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Replaced content with "{{Infobox ET}} 31edt divides the tritave into equal parts of 61.353 cents, corresponding to the non-octave third tone scale of 39edo where each degree gets ~.185..."
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=<span style="background-color: #ffffff;">Intervals</span>=
=<span style="background-color: #ffffff;">Intervals</span>=
 
See also: [[Specific intervals in 31edt]]
===<span style="background-color: #ffffff;">1\31 tritave- approx. <span style="font-size: 1.1em;">61.35¢ - Third tone</span></span>===
<span style="background-color: #ffffff;">A single step of 31-edt is about 61.35¢. Intervals around this size are called ''third tones''. In 31 it is equivalent to the difference between one tritave 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). The third tone is a defining sound of 31edt; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size.</span>
 
===<span style="background-color: #ffffff;">2\31 tritave - approx. 122.71¢ - Two-third tone or Small Minor Second</span>===
<span style="background-color: #ffffff;">The difference between a major and minor third and the closest thing to a 'half step'; in macromeantone, it is exactly analogus to the </span>''<span style="background-color: #ffffff;">chromatic semitone</span>''<span style="background-color: #ffffff;">, the interval that distinguishes major and minor intervals of the same generic interval class (eg. thirds).</span>
 
====<span style="background-color: #ffffff;">MOS Scales generated by 2\31:</span>====
 
{| class="wikitable"
|-
! | number of tones
! | MOS class
! | 0
! | 1
! | 2
! | 3
! | 4
! | 5
! | 6
! | 7
! | 8
! | 9
! | 10
! | 11
! | 12
! | 13
! | 14
! | 15
! | 16
! | 17
! | 18
! | 19
! | 20
! | 21
! | 22
! | 23
! | 24
! | 25
! | 26
! | 27
! | 28
! | 29
! | 30
|-
| | 15-tone ([[Maximal_evenness|ME]] or quasi-equal)
| | [[1L_14s|1L 14s]]
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 3
| |
| |
|-
| | 16-tone
| | [[15L_1s|15L 1s]]
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 1
|}
 
===<span style="background-color: #ffffff;">3\31 tritave - approx. 184.06¢- Whole tone or Large Minor Second</span>===
 
<span style="background-color: #ffffff;">A small whole tone only ~1.6 cents wide of 10:9 which is an interval sometimes called melodically dull; in macromeantone, it is exactly analogus to the </span>''<span style="background-color: #ffffff;">diatonic semitone</span>''<span style="background-color: #ffffff;"> which appears in the diatonic scale between, for instance, the major third and perfect fourth, and the major seventh and octave. </span>
 
====<span style="background-color: #ffffff;">MOS Scales generated by 3\31:</span>====
 
{| class="wikitable"
|-
! | 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|1L 8s]]
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
|-
| | decatonic (quasi-equal)
| | [[9L_1s|9L 1s]]
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 4
| |
| |
| |
|-
| | 11-tone
| | [[10L_1s|10L 1s]]
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 1
|-
| | 21-tone (silimlar to Blackjack)
| | [[10L 11s]]
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 1
|}
 
===<span style="background-color: #ffffff;">4\31 tritave - approx. 245.41¢ - Classical hemifourth or Neutral Second</span>===
<span style="background-color: #ffffff;">Exactly one half of the minor third and twice the minor semitone, 4\31 stands in for 15:13 (247.74¢). Although 31 is not extremely accurate with 5 or 13, it is notable that the inaccuracies of these harmonics cancel out so much, leaving the interval that distinguishes them (15/13) only about 2.3¢ off.</span>
 
====<span style="background-color: #ffffff;">MOS Scales generated by 4\31:</span>====
 
{| class="wikitable"
|-
! | 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|1L 6s]]
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
|-
| | octatonic (quasi-equal)
| | [[7L_1s|7L 1s]]
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 3
| |
| |
|-
| | 15-tone
| | [[8L_7s|8L 7s]]
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 3
| |
| |
|-
| | 23-tone
| | [[8L_15s|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
| |
|}
 
