36edo: Difference between revisions
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= Theory = | |||
<b>36edo</b>, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33⅓ cents. | <b>36edo</b>, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33⅓ cents. | ||
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That 36edo contains 12edo as a subset makes it compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33 cents, one can arrive at a 24-tone subset of 36 edo (see, for instance, Jacob Barton's piece for two clarinets, [http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/ De-quinin']). Three 12edo instruments could play the entire gamut. | That 36edo contains 12edo as a subset makes it compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33 cents, one can arrive at a 24-tone subset of 36 edo (see, for instance, Jacob Barton's piece for two clarinets, [http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/ De-quinin']). Three 12edo instruments could play the entire gamut. | ||
For those interested in approximations to just intonation, 36edo offers no improvement over 12edo in the 5-limit, since its nearest approximation to 5:4 is the overly-familiar 400-cent sharp third. However, it excels at approximations involving 3 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [http://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36 cents, and as 64:63, the so-called [http://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [http://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo|72edo]] does in the full [[17-limit|17-limit]]. | For those interested in approximations to just intonation, 36edo offers no improvement over 12edo in the 5-limit, since its nearest approximation to 5:4 is the overly-familiar 400-cent sharp third. However, it excels at approximations involving 3 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [http://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36 cents, and as 64:63, the so-called [http://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [http://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (which = 49:48 x 64:63). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo|72edo]] does in the full [[17-limit|17-limit]]. | ||
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Heinz Bohlen proposed it as a suitable temperament for approximating his 833-cents scale. | Heinz Bohlen proposed it as a suitable temperament for approximating his 833-cents scale. | ||
= | = Intervals = | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
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| style="text-align:center;" | ^1 | | style="text-align:center;" | ^1 | ||
| style="text-align:center;" | up unison | | style="text-align:center;" | up unison | ||
| style="text-align:center;" | D | | style="text-align:center;" | ^D | ||
|- | |- | ||
| style="text-align:center;" | 2 | | style="text-align:center;" | 2 | ||
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| style="text-align:center;" | vm2 | | style="text-align:center;" | vm2 | ||
| style="text-align:center;" | downminor 2nd | | style="text-align:center;" | downminor 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | vEb | ||
|- | |- | ||
| style="text-align:center;" | 3 | | style="text-align:center;" | 3 | ||
| Line 149: | Line 67: | ||
| style="text-align:center;" | ^m2 | | style="text-align:center;" | ^m2 | ||
| style="text-align:center;" | upminor 2nd | | style="text-align:center;" | upminor 2nd | ||
| style="text-align:center;" | Eb | | style="text-align:center;" | ^Eb | ||
|- | |- | ||
| style="text-align:center;" | 5 | | style="text-align:center;" | 5 | ||
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| style="text-align:center;" | vM2 | | style="text-align:center;" | vM2 | ||
| style="text-align:center;" | downmajor 2nd | | style="text-align:center;" | downmajor 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | vE | ||
|- | |- | ||
| style="text-align:center;" | 6 | | style="text-align:center;" | 6 | ||
| Line 173: | Line 91: | ||
| style="text-align:center;" | ^M2 | | style="text-align:center;" | ^M2 | ||
| style="text-align:center;" | upmajor 2nd | | style="text-align:center;" | upmajor 2nd | ||
| style="text-align:center;" | E | | style="text-align:center;" | ^E | ||
|- | |- | ||
| style="text-align:center;" | 8 | | style="text-align:center;" | 8 | ||
| Line 181: | Line 99: | ||
| style="text-align:center;" | vm3 | | style="text-align:center;" | vm3 | ||
