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'''[[Ed7|Division of the 7th harmonic]] into 24 equal parts''' (24ed7) is related to the [[Hemifamity temperaments|quanic temperament]], which tempers out 352/351, 540/539, 729/728, and 1331/1323 in the 13-limit; 352/351, 442/441, 540/539, 561/560, and 715/714 in the 17-limit; 253/252, 345/343, 352/351, 391/390, 442/441, and 460/459 in the 2.3.5.7.11.13.17.23 subgroup. The step size is about 140.3677 cents, corresponding to 8.5490 [[edo]].
{{Infobox ET}}
'''[[Ed7|Division of the 7th harmonic]] into 24 equal parts''' (24ed7) is related to the [[Hemifamity temperaments|quanic temperament]], which tempers out 352/351, 540/539, 729/728, and 1331/1323 in the 13-limit; 352/351, 442/441, 540/539, 561/560, and 715/714 in the 17-limit. The step size is about 140.3677 cents, corresponding to 8.5490 [[edo]].


{| class="wikitable"
== Intervals ==
{| class="wikitable mw-collapsible"
|+ Intervals of 24ed7
|-
|-
! | degree
! | degree
Line 25: Line 28:
| | 3
| | 3
| | 421.1032
| | 421.1032
| | [[14/11]], 23/18, [[32/25]]
| | [[14/11]], 65/51, 51/40, 23/18, [[32/25]]
| |  
| |  
|-
|-
Line 70: Line 73:
| | 12
| | 12
| | 1684.4130
| | 1684.4130
| |  
| | 119/45, 45/17
| |  
| |  
|-
|-
Line 85: Line 88:
| | 15
| | 15
| | 2105.5162
| | 2105.5162
| | [[27/16|27/8]]
| | 91/27, [[27/16|27/8]]
| |  
| |  
|-
|-
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| | 17
| | 17
| | 2386.2517
| | 2386.2517
| | 119/30, 250/63
| | 119/30, 143/36
| |  
| |  
|-
|-
| | 18
| | 18
| | 2526.6194
| | 2526.6194
| | 56/13
| | [[14/13|56/13]]
| |  
| |  
|-
|-
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|}
|}


[[Category:Ed7]]
== Harmonics ==
[[Category:Edonoi]]
{{Harmonics in equal|24|7|1}}
{{Harmonics in equal|24|7|1|collapsed=1|start=12}}
 
== Music ==
; [[Ambient Esoterica]]
* [https://www.youtube.com/watch?v=qVB1UFLMO4U ''SQRT(7):1 | Irrational Octave Microtones''] (2023)

Latest revision as of 23:51, 22 December 2024

← 23ed7 24ed7 25ed7 →
Prime factorization 23 × 3
Step size 140.368 ¢ 
Octave 9\24ed7 (1263.31 ¢) (→ 3\8ed7)
Twelfth 14\24ed7 (1965.15 ¢) (→ 7\12ed7)
Consistency limit 3
Distinct consistency limit 3
Special properties

Division of the 7th harmonic into 24 equal parts (24ed7) is related to the quanic temperament, which tempers out 352/351, 540/539, 729/728, and 1331/1323 in the 13-limit; 352/351, 442/441, 540/539, 561/560, and 715/714 in the 17-limit. The step size is about 140.3677 cents, corresponding to 8.5490 edo.

Intervals

Intervals of 24ed7
degree cents value corresponding
JI intervals 		comments
0 0.0000 exact 1/1
1 140.3677 13/12, 25/23
2 280.7355 20/17
3 421.1032 14/11, 65/51, 51/40, 23/18, 32/25
4 561.4710 18/13
5 701.8387 3/2
6 842.2065 13/8, 80/49
7 982.5742 30/17
8 1122.9420 21/11
9 1263.3097 56/27, 27/13
10 1403.6775 9/4
11 1544.0452 56/23, 39/16, 22/9, 120/49
12 1684.4130 119/45, 45/17
13 1824.7807 63/22, 112/39, 23/8
14 1965.1484 28/9
15 2105.5162 91/27, 27/8
16 2245.8839 11/3
17 2386.2517 119/30, 143/36
18 2526.6194 56/13
19 2666.9872 14/3
20 2807.3549 91/18
21 2947.7227 11/2
22 3088.0904 119/20
23 3228.4582 84/13
24 3368.8259 exact 7/1 harmonic seventh plus two octaves

Harmonics

Approximation of harmonics in 24ed7
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +63.3 +63.2 -13.7 +21.0 -13.9 +0.0 +49.6 -14.0 -56.0 +59.7 +49.4
Relative (%) +45.1 +45.0 -9.8 +15.0 -9.9 +0.0 +35.3 -10.0 -39.9 +42.5 +35.2
Steps
(reduced) 		9
(9) 		14
(14) 		17
(17) 		20
(20) 		22
(22) 		24
(0) 		26
(2) 		27
(3) 		28
(4) 		30
(6) 		31
(7)
Approximation of harmonics in 24ed7
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +51.2 +63.3 -56.1 -27.5 +7.9 +49.3 -44.3 +7.3 +63.2 -17.3 +46.1
Relative (%) +36.5 +45.1 -40.0 -19.6 +5.6 +35.1 -31.5 +5.2 +45.0 -12.4 +32.8
Steps
(reduced) 		32
(8) 		33
(9) 		33
(9) 		34
(10) 		35
(11) 		36
(12) 		36
(12) 		37
(13) 		38
(14) 		38
(14) 		39
(15)

Music

Ambient Esoterica
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