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'''24EDT''' is the [[Edt|equal division of the third harmonic]] into 24 parts of 79.2481 [[cent|cents]] each, corresponding to 15.1423 [[edo]] (similar to every seventh step of [[106edo]]). | {{Infobox ET}} | ||
'''24EDT''' is the [[Edt|equal division of the third harmonic]] into 24 parts of 79.2481 [[cent|cents]] each, corresponding to 15.1423 [[edo]] (similar to every seventh step of [[106edo]]). It is related to the rank-three temperament which tempers out 325/324, 625/624, and 468512/468195 in the 13-limit, which is supported by [[15edo|15]], [[106edo|106]], [[121edo|121]], [[212edo|212]], and [[227edo|227]] EDOs. | |||
== Theory == | |||
{{Harmonics in equal|24|3|1|prec=2|columns=15}} | |||
== Interval table == | |||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! | degree | ! | degree | ||
! | cents value | ! | cents value | ||
!hekts | |||
! | corresponding <br>JI intervals | ! | corresponding <br>JI intervals | ||
! | comments | ! | comments | ||
|- | |- | ||
! colspan="3" | 0 | |||
| | '''exact [[1/1]]''' | | | '''exact [[1/1]]''' | ||
| | | | | | ||
| Line 15: | Line 20: | ||
| | 1 | | | 1 | ||
| | 79.2481 | | | 79.2481 | ||
|54.167 | |||
| | [[22/21]] | | | [[22/21]] | ||
| | | | | | ||
| Line 20: | Line 26: | ||
| | 2 | | | 2 | ||
| | 158.4963 | | | 158.4963 | ||
|108.333 | |||
| | | | | | ||
| | | | |pseudo-12/11 | ||
|- | |- | ||
| | 3 | | | 3 | ||
| | 237.7444 | | | 237.7444 | ||
| | 39/34 | |162.5 | ||
| | 39/34, 8/7 | |||
| | | | | | ||
|- | |- | ||
| | 4 | | | 4 | ||
| | 316.9925 | | | 316.9925 | ||
|216.667 | |||
| | [[6/5]] | | | [[6/5]] | ||
| | | | | | ||
| Line 35: | Line 44: | ||
| | 5 | | | 5 | ||
| | 396.2406 | | | 396.2406 | ||
|270.833 | |||
| | 44/35 | | | 44/35 | ||
| | | | |pseudo-[[5/4]] | ||
|- | |- | ||
| | 6 | | | 6 | ||
| | 475.4888 | | | 475.4888 | ||
| | | |325 | ||
| | | | |21/16 | ||
| |pseudo-[[4/3]] | |||
|- | |- | ||
| | 7 | | | 7 | ||
| | 554.7369 | | | 554.7369 | ||
| | | |379.167 | ||
| |11/8 | |||
| | | | | | ||
|- | |- | ||
| | 8 | | | 8 | ||
| | 633. | | | 633.985 | ||
| | 75/52 | |433.333 | ||
| | 75/52, 13/9 | |||
| | | | | | ||
|- | |- | ||
| | 9 | | | 9 | ||
| | 713.2331 | | | 713.2331 | ||
|487.5 | |||
| | | | | | ||
| | | | |pseudo-[[3/2]] | ||
|- | |- | ||
| | 10 | | | 10 | ||
| | 792.4813 | | | 792.4813 | ||
|541.667 | |||
| | [[30/19]], [[19/12]] | | | [[30/19]], [[19/12]] | ||
| | | | | | ||
| Line 65: | Line 80: | ||
| | 11 | | | 11 | ||
| | 871.7294 | | | 871.7294 | ||