| style="text-align:center;" | downminor 3rd | | style="text-align:center;" | downminor 3rd | ||
| style="text-align:center;" | | | style="text-align:center;" | vF | ||
|- | |- | ||
| style="text-align:center;" | 9 | | style="text-align:center;" | 9 | ||
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| style="text-align:center;" | ^m3 | | style="text-align:center;" | ^m3 | ||
| style="text-align:center;" | upminor 3rd | | style="text-align:center;" | upminor 3rd | ||
| style="text-align:center;" | F | | style="text-align:center;" | ^F | ||
|- | |- | ||
| style="text-align:center;" | 11 | | style="text-align:center;" | 11 | ||
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| style="text-align:center;" | vM3 | | style="text-align:center;" | vM3 | ||
| style="text-align:center;" | downmajor 3rd | | style="text-align:center;" | downmajor 3rd | ||
| style="text-align:center;" | | | style="text-align:center;" | vF# | ||
|- | |- | ||
| style="text-align:center;" | 12 | | style="text-align:center;" | 12 | ||
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| style="text-align:center;" | ^M3 | | style="text-align:center;" | ^M3 | ||
| style="text-align:center;" | upmajor 3rd | | style="text-align:center;" | upmajor 3rd | ||
| style="text-align:center;" | F# | | style="text-align:center;" | ^F# | ||
|- | |- | ||
| style="text-align:center;" | 14 | | style="text-align:center;" | 14 | ||
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| style="text-align:center;" | v4 | | style="text-align:center;" | v4 | ||
| style="text-align:center;" | down 4th | | style="text-align:center;" | down 4th | ||
| style="text-align:center;" | | | style="text-align:center;" | vG | ||
|- | |- | ||
| style="text-align:center;" | 15 | | style="text-align:center;" | 15 | ||
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| style="text-align:center;" | ^4 | | style="text-align:center;" | ^4 | ||
| style="text-align:center;" | up 4th | | style="text-align:center;" | up 4th | ||
| style="text-align:center;" | G | | style="text-align:center;" | ^G | ||
|- | |- | ||
| style="text-align:center;" | 17 | | style="text-align:center;" | 17 | ||
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| style="text-align:center;" | vA4 | | style="text-align:center;" | vA4 | ||
| style="text-align:center;" | downaug 4th | | style="text-align:center;" | downaug 4th | ||
| style="text-align:center;" | | | style="text-align:center;" | vG# | ||
|- | |- | ||
| style="text-align:center;" | 18 | | style="text-align:center;" | 18 | ||
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| style="text-align:center;" | ^d5 | | style="text-align:center;" | ^d5 | ||
| style="text-align:center;" | updim 5th | | style="text-align:center;" | updim 5th | ||
| style="text-align:center;" | Ab | | style="text-align:center;" | ^Ab | ||
|- | |- | ||
| style="text-align:center;" | 20 | | style="text-align:center;" | 20 | ||
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| style="text-align:center;" | v5 | | style="text-align:center;" | v5 | ||
| style="text-align:center;" | down 5th | | style="text-align:center;" | down 5th | ||
| style="text-align:center;" | | | style="text-align:center;" | vA | ||
|- | |- | ||
| style="text-align:center;" | 21 | | style="text-align:center;" | 21 | ||
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| style="text-align:center;" | ^5 | | style="text-align:center;" | ^5 | ||
| style="text-align:center;" | up fifth | | style="text-align:center;" | up fifth | ||
| style="text-align:center;" | A | | style="text-align:center;" | ^A | ||
|- | |- | ||
| style="text-align:center;" | 23 | | style="text-align:center;" | 23 | ||
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| style="text-align:center;" | vm6 | | style="text-align:center;" | vm6 | ||
| style="text-align:center;" | downminor 6th | | style="text-align:center;" | downminor 6th | ||
| style="text-align:center;" | | | style="text-align:center;" | vBb | ||
|- | |- | ||
| style="text-align:center;" | 24 | | style="text-align:center;" | 24 | ||
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| style="text-align:center;" | ^m6 | | style="text-align:center;" | ^m6 | ||
| style="text-align:center;" | upminor 6th | | style="text-align:center;" | upminor 6th | ||
| style="text-align:center;" | Bb | | style="text-align:center;" | ^Bb | ||
|- | |- | ||
| style="text-align:center;" | 26 | | style="text-align:center;" | 26 | ||
