31edo: Difference between revisions

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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|Adriaan Fokker]]. Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents|cents]]. 31's perfect fifth is flat of the just interval 3:2 (over five cents), as befits a tuning supporting meantone, but the major third is less than a cent sharp (of just 5:4). 31's approximation of 7:4, a cent flat, is also very close to just. Because of these near-just values 31-et is relatively quite accurate and is in fact the sixth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]]. Many [[7-limit|7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also deals with the [[11-limit|11-limit]] fairly well, and is consistent through it, but is the [[Optimal_patent_val|optimal patent val]] for the rank five temperament tempering out the 13-limit comma 66/65. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.

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

[[List_of_31et_rank_two_temperaments_by_badness|List of 31et rank two temperaments by badness]]

[[List_of_edo-distinct_31et_rank_two_temperaments|List of edo-distinct 31et rank two temperaments]]

! | [[pergen|Pergen]]

| | [[Slender|Slender]]

| | [[Valentine|Valentine]]/[[Lupercalia|Lupercalia]]

| | [[Miracle|Miracle]]

| | [[Nusecond|Nusecond]]

| | [[Luna|Luna]]/[[Hemithirds|Hemithirds]]/[[Hemiwürschmidt|Hemiwürschmidt]]

| | [[Mothra|Mothra]]/[[Mosura|Mosura]]

| | [[Orson|Orson]]/[[Orwell|Orwell]]/[[Semicomma_family#Orwell-Winston|Winston]]

| | [[Myna|Myna]]

| | [[Vicentino|Vicentino]]/[[Mohajira|Mohajira]]/[[Migration|Migration]]

| | [[Würschmidt|Würschmidt]]/[[Worschmidt|Worschmidt]]

| | [[Squares|Squares]]/[[Sentinel|Sentinel]]

| | [[Semisept|Semisept]]

| | [[Meantone|Meantone]]/[[Meanpop|Meanpop]]

| | [[casablanca|Casablanca]]/[[Cypress|Cypress]]/[[Oracle|Oracle]]

| | [[Tritonic|Tritonic]]/[[Tritoni|Tritoni]]

! | Error (abs., in [[cent|cents]])

| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]

| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]

| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]

| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]

| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]

| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]

| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]

| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]

| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]

| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]

| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]

| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]

| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]

| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]

| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]

| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]

| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]

| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]

| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]

| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]

| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]

| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]

| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]

| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]

! | Error (abs., in [[cent|cents]])

| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]

| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]

| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]

| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]

| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]

| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]

| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]

| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]

| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]

| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]

| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]

| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]

| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]

| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]

| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]

| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]

| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]

| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]

| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]

| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]

| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]

| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]

| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]

| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]

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 B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). In the [[11-limit|11-limit]], the diesis 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. 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|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 (eg. thirds). 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|valentine temperament]] - aka [[Armodue_theory#Semi-equalized Armodue|semi-equalized Armodue]].

====MOS Scales generated by 2\31:====

| | 15-tone ([[Maximal_evenness|ME]] or quasi-equal)

| | [[1L_14s|1L 14s]]

| | [[15L_1s|15L 1s]]

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 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 [[Gamelismic_clan|miracle temperament]].

====MOS Scales generated by 3\31:====

| | [[1L_8s|1L 8s]]

| | [[9L_1s|9L 1s]]

| | [[10L_1s|10L 1s]]

| | [[11L_10s|11L 10s]]

Exactly one half of the minor third and twice the minor semitone. 4\31 stands in for 12:11 (150.64¢); 35:32 (155.14¢); 11:10 (165.00¢) and others. Although neutral seconds are typically associated with the 11-limit, 4\31 approximates the [[7-limit|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 [[Starling_temperaments|nusecond temperament]].

====MOS Scales generated by 4\31:====

| | [[1L_6s|1L 6s]]

| | [[7L_1s|7L 1s]]

| | [[8L_7s|8L 7s]]

| | [[8L_15s|8L 15s]]

A rather smallish whole tone. Sometimes 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. 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 0-18-36-45). Generates [[Gamelismic_clan|hemithirds temperament]] and [[Wuerschmidt_family|hermiwuerschmidt temperament]].

====MOS Scales generated by 5\31:====

| | [[1L_5s|1L 5s]]

| | [[6L_1s|6L 1s]]

| | [[6L_7s|6L 7s]]

| | [[6L_13s|6L 13s]]

| | [[6L_19s|6L 19s]]

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, eg. C to Ebb. Generates [[Meantone_family|mothra temperament]].

====MOS Scales generated by 6\31:====

| | [[1L_4s|1L 4s]]

| | [[5L_1s|5L 1s]]

| | [[5L_6s|5L 6s]]

| | [[5L_11s|5L 11s]]

| | [[5L_16s|5L 16s]]

| | [[5L_21s|5L 21s]]

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, eg. C to D#. Generates [[Semicomma_family|orwell temperament]].

====MOS Scales generated by 7\31:====

| | [[4L_1s|4L 1s]]

| | [[4L_5s|4L 5s]]

| | [[9L_4s|9L 4s]]

| | [[9L_13s|9L 13s]]

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 [[Starling_temperaments|myna temperament]].

====MOS Scales generated by 8\31:====

| | [[3L_1s|3L 1s]]

| | [[4L_3s|4L 3s]]

| | [[4L_7s|4L 7s]]

| | [[4L_11s|4L 11s]]

| | [[4L_15s|4L 15s]]

| | [[4L_19s|4L 19s]]

| | [[4L_23s|4L 23s]]

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|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|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.

