31edo/Individual degrees: Difference between revisions

This article explains individual degrees of [[31edo]] in depth.  

A single step of 31-edo is about 38.71¢. Intervals around this size are called ''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 {{nowrap|B♯ ≠ C}}), or four minor thirds (C to E♭ to G♭ to B𝄫 to {{nowrap|D𝄫 ≠ C}}). In the [[11-limit]], the diesis stands in for just ratios {{nowrap|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. The diesis is a defining sound of 31edo; 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. Demonstrated in [[SpiralProgressions]].

The difference between a major and minor third. The more "expressive" of the "half steps," and the larger of 31's two "microtones". In meantone, it is the ''chromatic semitone'', the interval that distinguishes major and minor intervals of the same generic interval class (e.g. thirds). 2\31 stands in for just ratios {{nowrap|28:27 (62.96¢)|25:24 (70.67¢)|22:21 (80.54¢)|21:20 (84.45¢)}}, and others. Generates [[Valentine|valentine temperament]]—aka [[Armodue theory#Semi-equalized Armodue|semi-equalized Armodue]].

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The larger and clunkier of the 31edo semitones. In meantone, it is the ''diatonic semitone'' which appears in the diatonic scale between, for instance, the major third and perfect fourth, and the major seventh and octave. 3\31 stands in for just ratios {{nowrap|16:15 (111.73¢)|15:14 (119.44¢)}}, and others. It is notable that two of these make an 8/7; this implies that the 3\31 is a ''secor'' and generates [[Miracle|miracle temperament]]. It represents 343:320 (120.16¢) and 14:13 (128.29¢), and five of these give a 7/5, meaning it generates [[Quince clan#Mercy|mercy temperament]]. The Pythagorean apotome, 2187:2048 (113.69¢) is close to 3\31 in value, but is not consistent with the mapping of the primes 2 and 3 in 31edo (in fact the apotome of 31edo is the previous degree 2\31).

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Exactly one half of the minor third and twice the minor semitone. 4\31 stands in for {{nowrap|12:11 (150.64¢)|35:32 (155.14¢)|11:10 (165¢)}}, and others. Although neutral seconds are typically associated with the 11-limit, 4\31 approximates the [[7-limit]] interval 35/32 quite well, as the 5th harmonic of the 7th harmonic or vice versa, both of which are closely approximated in 31edo. And although 31 is not extremely accurate in the 11-limit, it is notable that since 11 and 3 are both flat, the interval that distinguishes them (12/11) is only about 4.5¢ off. Generates [[Nusecond|nusecond temperament]].

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A rather smallish whole tone, 0.99¢ sharp of [[19/17]] (192.56¢). Sometimes called melodically dull, 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. Perhaps it is worth noting that its relative narrowness (to JI 9/8) makes it easier to distinguish from the 8/7 approximation. And although it is over 10¢ flat of 9/8, 5\31 can function as a somewhat "active" (as opposed to perfectly stable) harmonic ninth, and it can be effective in combination with the also-narrow 11th harmonic. Indeed, the 11/9 approximation is excellent. Try, for instance 31's version of a 4:6:9:11 chord (steps {{dash|0, 18, 36, 45|med}}). Generates [[hemithirds]] and [[hemiwürschmidt]] temperaments.

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Exactly one half of a narrow fourth, twice a major semitone, or thrice a minor semitone. In 7-limit tonal music, 6\31 closely represents 8:7 (231.17¢). In meantone, it is a diminished third, e.g. C to E𝄫. Generates [[Mothra|mothra temperament]].

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Exactly one half of a superfourth (11:8 approximation). In 7-limit tonal music, 7\31 stands in for 7:6 (266.87¢). In meantone temperament, it is an augmented 2nd, e.g. C to D♯. Generates [[Orwell|orwell]] temperament.

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A minor third, closer to the just 6:5 (315.64¢) than 12-edo, but still on the flat side. Exactly twice a neutral second, four times a minor semitone, and half of a large tritone. Generates [[Myna|myna temperament]].

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A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for [[mohajira]] temperament), and also thrice a major semitone. It is closer in quality to a minor third than a major third, but indeed, it is distinct. It is 11¢ shy of 16/13 (359.47¢), suggesting a [[13-limit]] interpretation for 31edo. However, its close proximity to 11/9 makes it hard to hear it as 16/13, which in JI has a different quality (and, as a neutral third, is more "major-like" than "minor-like"). Also, its inversion, 22\31 (851.61¢) is wide of the 13th harmonic by about 11¢, which leaves the 143rd harmonic only about 2¢ wide after cancelling with the narrow 11th harmonic, while all the lower harmonics are either near-just or narrow. This means the errors can accumulate, for instance, with 13/9 (636.62¢) represented by 17\31 (658.06¢), a good 21.4¢ sharp.

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A near-just major 3rd (compare with {{nowrap|5:4 {{=}} 386.31¢}}). Has led to the characterization of 31-edo as "smooth". Generates [[Würschmidt|würschmidt and worschmidt]] temperaments.

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11\31 functions as {{nowrap|14:11 (417.51¢)|23:18 (424.36¢)|32:25 (427.37¢)|9:7 (435.08¢)}}, and others. In meantone temperament, it is a diminished fourth, e.g. C to F♭. It is notable as closely approximating an interval of the [[23-limit]], suggesting the possibility of treating 16\31 (619.35¢) as a flat version of 23/16 (628.27¢). It is perhaps also notable for being close to 6\17, the bright major third of the ever-popular [[17edo]]. Generates [[Squares]] temperament.

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Exactly twice a supermajor second, thrice a neutral second, or four times a minor second. In the 7-limit, 12\31 functions as 21:16 (470.78¢). Although 31edo does not offer reasonable approximations of the 17th or 13th harmonics, 12\31 is less than 0.09¢ flat of the [[17-limit]] interval [[17/13]] (464.43¢); combining this with 17\31's 1.08-cent-sharp approximation to [[19/13]] yields a good 13:17:19, which helps make this identity clear. This interval and its inversion 19\31 (735.48¢, a superfifth) are notable for being the only intervals in the 31edo octave larger than the 3\31 diatonic semitone (and smaller than its inversion, 28\31) that are not 11-odd-limit consonances, and the only intervals larger than 2\31 and smaller than 29\31 that are not 15-odd-limit consonances. Generates [[A-Team]] and [[semisept]] temperaments.

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A slightly wide perfect fourth (compare to {{nowrap|4:3 {{=}} 498.04¢}}). As such, it functions marvelously as a generator for [[meantone]] temperament.

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Exactly twice a subminor third, and 1.08c flat of [[26/19]] (543.02¢). When viewing 31edo as a 11-limit temperament, the interval functions as both the 11:8 (551.32¢) and 15:11 (536.95¢) undecimal superfourths, resulting in [[121/120]] being tempered out. Thus it makes possible a symmetrical tempered version of an 8:11:15 triad. As 11/8, 14\31 is about 9¢ flat; however, it fits nicely with the also-flat 9/8, allowing a near-just 11/9. Nonetheless, most 11-limit chords in 31edo have a somewhat unstable quality which distinguishes them from their just counterparts. Generates [[casablanca]] and [[joan]] temperaments.

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In 7-limit tonal music, functions quite well as 7:5 (582.51¢). Exactly thrice a whole tone. Generates [[tritonic]] temperament.