===<span style="background-color: #ffffff;">5\31 tritave - approx. 306.77¢ - <span style="font-size: 1.1em;">Sesquitone or Major Second</span></span>===
<span style="background-color: #ffffff;">At ~8.8 cents flat of a just 6:5, 5\31 is considered a "sesquitone". Two of this sesquitone make a near-just 10:7 tritone. Because it is fairly close to no intelligibly small integer ratio but 6/5, 5\31 can function as a semi-stabilized harmonic ninth.</span>
 
====<span style="background-color: #ffffff;">MOS Scales generated by 5\31:</span>====
 
{| class="wikitable"
|-
! | 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|1L 5s]]
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 6
| |
| |
| |
| |
| |
|-
| | heptatonic
| | [[6L_1s|6L 1s]]
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 1
|-
| | 13-tone
| | [[6L_7s|6L 7s]]
| | 4
| |
| |
| |
| | 1
| | 4
| |
| |
| |
| | 1
| | 4
| |
| |
| |
| | 1
| | 4
| |
| |
| |
| | 1
| | 4
| |
| |
| |
| | 1
| | 4
| |
| |
| |
| | 1
| | 1
|-
| | 19-tone
| | [[6L_13s|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|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
|}
 
===<span style="background-color: #ffffff;">6\31 tritave - approx. 368.12¢ - Supermajor Second</span>===
<span style="background-color: #ffffff;">Exactly one half of a narrow fourth, twice a tone, or thrice a two-third tone. In 17-limit tonal music, 6\31 closely represents 21:17 (365.825¢). In macromeantone, it is a diminished third, eg. C to Ebb.</span>
 
====<span style="background-color: #ffffff;">MOS Scales generated by 6\31:</span>====
 
{| class="wikitable"
|-
! | 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|1L 4s]]
| | 6
| |
| |
| |
| |
| |
| | 6
| |
| |
| |
| |
| |
| | 6
| |
| |
| |
| |
| |
| | 6
| |
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
|-
| | hexatonic
| | [[5L_1s|5L 1s]]
| | 6
| |
| |
| |
| |
| |
| | 6
| |
| |
| |
| |
| |
| | 6
| |
| |
| |
| |
| |
| | 6
| |
| |
| |
| |
| |
| | 6
| |
| |
| |
| |
| |
| | 1
|-
| | 11-tone
| | [[5L_6s|5L 6s]]
| | 5
| |
| |
| |
| |
| | 1
| | 5
| |
| |
| |
| |
| | 1
| | 5
| |
| |
| |
| |
| | 1
| | 5
| |
| |
| |
| |
| | 1
| | 5
| |
| |
| |
| |
| | 1
| | 1
|-
| | 16-tone
| | [[5L_11s|5L 11s]]
| | 4
| |
| |
| |
| | 1
| | 1
| | 4
| |
| |
| |
| | 1
| | 1
| | 4
| |
| |
| |
| | 1
| | 1
| | 4
| |
| |
| |
| | 1
| | 1
| | 4
| |
| |
| |
| | 1
| | 1
| | 1
|-
| | 21-tone
| | [[5L_16s|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|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
|}
 
===<span style="background-color: #ffffff;">7\31 tritave - approx. 429.47¢ - Subminor Third</span>===
<span style="background-color: #ffffff;">Exactly one half of a superfourth. In 7-limit tonal music, 7\31 stands in for 9:7 (435.08¢). In macromeantone temperament, it is an augmented 2nd, eg. C to D#.</span>
 
====<span style="background-color: #ffffff;">MOS Scales generated by 7\31:</span>====
 
{| class="wikitable"
|-
! | 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|4L 1s]]
| | 7
| |
| |
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
| | 3
| |
| |
|-
| | nonatonic (quasi-equal; similar to Orwell[9])
| | [[4L_5s|4L 5s]]
| | 4
| |
| |
| |
| | 3
| |
| |
| | 4
| |
| |
| |
| | 3
| |
| |
| | 4
| |
| |
| |
| | 3
| |
| |
| | 4
| |
| |
| |
| | 3
| |
| |
| | 3
| |
| |
|-
| | 13-tone (similar to Orwell[13])
| | [[9L_4s|9L 4s]]
| | 1
| | 3
| |
| |
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
|-
| | 22-tone (similar to Orwell[22])
| | [[9L_13s|9L 13s]]
| | 1
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
|}
 