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| style="text-align:center;" | vM6 | | style="text-align:center;" | vM6 | ||
| style="text-align:center;" | downmajor 6th | | style="text-align:center;" | downmajor 6th | ||
| style="text-align:center;" | | | style="text-align:center;" | vB | ||
|- | |- | ||
| style="text-align:center;" | 27 | | style="text-align:center;" | 27 | ||
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| style="text-align:center;" | ^M6 | | style="text-align:center;" | ^M6 | ||
| style="text-align:center;" | upmajor 6th | | style="text-align:center;" | upmajor 6th | ||
| style="text-align:center;" | B | | style="text-align:center;" | ^B | ||
|- | |- | ||
| style="text-align:center;" | 29 | | style="text-align:center;" | 29 | ||
| Line 349: | Line 267: | ||
| style="text-align:center;" | vm7 | | style="text-align:center;" | vm7 | ||
| style="text-align:center;" | downminor 7th | | style="text-align:center;" | downminor 7th | ||
| style="text-align:center;" | | | style="text-align:center;" | vC | ||
|- | |- | ||
| style="text-align:center;" | 30 | | style="text-align:center;" | 30 | ||
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| style="text-align:center;" | ^m7 | | style="text-align:center;" | ^m7 | ||
| style="text-align:center;" | upminor 7th | | style="text-align:center;" | upminor 7th | ||
| style="text-align:center;" | C | | style="text-align:center;" | ^C | ||
|- | |- | ||
| style="text-align:center;" | 32 | | style="text-align:center;" | 32 | ||
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| style="text-align:center;" | vM7 | | style="text-align:center;" | vM7 | ||
| style="text-align:center;" | downmajor 7th | | style="text-align:center;" | downmajor 7th | ||
| style="text-align:center;" | | | style="text-align:center;" | vC# | ||
|- | |- | ||
| style="text-align:center;" | 33 | | style="text-align:center;" | 33 | ||
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| style="text-align:center;" | ^M7 | | style="text-align:center;" | ^M7 | ||
| style="text-align:center;" | upmajor 7th | | style="text-align:center;" | upmajor 7th | ||
| style="text-align:center;" | C# | | style="text-align:center;" | ^C# | ||
|- | |- | ||
| style="text-align:center;" | 35 | | style="text-align:center;" | 35 | ||
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| style="text-align:center;" | v8 | | style="text-align:center;" | v8 | ||
| style="text-align:center;" | down 8ve | | style="text-align:center;" | down 8ve | ||
| style="text-align:center;" | | | style="text-align:center;" | vD | ||
|- | |- | ||
| style="text-align:center;" | 36 | | style="text-align:center;" | 36 | ||
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|} | |} | ||
Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[ | Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]]. | ||
=Relation to 12edo= | |||
For people accustomed to 12edo, 36edo is one of the easiest (if not ''the'' easiest) higher edo to become accustomed to. This is because one way to envision it is as an extended 12edo to which [https://en.wikipedia.org/wiki/Blue_note blue notes] (which are a sixth-tone lower than normal) and "red notes" (a sixth-tone higher) have been added. | |||
The intervals in 36edo are all either the familiar 12edo intervals, or else "red" and "blue" versions of them. Unlike [[24edo|24edo]], which has genuinely foreign intervals such as 250 cents (halfway between a tone and a third) and 450 cents (halfway between a fourth and a third), the new intervals in 36edo all variations on existing ones. Unlike 24edo, 36edo is also relatively free of what Easley Blackwood called "discordant" intervals. | |||
An easy way of notating 36edo (at least for people who aren't colorblind) is to use colors. For example, '''A''' is 33.333 cents above '''<span style="background-color: #6ee8e8; color: #071ac7;">A</span>''' and 33.333 cents below '''<span style="background-color: #eda2a2; color: #ff0000;">A</span>'''. Or the colors could be written out (red A, blue C#, etc.) or abbreviated as rA, bC#, etc. This use of red and blue is consistent with [[Kite's_color_notation|color notation]] (ru and zo). | |||
