24edt: Difference between revisions

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{{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.
'''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
|-
|-
| | 0
! colspan="3" | 0
| | 0.0000
| | '''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
| | 633.9850
| | 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
| | 1267.9700
| | 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
| | 1901.9550
| | 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 temperament==
==Related regular temperaments==
===11-limit 15&amp;106&amp;212===
===11-limit 15&amp;106&amp;212===
Commas: 15625/15552, 585640/583443
Commas: 15625/15552, 585640/583443
Line 140: Line 169:
POTE generators: ~7/4 = 968.8778, ~22/21 = 79.2597
POTE generators: ~7/4 = 968.8778, ~22/21 = 79.2597


Map: [&lt;1 0 1 0 -1|, &lt;0 24 20 0 25|, &lt;0 0 0 1 1|]
Mapping: [&lt;1 0 1 0 -1|, &lt;0 24 20 0 25|, &lt;0 0 0 1 1|]


EDOs: 15, 106, 121, 212, 227
EDOs: 15, 106, 121, 212, 227
Line 149: Line 178:
POTE generators: ~7/4 = 968.8187, ~22/21 = 79.2727
POTE generators: ~7/4 = 968.8187, ~22/21 = 79.2727


Map: [&lt;1 0 1 0 -1 0|, &lt;0 24 20 0 25 56|, &lt;0 0 0 1 1 0|]
Mapping: [&lt;1 0 1 0 -1 0|, &lt;0 24 20 0 25 56|, &lt;0 0 0 1 1 0|]


EDOs: 15, 106, 121, 212, 227
EDOs: 15, 106, 121, 212, 227


[[Category:Edt]]
{{todo|expand}}
[[Category:Edonoi]]

Latest revision as of 19:21, 1 August 2025

← 23edt 24edt 25edt →
Prime factorization 23 × 3
Step size 79.2481 ¢ 
Octave 15\24edt (1188.72 ¢) (→ 5\8edt)
Consistency limit 6
Distinct consistency limit 6
Special properties

24EDT is the equal division of the third harmonic into 24 parts of 79.2481 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 15, 106, 121, 212, and 227 EDOs.

Theory

Approximation of harmonics in 24edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Error Absolute (¢) -11.28 +0.00 -22.56 -12.63 -11.28 +38.84 -33.83 +0.00 -23.91 -30.42 -22.56 -2.63 +27.57 -12.63 +34.14
Relative (%) -14.2 +0.0 -28.5 -15.9 -14.2 +49.0 -42.7 +0.0 -30.2 -38.4 -28.5 -3.3 +34.8 -15.9 +43.1
Steps
(reduced) 		15
(15) 		24
(0) 		30
(6) 		35
(11) 		39
(15) 		43
(19) 		45
(21) 		48
(0) 		50
(2) 		52
(4) 		54
(6) 		56
(8) 		58
(10) 		59
(11) 		61
(13)

Interval table

degree cents value hekts corresponding
JI intervals 		comments
0 exact 1/1
1 79.2481 54.167 22/21
2 158.4963 108.333 pseudo-12/11
3 237.7444 162.5 39/34, 8/7
4 316.9925 216.667 6/5
5 396.2406 270.833 44/35 pseudo-5/4
6 475.4888 325 21/16 pseudo-4/3
7 554.7369 379.167 11/8
8 633.985 433.333 75/52, 13/9
9 713.2331 487.5 pseudo-3/2
10 792.4813 541.667 30/19, 19/12
11 871.7294 595.833 pseudo-5/3
12 950.9775 650 45/26, 26/15
13 1030.2256 704.167 pseudo-9/5
14 1109.4738 758.333 36/19, 19/10
15 1188.7219 812.5 pseudooctave
16 1267.97 866.667 52/25, 27/13
17 1347.2181 920.833 24/11
18 1426.4663 975 16/7 pseudo-9/4
19 1505.7144 1029.167 105/44 pseudo-12/5 (6/5 plus pseudooctave)
20 1584.9625 1083.333 5/2
21 1664.2106 1137.5 34/13, 21/8 φ2
22 1743.4588 1191.667 pseudo-11/4 (11/8 plus pseudooctave)
23 1822.7069 1245.833. 63/22
24 1901.955 1300 exact 3/1 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

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