ws:end:WikiTextHeadingRule:60 -->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:62:&lt;h4&gt; --><h4 id="toc31"><!-- ws:end:WikiTextHeadingRule:62 --> </h4>
 <!-- ws:start:WikiTextHeadingRule:64:&lt;h4&gt; --><h4 id="toc32"><a name="Intervals--13\31 octave - approx. 503.23¢ - Perfect Fourth-MOS Scales generated by 13\31:"></a><!-- ws:end:WikiTextHeadingRule:64 -->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:66:&lt;h3&gt; --><h3 id="toc33"><a name="Intervals--14\31 octave - approx. 541.94¢ - Superfourth"></a><!-- ws:end:WikiTextHeadingRule:66 -->14\31 octave - approx. 541.94¢ - Superfourth</h3>
 10¢ off from a just 11:8 (551.32¢); barely functional as such. Exactly twice a subminor third. Generates <a class="wiki_link" href="/Starling%20temperaments">casablanca temperament</a>.<br />
<!-- ws:start:WikiTextHeadingRule:68:&lt;h4&gt; --><h4 id="toc34"><!-- ws:end:WikiTextHeadingRule:68 --> </h4>
 <!-- ws:start:WikiTextHeadingRule:70:&lt;h4&gt; --><h4 id="toc35"><a name="Intervals--14\31 octave - approx. 541.94¢ - Superfourth-MOS Scales generated by 14\31:"></a><!-- ws:end:WikiTextHeadingRule:70 -->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:72:&lt;h3&gt; --><h3 id="toc36"><a name="Intervals--15\31 octave - approx. 580.65¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth"></a><!-- ws:end:WikiTextHeadingRule:72 -->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:74:&lt;h4&gt; --><h4 id="toc37"><!-- ws:end:WikiTextHeadingRule:74 --> </h4>
 <!-- ws:start:WikiTextHeadingRule:76:&lt;h4&gt; --><h4 id="toc38"><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:76 -->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:78:&lt;h3&gt; --><h3 id="toc39"><a name="Intervals--16\31 octave"></a><!-- ws:end:WikiTextHeadingRule:78 -->16\31 octave</h3>
 The large tritone.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:80:&lt;h1&gt; --><h1 id="toc40"><a name="Modes"></a><!-- ws:end:WikiTextHeadingRule:80 -->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:82:&lt;h4&gt; --><h4 id="toc41"><a name="Modes---Some 31 tone equal modes:"></a><!-- ws:end:WikiTextHeadingRule:82 -->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:84:&lt;h1&gt; --><h1 id="toc42"><a name="Music in 31-edo"></a><!-- ws:end:WikiTextHeadingRule:84 -->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:86:&lt;h2&gt; --><h2 id="toc43"><a name="Music in 31-edo-Thirty-one tone pedagogy"></a><!-- ws:end:WikiTextHeadingRule:86 -->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:88:&lt;h1&gt; --><h1 id="toc44"><a name="Practical Theory / Books"></a><!-- ws:end:WikiTextHeadingRule:88 -->Practical Theory / Books</h1>
 <br />
<!-- ws:start:WikiTextRemoteImageRule:7306:&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:7306 --><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:90:&lt;h1&gt; --><h1 id="toc45"><a name="Other Articles"></a><!-- ws:end:WikiTextHeadingRule:90 -->Other Articles</h1>
 <ul><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.xs4all.nl/%7Ehuygensf/english/index.html" rel="nofollow">de Beer, Anton, ''The Development of 31-tone Music</a></span></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.xs4all.nl/%7Ehuygensf/doc/fokkerorg.html" rel="nofollow">Fokker, Adriaan Daniël, ''Equal Temperament and the Thirty-one-keyed organ</a></span></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.xs4all.nl/%7Ehuygensf/doc/rap31.html" rel="nofollow">Rapoport, Paul, ''About 31-tone Equal Temperament</a></span></li><li><span class="wiki_link_ext"><a class="wiki_link_ext" href="http://www.xs4all.nl/%7Ehuygensf/doc/terp31.html" rel="nofollow">Terpstra, Siemen, ''Toward a Theory of Meantone (and 31-et) Harmony''</a></span></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></li></ul></body></html>