===<span style="background-color: #ffffff;">8\31 octave - approx. 490.83¢ - Minor Third</span>===
<span style="background-color: #ffffff;">A 4:3 ~1/3 syntonic comma flat. Exactly twice a neutral second, four times a minor semitone, and half of a large tritone. Generates a Fair Sigma scale</span>
 
====<span style="background-color: #ffffff;">MOS Scales generated by 8\31:</span>====
 
{| class="wikitable"
|-
! | 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|3L 1s]]
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
|-
| | heptatonic
| | [[4L_3s|4L 3s]]
| | 1
| | 7
| |
| |
| |
| |
| |
| |
| | 1
| | 7
| |
| |
| |
| |
| |
| |
| | 1
| | 7
| |
| |
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
|-
| | 11-tone
| | [[4L_7s|4L 7s]]
| | 1
| | 1
| | 6
| |
| |
| |
| |
| |
| | 1
| | 1
| | 6
| |
| |
| |
| |
| |
| | 1
| | 1
| | 6
| |
| |
| |
| |
| |
| | 1
| | 6
| |
| |
| |
| |
| |
|-
| | 15-tone
| | [[4L_11s|4L 11s]]
| | 1
| | 1
| | 1
| | 5
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 5
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 5
| |
| |
| |
| |
| | 1
| | 1
| | 5
| |
| |
| |
| |
|-
| | 19-tone
| | [[4L_15s|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|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|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
| |
|}
 
===<span style="background-color: #ffffff;">9\31 tritave - approx. 552.18¢ - Neutral Third</span>===
<span style="background-color: #ffffff;">A neutral 3rd, about 1¢ away from 11:8 (551.32¢). 9\31 is half a perfect fifth (making it a suitable generator for macro</span>mohajira temperament<span style="background-color: #ffffff;">), and also a very small tritone. It is closer in quality to a minor third than a major third, but indeed, it is distinct. It is 11¢ shy of 18/13 (563.38¢), suggesting a </span>[[13-limit|13-limit]]<span style="background-color: #ffffff;"> interpretation for 31edt. However, its close proximity to 11/8 makes it hard to hear it as 18/13, which in JI has a different quality (and, as a neutral third, is more "major-like" than "minor-like")..</span>
 
====<span style="background-color: #ffffff;">MOS Scales generated by 9\31:</span>====
 
{| class="wikitable"
|-
! | 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|3L 1s]]
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
| | 4
| |
| |
| |
|-
| | heptatonic (quasi-equal)
| | [[3L_4s|3L 4s]]
| | 5
| |
| |
| |
| |
| | 4
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 4
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
|-
| | 10-tone
| | [[7L_3s|7L 3s]]
| | 1
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 1
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 1
| | 4
| |
| |
| |
| | 4
| |
| |
| |
| | 4
| |
| |
| |
|-
| | 17-tone
| | [[7L_10s|7L 10s]]
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 1
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
| | 1
| | 3
| |
| |
|-
| | 24-tone
| | [[7L_17s|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
| |
|}
 
===<span style="background-color: #ffffff;">10\31 tritave - approx. 613.53¢ - Major Third</span>===
<span style="background-color: #ffffff;">A near-enough-just greater septimal tritone (compare with 10:7 = 617.49¢). Generates </span>[[Wuerschmidt_family|wurshmidt/worshmidt temperaments]]<span style="background-color: #ffffff;">.</span>
 
====<span style="background-color: #ffffff;">MOS Scales generated by 10\31:</span>====
 