Because of the presence of blue notes, and the closeness with which intervals such as 4:7 are matched, 36edo is an ideal scale to use for African-American styles of music such as blues and jazz, in which chords containing the seventh harmonic are frequently used. The 5th and 11th harmonic fall almost halfway in between scale degrees of 36edo, and thus intervals containing them can be approximated two different ways, one of which is significantly sharp and the other significantly flat. The 333.333-cent interval (the "red minor third") sharply approximates 5:6 and flatly approximates 9:11, for instance, whereas the sharp 9:11 is 366.667 cents and the flat 5:6 is 300 cents. However, 10:11 and 11:15 each have a single (very close) approximation since they contain both the 5th and 11th harmonic. | |||
36edo is fairly cosmopolitan because many other genres of world music can be played in it too. Because it contains 9edo as a subset, pelog (and mavila) easily adapt to it. Slendro can be approximated in several different ways. 36edo can function as a "bridge" between these genres and Western music. Arabic music does not adapt as well, however, since many versions contain quarter tones. | |||
The "red unison" and "blue unison" are in fact the same interval (33.333 cents), which is actually fairly consonant as a result of being so narrow (it is perceived as a unison, albeit noticeably "out of tune", but still pleasing). In contrast, the smallest interval in 24edo, which is 50 cents, sounds very bad to most ears. | |||
People with perfect (absolute) pitch often have a harder time listening to xenharmonic and non-12edo scales, which is due to their ability to memorize and become accustomed to the pitches and intervals of 12edo (which results in other pitches and intervals sounding bad). This is not as much of a problem with 36edo, due to its similarity to 12. With practice, it might even be possible to extend one's perfect pitch to be able to recognize blue and red notes too. | |||
==="Quark"=== | |||
In particle physics, [https://en.wikipedia.org/wiki/Baryon baryons] , which are the main building blocks of atomic nuclei, are always comprised of three quarks. One could draw an analogy between baryons and semitones (the main building block of 12edo); the baryon is comprised of three quarks and the semitone of three sixth-tones. The number of quarks in a [https://en.wikipedia.org/wiki/Color_charge colorless] particle is always a multiple of three; similarly, the width of "colorless" intervals (the 12-edo intervals, which are neither red nor blue), expressed in terms of sixth-tones, is always a multiple of three. Because of this amusing coincidence, I (Mason Green) propose referring to the 33.333-cent sixth-tone interval as a "quark". | |||
=JI Approximations= | |||
==3-limit (Pythagorean) approximations (same as 12edo):== | |||
3/2 = 701.955... cents; 21 degrees of 36edo = 700 cents. | |||
4/3 = 498.045... cents; 15 degrees of 36edo = 500 cents. | |||
9/8 = 203.910... cents; 6 degrees of 36edo = 200 cents. | |||
16/9 = 996.090... cents; 30 degrees of 36edo = 1000 cents. | |||
27/16 = 905.865... cents; 27 degrees of 36edo = 900 cents. | |||
32/27 = 294.135... cents; 9 degrees of 36edo = 300 cents. | |||
81/64 = 407.820... cents; 12 degrees of 36edo = 400 cents. | |||
128/81 = 792.180... cents; 24 degrees of 36edo = 800 cents. | |||
==7-limit approximations:== | |||
===7 only:=== | |||
7/4 = 968.826... cents; 29 degrees of 36edo = 966.666... cents. | |||
8/7 = 231.174... cents; 7 degrees of 36edo = 233.333... cents. | |||
49/32 = 737.652... cents; 22 degrees of 36edo = 733.333... cents. | |||
64/49 = 462.348... cents; 14 degrees of 36edo = 466.666... cents. | |||
===3 and 7:=== | |||
7/6 = 266.871... cents; 8 degrees of 36edo = 266.666... cents. | |||
12/7 = 933.129... cents; 28 degrees of 36 = 933.333... cents. | |||
9/7 = 435.084... cents; 13 degrees of 36edo = 433.333... cents. | |||
14/9 = 764.916... cents; 23 degrees of 36edo = 766.666... cents. | |||
28/27 = 62.961... cents; 2 degrees of 36edo = 66.666... cents. | |||
27/14 = 1137.039... cents; 34 degrees of 36edo = 1133.333... cents. | |||
21/16 = 470.781... cents; 14 degrees of 36edo = 466.666... cents. | |||
32/21 = 729.219... cents; 22 degrees of 36edo = 733.333... cents. | |||
49/48 = 35.697... cents; 1 degree of 36edo = 33.333... cents. | |||
96/49 = 1164.303... cents; 35 degrees of 36edo = 1166.666... cents. | |||
49/36 = 533.742... cents; 16 degrees of 36edo = 533.333... cents. | |||
72/49 = 666.258... cents; 20 degrees of 36edo = 666.666... cents. | |||
64/63 = 27.264... cents; 1 degree of 36edo = 33.333... cents. | |||
63/32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents. | |||
=Music= | =Music= | ||