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'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58. | {{Infobox ET}} | ||
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family#Dodecacot|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit. | |||
It is a strong [[half-prime subgroup|3/2.5/2.7/2 subgroup]] system, a fact first noted by [[User:CompactStar|CompactStar]], tempering out the commas [[10976/10935]] and [[3125/3087]], although the representation of [[11/2]] is more questionable. [[24edf]] (effectively 41edo) provides a correction for 11/2. It contains the [[macrodiatonic and microdiatonic scales|microdiatonic]] scale that corresponds to 12edo's [[5L 2s|diatonic scale]] with [[2/1]] compressed to [[3/2]]. | |||
==Harmonics== | |||
{{Harmonics in equal|12|3|2}} | |||
{{Harmonics in equal|12|3|2|start=12|collapsed=1}} | |||
==Intervals== | ==Intervals== | ||
| Line 9: | Line 16: | ||
! | comments | ! | comments | ||
|- | |- | ||
! colspan="2" | 0 | |||
| | '''exact [[1/1]]''' | | | '''exact [[1/1]]''' | ||
| | | | | | ||
|- | |- | ||
| | 1 | | | 1 | ||
| | 58. | | | 58.49625 | ||
| | 91/88, 88/85 | | | [[28/27]], 91/88, 88/85 | ||
| | | | | | ||
|- | |- | ||
| Line 24: | Line 31: | ||
|- | |- | ||
| | 3 | | | 3 | ||
| | 175. | | | 175.48875 | ||
| | [[21/19]] | | | [[10/9]], [[21/19]] | ||
| | | | | | ||
|- | |- | ||
| Line 34: | Line 41: | ||
|- | |- | ||
| | 5 | | | 5 | ||
| | 292. | | | 292.48125 | ||
| | 45/38 | | | 45/38 | ||
| | | | | | ||
| Line 44: | Line 51: | ||
|- | |- | ||
| | 7 | | | 7 | ||
| | 409. | | | 409.47375 | ||
| | [[19/15]] | | | [[19/15]], [[63/50]] | ||
| | | | | | ||
|- | |- | ||
| Line 54: | Line 61: | ||
|- | |- | ||
| | 9 | | | 9 | ||
| | 526. | | | 526.46625 | ||
| | [[19/14]] | | | [[19/14]] | ||
| | | | | | ||
| Line 65: | Line 72: | ||
| | 11 | | | 11 | ||
| | 643.4588 | | | 643.4588 | ||
| |13/9 | | | [[13/9]] | ||
| | | | | | ||
|- | |- | ||
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|} | |} | ||
{{todo|expand}} | |||
Latest revision as of 19:20, 1 August 2025
| ← 11edf | 12edf | 13edf → |
12EDF is the equal division of the just perfect fifth into 12 parts of 58.49625 cents each, corresponding to 20.5141 edo (similar to every second step of 41edo). It is an intersection of 3edf~5edo and 4edf~7edo relations, and could pass as both 20edo and 21edo, with both relations nearly breaking down by this point. It is related to the dodecacot temperament, which tempers out 3087/3125 and 10976/10935 in the 7-limit.
It is a strong 3/2.5/2.7/2 subgroup system, a fact first noted by CompactStar, tempering out the commas 10976/10935 and 3125/3087, although the representation of 11/2 is more questionable. 24edf (effectively 41edo) provides a correction for 11/2. It contains the microdiatonic scale that corresponds to 12edo's diatonic scale with 2/1 compressed to 3/2.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +28.4 | +28.4 | -1.7 | +21.5 | -1.7 | +24.0 | +26.8 | -1.7 | -8.6 | +1.9 | +26.8 |
| Relative (%) | +48.6 | +48.6 | -2.8 | +36.8 | -2.8 | +41.0 | +45.8 | -2.8 | -14.6 | +3.3 | +45.8 | |
| Steps (reduced) |
21 (9) |
33 (9) |
41 (5) |
48 (0) |
53 (5) |
58 (10) |
62 (2) |
65 (5) |
68 (8) |
71 (11) |
74 (2) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.2 | -6.1 | -8.6 | -3.3 | +8.7 | +26.8 | -8.3 | +19.9 | -6.1 | -28.2 | +11.9 |
| Relative (%) | +8.9 | -10.5 | -14.6 | -5.7 | +14.9 | +45.8 | -14.3 | +33.9 | -10.5 | -48.1 | +20.3 | |
| Steps (reduced) |
76 (4) |
78 (6) |
80 (8) |
82 (10) |
84 (0) |
86 (2) |
87 (3) |
89 (5) |
90 (6) |
91 (7) |
93 (9) | |
Intervals
| degree | cents value | corresponding JI intervals |
comments |
|---|---|---|---|
| 0 | exact 1/1 | ||
| 1 | 58.49625 | 28/27, 91/88, 88/85 | |
| 2 | 116.9925 | 15/14 | |
| 3 | 175.48875 | 10/9, 21/19 | |
| 4 | 233.9850 | 8/7 | |
| 5 | 292.48125 | 45/38 | |
| 6 | 350.9775 | 11/9, 27/22 | |
| 7 | 409.47375 | 19/15, 63/50 | |
| 8 | 467.9700 | 21/16 | |
| 9 | 526.46625 | 19/14 | |
| 10 | 584.9625 | 7/5 | |
| 11 | 643.4588 | 13/9 | |
| 12 | 701.9550 | exact 3/2 | just perfect fifth |
| 13 | 760.45125 | 273/176, 132/85 | |
| 14 | 818.9475 | 8/5 | |
| 15 | 877.44375 | 63/38 | |
| 16 | 935.94 | 12/7 | |
| 17 | 994.43625 | 135/76 | |
| 18 | 1052.9325 | 11/6, 81/44 | |
| 19 | 1111.42875 | 19/10 | |
| 20 | 1169.925 | 63/32 | |
| 21 | 1228.42125 | 57/28 | |
| 22 | 1286.9175 | 21/10 | |
| 23 | 1345.41375 | 13/6 | |
| 24 | 1403.91 | exact 9/4 | |