{| class="wikitable"
|-
! | 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|1L 2s]]
| | 10
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 10
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 11
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
|-
| | tetratonic
| | [[3L_1s|3L 1s]]
| | 10
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 10
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 10
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
|-
| | heptatonic
| | [[3L_4s|3L 4s]]
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
|-
| | 10-tone
| | [[3L_7s|3L 7s]]
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
|-
| | 13-tone
| | [[3L_10s|3L 10s]]
| | 7
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 7
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 7
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
|-
| | 16-tone
| | [[3L_13s|3L 13s]]
| | 6
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 6
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 6
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
|-
| | 19-tone
| | [[3L_16s|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|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|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|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
|}
 
===<span style="background-color: #ffffff;">11\31 tritave - approx. 674.89¢ - Supermajor Third</span>===
<span style="background-color: #ffffff;">11\31 functions as 126:85 (681.47¢). In macromeantone temperament, it is a diminished fourth, eg. C to Fb. It is notable as closely approximating the 9/16edo Armodue sixth. Generates the Unfair Mu scale.</span>
 
====<span style="background-color: #ffffff;">MOS Scales generated by 11\31:</span>====
 
{| class="wikitable"
|-
! | 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|2L 1s]]
| | 11
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 11
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
|-
| | pentatonic
| | [[3L_2s|3L 2s]]
| | 2
| |
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
| | 2
| |
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
|-
| | octatonic
| | [[3L_5s|3L 5s]]
| | 2
| |
| | 2
| |
| | 7
| |
| |
| |
| |
| |
| |
| | 2
| |
| | 2
| |
| | 7
| |
| |
| |
| |
| |
| |
| | 2
| |
| | 7
| |
| |
| |
| |
| |
| |
|-
| | 11-tone
| | [[3L_8s|3L 8s]]
| | 2
| |
| | 2
| |
| | 2
| |
| | 5
| |
| |
| |
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 5
| |
| |
| |
| |
| | 2
| |
| | 2
| |
| | 5
| |
| |
| |
| |
|-
| | 14-tone (quasi-equal)
| | [[3L_11s|3L 11s]]
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 3
| |
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 3
| |
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 3
| |
| |
|-
| | 17-tone
| | [[3L_14s|3L 14s]]
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 2
| | 1
| |
| | 2
| |
| | 2
| |
| | 2
| |
| | 1
|}
 
===<span style="background-color: #ffffff;">12\31 tritave - approx. 736.52¢ - Narrow Fourth or Subfourth</span>===
<span style="background-color: #ffffff;">Exactly twice a supermajor second, thrice a neutral second, or four times a minor second. In the 7-limit, 12\31 functions as 32:21 (729.22¢). It is also quite close to the </span>[[17-limit|17-limit]] <span style="background-color: #ffffff;">interval 26/17 (735.57¢) and 19\31edo (735.48¢). However, although 31edt offers up a reasonable approximation of the 17th harmonic (18\31), no such approximation of the 13th comes with it to help make this identity clear. Generates f</span>alse Father<span style="background-color: #ffffff;"> temperament.</span>
 
====<span style="background-color: #ffffff;">MOS Scales generated by 12\31:</span>====
 
{| class="wikitable"
|-
! | 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|2L 1s]]
| | 12
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 12
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
|-
| | pentatonic
| | [[3L_2s|3L 2s]]
| | 5
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
| | 7
| |
| |
| |
| |
| |
| |
|-
| | octatonic
| | [[5L_3s|5L 3s]]
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 2
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 2
| |
| | 5
| |
| |
| |
| |
| | 2
| |
|-
| | 13-tone (quasi-equal)
| | [[5L_8s|5L 8s]]
| | 3
| |
| |
| | 2
| |
| | 3
| |
| |
| | 2
| |
| | 2
| |
| | 3
| |
| |
| | 2
| |
| | 3
| |
| |
| | 2
| |
| | 2
| |
| | 3
| |
| |
| | 2
| |
| | 2
| |
|-
| | 18-tone
| | [[13L_5s|13L 5s]]
| | 1
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 2
| |
|}
 
