12edt: Difference between revisions
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{{Harmonics in equal|12|3|1|intervals=prime}} | {{Harmonics in equal|12|3|1|intervals=prime}} | ||
==Scala file== | == Scala file == | ||
<pre> | <pre> | ||
! C:\Cakewalk\scales\tritave-in-12.scl | ! C:\Cakewalk\scales\tritave-in-12.scl | ||
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</pre> | </pre> | ||
==Theory== | == Theory == | ||
In octave land, 12edo handles the 2.3.5 subgroup and [[11edo]] handles the 2.7.11 | In octave land, 12edo handles the 2.3.5 subgroup and [[11edo]] handles the 2.7.11 subgroup—ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen–Pierce) and 12edt handles the 2.3.5.13.17.19—and, it is a multiple of 4edt which is the simplest BP equal temperament. | ||
===Macrodiatonic and macromeantone=== | === Macrodiatonic and macromeantone === | ||
12edt can be viewed as a version of [[12edo]] with octaves stretched out to [[3/1|tritaves]], so it contains an extremely stretched diatonic scale or [[macrodiatonic and microdiatonic|macrodiatonic]] scale ([[5L 2s (3/1-equivalent)|5L 2s<3/1>]]). This scale has an identical structure to diatonic, but with everything stretched out to be | 12edt can be viewed as a version of [[12edo]] with octaves stretched out to [[3/1|tritaves]], so it contains an extremely stretched diatonic scale or [[macrodiatonic and microdiatonic|macrodiatonic]] scale ([[5L 2s (3/1-equivalent)|5L 2s<3/1>]]). This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable—for example, the "perfect fifth" is inflated to the size of a major seventh. The stretched perfect fifth can be approximated by [[17/9]] and the stretched major third by [[13/9]]. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 subgroup, equating 4 [[17/9]] to [[13/9]] tritave-reduced, rather than 4 [[3/2]] to [[5/4]] octave-reduced (although this is not a completely exact stretching of meantone, unlike some macromeantones like [[meansquared]] which repeats at [[4/1]]). | ||
Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord. | Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord. | ||
==Compositions== | == Compositions == | ||
[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]] | [https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]] | ||
Revision as of 18:26, 25 February 2025
| ← 11edt | 12edt | 13edt → |
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for hemikleismic temperament. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.
Interval table
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 158.5 | 108.3 | 21/19, 23/21 |
| 2 | 317 | 216.7 | 6/5, 13/11, 17/14, 23/19 |
| 3 | 475.5 | 325 | 17/13 |
| 4 | 634 | 433.3 | 13/9, 19/13 |
| 5 | 792.5 | 541.7 | 11/7, 14/9 |
| 6 | 951 | 650 | 19/11 |
| 7 | 1109.5 | 758.3 | 17/9, 21/11 |
| 8 | 1268 | 866.7 | 19/9, 23/11 |
| 9 | 1426.5 | 975 | |
| 10 | 1585 | 1083.3 | 5/2 |
| 11 | 1743.5 | 1191.7 | 19/7 |
| 12 | 1902 | 1300 | 3/1 |
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +68.0 | +0.0 | +66.6 | -40.4 | -30.4 | -2.6 | +8.4 | -25.6 | -39.4 | +34.8 | +77.8 |
| Relative (%) | +42.9 | +0.0 | +42.0 | -25.5 | -19.2 | -1.7 | +5.3 | -16.2 | -24.9 | +21.9 | +49.1 | |
| Steps (reduced) |
8 (8) |
12 (0) |
18 (6) |
21 (9) |
26 (2) |
28 (4) |
31 (7) |
32 (8) |
34 (10) |
37 (1) |
38 (2) | |
Scala file
! C:\Cakewalk\scales\tritave-in-12.scl ! 3/1 in 12 12 ! 158.49625 316.99250 475.48875 633.98500 792.48125 950.97750 1109.47375 1267.97000 1426.46625 1584.96250 1743.45875 3/1
Theory
In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup—ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen–Pierce) and 12edt handles the 2.3.5.13.17.19—and, it is a multiple of 4edt which is the simplest BP equal temperament.
Macrodiatonic and macromeantone
12edt can be viewed as a version of 12edo with octaves stretched out to tritaves, so it contains an extremely stretched diatonic scale or macrodiatonic scale (5L 2s<3/1>). This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable—for example, the "perfect fifth" is inflated to the size of a major seventh. The stretched perfect fifth can be approximated by 17/9 and the stretched major third by 13/9. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 subgroup, equating 4 17/9 to 13/9 tritave-reduced, rather than 4 3/2 to 5/4 octave-reduced (although this is not a completely exact stretching of meantone, unlike some macromeantones like meansquared which repeats at 4/1).
Another example of a macrodiatonic scale is hyperpyth which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.