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31edt divides the tritave into equal parts of 61.353 cents, corresponding to the non-octave third tone scale of [[39edo|39edo]] where each degree gets ~.185 cents flatter than the corresponding degree of 39edo. It supports the same higher-limit meantone temperament as 12 edt with better intonation of triads. It also contains a flat version of the BP nonatonic scale and the fair Sigma and false Father scales.
{{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 ~0.185 cents flatter than the corresponding degree of 39edo. It [[support]]s the same higher-limit meantone temperament as 12edt with better intonation of triads. It also contains a flat version of the BP nonatonic scale and the fair Sigma and false Father scales.


=<span style="background-color: #ffffff;">Intervals</span>=
== Intervals ==
{{See also|Specific intervals in 31edt}}
{{Interval table}}


===<span style="background-color: #ffffff;">1\31 tritave- approx. <span style="font-size: 1.1em;">61.35¢ - Third tone</span></span>===
== Harmonics ==
<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>
{{Harmonics in equal
| steps = 31
| num = 3
| denom = 1
| intervals = integer
}}
{{Harmonics in equal
| steps = 31
| num = 3
| denom = 1
| start = 12
| collapsed = 1
| intervals = integer
}}


===<span style="background-color: #ffffff;">2\31 tritave - approx. 122.71¢ - Two-third tone or Small Minor Second</span>===
{{todo|expand}}
<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>====
[[Category:31edt| ]]
 
{| 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)
| | [[11L_10s|11L 10s]]
| | 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
|}

Latest revision as of 16:01, 31 July 2025

← 30edt 31edt 32edt →
Prime factorization 31 (prime)
Step size 61.3534 ¢ 
Octave 20\31edt (1227.07 ¢)
Consistency limit 3
Distinct consistency limit 3

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 ~0.185 cents flatter than the corresponding degree of 39edo. It supports the same higher-limit meantone temperament as 12edt with better intonation of triads. It also contains a flat version of the BP nonatonic scale and the fair Sigma and false Father scales.

Intervals

See also: Specific intervals in 31edt
Steps Cents Hekts Approximate ratios
0 0 0 1/1
1 61.4 41.9 27/26
2 122.7 83.9
3 184.1 125.8 10/9, 19/17
4 245.4 167.7 15/13
5 306.8 209.7
6 368.1 251.6 21/17, 26/21
7 429.5 293.5 9/7
8 490.8 335.5
9 552.2 377.4 26/19
10 613.5 419.4 10/7, 27/19
11 674.9 461.3
12 736.2 503.2 23/15, 26/17
13 797.6 545.2 27/17
14 858.9 587.1 18/11
15 920.3 629 17/10
16 981.7 671 23/13
17 1043 712.9 11/6, 20/11
18 1104.4 754.8 17/9, 19/10
19 1165.7 796.8
20 1227.1 838.7
21 1288.4 880.6 19/9, 21/10
22 1349.8 922.6
23 1411.1 964.5
24 1472.5 1006.5 7/3
25 1533.8 1048.4 17/7
26 1595.2 1090.3
27 1656.5 1132.3 13/5
28 1717.9 1174.2 27/10
29 1779.2 1216.1
30 1840.6 1258.1 26/9
31 1902 1300 3/1

Harmonics

Approximation of harmonics in 31edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +27.1 +0.0 -7.2 -25.4 +27.1 +5.6 +19.8 +0.0 +1.7 +20.7 -7.2
Relative (%) +44.1 +0.0 -11.8 -41.4 +44.1 +9.1 +32.4 +0.0 +2.7 +33.8 -11.8
Steps
(reduced) 		20
(20) 		31
(0) 		39
(8) 		45
(14) 		51
(20) 		55
(24) 		59
(28) 		62
(0) 		65
(3) 		68
(6) 		70
(8)
Approximation of harmonics in 31edt
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -23.1 -28.7 -25.4 -14.4 +3.3 +27.1 -5.2 +28.7 +5.6 -13.6 -29.2
Relative (%) -37.6 -46.7 -41.4 -23.5 +5.4 +44.1 -8.4 +46.8 +9.1 -22.1 -47.6
Steps
(reduced) 		72
(10) 		74
(12) 		76
(14) 		78
(16) 		80
(18) 		82
(20) 		83
(21) 		85
(23) 		86
(24) 		87
(25) 		88
(26)
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