===<span style="background-color: #ffffff;">13\31 tritave - approx. 797.54¢ - Perfect Fourth</span>===
<span style="background-color: #ffffff;">A slightly narrow perfect fourth (compare to 27:17 = 800.91¢). As such, it functions marvelously as a generator for macro</span>meantone<span style="background-color: #ffffff;"> temperament.</span>
 
====<span style="background-color: #ffffff;">MOS Scales generated by 13\31:</span>====
 
{| class="wikitable"
|-
! | 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|2L 1s]]
| | 13
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 13
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
|-
| | pentatonic
| | [[2L_3s|2L 3s]]
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
|-
| | heptatonic
| | [[5L_2s|5L 2s]]
| | 3
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 3
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
| | 5
| |
| |
| |
| |
|-
| | 12-tone (quasi-equal)
| | [[7L_5s|7L 5s]]
| | 3
| |
| |
| | 3
| |
| |
| | 2
| |
| | 3
| |
| |
| | 2
| |
| | 3
| |
| |
| | 3
| |
| |
| | 2
| |
| | 3
| |
| |
| | 2
| |
| | 3
| |
| |
| | 2
| |
|-
| | 19-tone
| | [[12L_7s|12L 7s]]
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
|}
 
<span style="background-color: #ffffff; font-size: 1.1em;">'''14\31 tritave - approx. 858.95¢ - Superfourth'''</span>
 
<span style="background-color: #ffffff;">Exactly twice a subminor third, this interval functions as both the 28:17 (863.86¢) septendecimal and 23:14 (859.44¢) vicesmotertial superfourths (392/391 is tempered out). Thus it makes possible a symmetrical tempered version of a 17:28:46 triad. As either, 14\31 is flat (about 5¢ or about .5¢); however, it fits nicely with the sharp 17, allowing a even-nearer-just 28/27.</span>
 
====<span style="background-color: #ffffff;">MOS Scales generated by 14\31:</span>====
 
{| class="wikitable"
|-
! | 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|2L 1s]]
| | 14
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 14
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 3
| |
| |
|-
| | pentatonic
| | [[2L_3s|2L 3s]]
| | 11
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 3
| |
| |
| | 11
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 3
| |
| |
| | 3
| |
| |
|-
| | heptatonic
| | [[2L_5s|2L 5s]]
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
|-
| | nonatonic
| | [[2L_7s|2L 7s]]
| | 5
| |
| |
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 5
| |
| |
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
|-
| | 11-tone (quasi-equal)
| | [[9L_2s|9L 2s]]
| | 2
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 2
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
| | 3
| |
| |
|-
| | 20-tone
| | [[11L_9s|11L 9s]]
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
| | 2
| |
| | 1
|}
 
===<span style="background-color: #ffffff;">15\31 tritave - approx. 920.3¢ - Small Tritone or Augmented Fourth or Subdiminished Fifth</span>===
<span style="background-color: #ffffff;">In 23-limit tonal music, functions quite well as 46:27 (922.41¢). Exactly thrice a whole tone. Generates Trans-[[Arcturus|Arcturus]] temperament.</span>
 
====<span style="background-color: #ffffff;">MOS Scales generated by 15\31:</span>====
 
{| class="wikitable"
|-
! | 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|2L 1s]]
| | 15
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 15
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
|-
| | pentatonic
| | [[2L_3s|2L 3s]]
| | 14
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 14
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
|-
| | heptatonic
| | [[2L_5s|2L 5s]]
| | 13
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 13
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
|-
| | nonatonic
| | [[2L_7s|2L 7s]]
| | 12
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 12
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
|-
| | 11-tone
| | [[2L_9s|2L 9s]]
| | 11
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 11
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
|-
| | 13-tone
| | [[2L_11s|2L 11s]]
| | 10
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 10
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
|-
| | 15-tone
| | [[2L_13s|2L 13s]]
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 9
| |
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
|-
| | 17-tone
| | [[2L_15s|2L 15s]]
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 8
| |
| |
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
| | 1
|-
| | 19-tone
| | [[2L_17s|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|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|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|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|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|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
|}
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