|595.833 | |||
| | | | | | ||
| | | | |pseudo-[[5/3]] | ||
|- | |- | ||
| | 12 | | | 12 | ||
| | 950.9775 | | | 950.9775 | ||
|650 | |||
| | 45/26, [[26/15]] | | | 45/26, [[26/15]] | ||
| | | | | | ||
| Line 75: | Line 92: | ||
| | 13 | | | 13 | ||
| | 1030.2256 | | | 1030.2256 | ||
|704.167 | |||
| | | | | | ||
| | | | |pseudo-[[9/5]] | ||
|- | |- | ||
| | 14 | | | 14 | ||
| | 1109.4738 | | | 1109.4738 | ||
|758.333 | |||
| | 36/19, [[19/10]] | | | 36/19, [[19/10]] | ||
| | | | | | ||
| Line 85: | Line 104: | ||
| | 15 | | | 15 | ||
| | 1188.7219 | | | 1188.7219 | ||
|812.5 | |||
| | | | | | ||
| | | | |pseudooctave | ||
|- | |- | ||
| | 16 | | | 16 | ||
| | 1267. | | | 1267.97 | ||
| | [[26/25|52/25]] | |866.667 | ||
| | [[26/25|52/25]], 27/13 | |||
| | | | | | ||
|- | |- | ||
| | 17 | | | 17 | ||
| | 1347.2181 | | | 1347.2181 | ||
| | | |920.833 | ||
| |24/11 | |||
| | | | | | ||
|- | |- | ||
| | 18 | | | 18 | ||
| | 1426.4663 | | | 1426.4663 | ||
| | | |975 | ||
| | | | |16/7 | ||
| |pseudo-9/4 | |||
|- | |- | ||
| | 19 | | | 19 | ||
| | 1505.7144 | | | 1505.7144 | ||
|1029.167 | |||
| | 105/44 | | | 105/44 | ||
| | | | |pseudo-12/5 (6/5 plus pseudooctave) | ||
|- | |- | ||
| | 20 | | | 20 | ||
| | 1584.9625 | | | 1584.9625 | ||
|1083.333 | |||
| | [[5/2]] | | | [[5/2]] | ||
| | | | | | ||
| Line 115: | Line 140: | ||
| | 21 | | | 21 | ||
| | 1664.2106 | | | 1664.2106 | ||
| | [[17/13|34/13]] | |1137.5 | ||
| | | | | [[17/13|34/13]], 21/8 | ||
| | φ<sup>2</sup> | |||
|- | |- | ||
| | 22 | | | 22 | ||
| | 1743.4588 | | | 1743.4588 | ||
|1191.667 | |||
| | | | | | ||
| | | | |pseudo-11/4 (11/8 plus pseudooctave) | ||
|- | |- | ||
| | 23 | | | 23 | ||
| | 1822.7069 | | | 1822.7069 | ||
|1245.833. | |||
| | 63/22 | | | 63/22 | ||
| | | | | | ||
|- | |- | ||
| | 24 | | | 24 | ||
| | 1901. | | | 1901.955 | ||
|1300 | |||
| | '''exact [[3/1]]''' | | | '''exact [[3/1]]''' | ||
| | [[3/2|just perfect fifth]] plus an octave | | | [[3/2|just perfect fifth]] plus an octave | ||
|} | |} | ||
[[ | ==Related regular temperaments== | ||
===11-limit 15&106&212=== | |||
Commas: 15625/15552, 585640/583443 | |||
POTE generators: ~7/4 = 968.8778, ~22/21 = 79.2597 | |||
Mapping: [<1 0 1 0 -1|, <0 24 20 0 25|, <0 0 0 1 1|] | |||
EDOs: 15, 106, 121, 212, 227 | |||
===13-limit 15&106&212=== | |||
Commas: 325/324, 625/624, 468512/468195 | |||
POTE generators: ~7/4 = 968.8187, ~22/21 = 79.2727 | |||
Mapping: [<1 0 1 0 -1 0|, <0 24 20 0 25 56|, <0 0 0 1 1 0|] | |||
EDOs: 15, 106, 121, 212, 227 | |||
{{todo|expand}} | |||