====MOS Scales generated by 9\31:====

| | [[3L_1s|3L 1s]]

| | [[3L_4s|3L 4s]]

| | [[7L_3s|7L 3s]]

| | [[7L_10s|7L 10s]]

| | [[7L_17s|7L 17s]]

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:====

| | [[1L_2s|1L 2s]]

| | [[3L_1s|3L 1s]]

| | [[3L_4s|3L 4s]]

| | [[3L_7s|3L 7s]]

| | [[3L_10s|3L 10s]]

| | [[3L_13s|3L 13s]]

| | [[3L_16s|3L 16s]]

| | [[3L_19s|3L 19s]]

| | [[3L_22s|3L 22s]]

| | [[3L_25s|3L 25s]]

11\31 functions as 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, eg. C to Fb. It is notable as closely approximating an interval of the [[23-limit|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|17edo]]. Generates [[Meantone_family|squares temperament]].

====MOS Scales generated by 11\31:====

| | [[2L_1s|2L 1s]]

| | [[3L_2s|3L 2s]]

| | [[3L_5s|3L 5s]]

| | [[3L_8s|3L 8s]]

| | [[3L_11s|3L 11s]]

| | [[3L_14s|3L 14s]]

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¢). It is also quite close to the [[17-limit|17-limit]] interval 17/13 (464.43¢), although 31edo does not offer up reasonable approximations of the 17th or 13th harmonics to help 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-limit consonances, and the only intervals in the 31edo octave that are not 15-limit consonances. Generates [[Semisept|semisept]] temperament.

====MOS Scales generated by 12\31:====

| | [[2L_1s|2L 1s]]

| | [[3L_2s|3L 2s]]

| | [[5L_3s|5L 3s]]

| | [[5L_8s|5L 8s]]

| | [[13L_5s|13L 5s]]

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

====MOS Scales generated by 13\31:====

| | [[2L_1s|2L 1s]]

| | [[2L_3s|2L 3s]]

| | [[5L_2s|5L 2s]]

| | [[7L_5s|7L 5s]]

| | [[12L_7s|12L 7s]]

Exactly twice a subminor third. Functions as both the 11:8 (551.32¢) and 15:11 (536.95¢) undecimal superfourths (121/120 is 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 [[Starling_temperaments|casablanca temperament]].

====MOS Scales generated by 14\31:====

| | [[2L_1s|2L 1s]]

| | [[2L_3s|2L 3s]]

| | [[2L_5s|2L 5s]]

| | [[2L_7s|2L 7s]]

| | [[9L_2s|9L 2s]]

| | [[11L_9s|11L 9s]]

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

====MOS Scales generated by 15\31:====

| | [[2L_1s|2L 1s]]

| | [[2L_3s|2L 3s]]

| | [[2L_5s|2L 5s]]

| | [[2L_7s|2L 7s]]

| | [[2L_9s|2L 9s]]

| | [[2L_11s|2L 11s]]

| | [[2L_13s|2L 13s]]

| | [[2L_15s|2L 15s]]

| | [[2L_17s|2L 17s]]

| | [[2L_19s|2L 19s]]

| | [[2L_21s|2L 21s]]

| | [[2L_23s|2L 23s]]

| | [[2L_25s|2L 25s]]

| | [[2L_27s|2L 27s]]

31edo can be notated with [[Ups_and_Downs_Notation|ups and downs notation]] like so:

Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:

For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].

31edo approximates Mode 8 of the [[OverToneSeries|harmonic series]] O.K., but many intervals between the harmonics aren't distinguished, most importantly 9/8 (major tone) and 10/9 (minor tone), as 31EDO is a meantone temperament. The interval between the 8th and 11th harmonics is approximated O.K., but the intervals between the 11th harmonic and closer harmonics such as the 12th and 9th harmonics are approximated even better. 31's version of 13/8 is quite wide and only vaguely suggests the [[13-limit|13-limit]].

| style="text-align:center;" | [[Syntonic_Comma|Syntonic Comma]]

| style="text-align:center;" | [[Würschmidt_comma|Würschmidt Comma]]

| style="text-align:center;" | [[semicomma|Semicomma]]

A large open list of modes (subsets) from 31edo that people have named: [[31edo_modes|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|David Rothenberg]]. Interesting (to somebody) [[9-note_31edo_scales|9-note 31edo scales]]. See also [[31edo_MOS_scales|31edo MOS scales]]. Some of the popular ones:

<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 3 3 3</li><li>the [[Euler-Fokker_genera|Euler-Fokker genera]] (technically [[JI|JI]] but representable in 31)</li></ul>

====Some 31 tone equal modes:====

[[31-edo_compositions|An alphabetical list of Tricesimoprimal Compositions]].

[http://archive.org/download/Aire2In31-equalTemperament/Aire2In31.mp3 Aire #2 in 31-equal temperament] by [[Jon_Lyle_Smith|Jon Lyle Smith]]

[http://soonlabel.com/xenharmonic/wp-content/uploads/2012/03/Claudi_Meneghin_Chaconne_G_001.mp3 Chaconna en G=, La Padana, ou la septimala (‘The Padanian, or the septimal’)] by [[Claudi_Meneghin|Claudi Meneghin]]

[[earwig|earwig]] by [[Andrew_Heathwaite|Andrew Heathwaite]]

[https://www.youtube.com/watch?v=r1mat9f1DZ0 Fanfare and Toccata] by [[Juhani_Nuorvala|Juhani Nuorvala]]

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

See also: [[31edo_solfege|31edo solfege]], [[Tricesimoprimal_Tetrachordal_Tesseract|Tricesimoprimal Tetrachordal Tesseract]], [[Pentachords_of_31edo|Pentachords of 31edo]].