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{{Infobox ET}}
{{Infobox ET}}
'''[[Ed7|Division of the 7th harmonic]] into 23 equal parts''' (23ED7) is related to the [[13edt|Bohlen-Pierce scale]], but with the 7/1 rather than the 3/1 being just. The step size is about 146.4707 cents, corresponding to 8.1928 [[EDO]]. It is almost identical to [[POTE generator]] for 7-limit [[bohpier|bohpier temperament]].
{{ED intro}}
23ED7 is related to the [[13edt|Bohlen–Pierce scale]], but with the 7/1 rather than the 3/1 being just. The step size is about 146.4707{{c}}, corresponding to 8.1928 [[EDO]]. It is almost identical to [[POTE generator]] for 7-limit [[bohpier|bohpier temperament]].


== Intervals ==
{| class="wikitable"
{| class="wikitable"
|-
|-
! | degree
! Degree
! | cents value
! Cents
! | corresponding <br>JI intervals
! Corresponding<br />JI intervals
! | comments
! Comments
|-
|-
| | 0
| 0
| | 0.0000
| 0.0000
| | '''exact [[1/1]]'''
| '''exact [[1/1]]'''
| |  
|  
|-
|-
| | 1
| 1
| | 146.4707
| 146.4707
| | [[49/45]]
| [[49/45]]
| |  
|  
|-
|-
| | 2
| 2
| | 292.9414
| 292.9414
| | [[13/11]]
| [[13/11]]
| |  
|  
|-
|-
| | 3
| 3
| | 439.4121
| 439.4121
| | [[9/7]]
| [[9/7]]
| |  
|  
|-
|-
| | 4
| 4
| | 585.8828
| 585.8828
| | [[7/5]]
| [[7/5]]
| |  
|  
|-
|-
| | 5
| 5
| | 732.3535
| 732.3535
| | [[49/32]]
| [[49/32]]
| |  
|  
|-
|-
| | 6
| 6
| | 878.8241
| 878.8241
| | [[5/3]]
| [[5/3]]
| |  
|  
|-
|-
| | 7
| 7
| | 1025.2948
| 1025.2948
| | [[9/5]]
| [[9/5]]
| |  
|  
|-
|-
| | 8
| 8
| | 1171.7655
| 1171.7655
| | 63/32
| 63/32
| |  
|  
|-
|-
| | 9
| 9
| | 1318.2362
| 1318.2362
| | [[15/7]]
| [[15/7]]
| |  
|  
|-
|-
| | 10
| 10
| | 1464.7069
| 1464.7069
| | [[7/3]]
| [[7/3]]
| |  
|  
|-
|-
| | 11
| 11
| | 1611.1776
| 1611.1776
| | [[14/11|28/11]]
| [[14/11|28/11]]
| |  
|  
|-
|-
| | 12
| 12
| | 1757.6483
| 1757.6483
| | [[11/4]]
| [[11/4]]
| |  
|  
|-
|-
| | 13
| 13
| | 1904.1190
| 1904.1190
| | [[3/1]]
| [[3/1]]
| |  
|  
|-
|-
| | 14
| 14
| | 2050.5897
| 2050.5897
| | 49/15
| 49/15
| |  
|  
|-
|-
| | 15
| 15
| | 2197.0604
| 2197.0604
| | 32/9
| 32/9
| |  
|  
|-
|-
| | 16
| 16
| | 2343.5311
| 2343.5311
| | 35/9
| 35/9
| |  
|  
|-
|-
| | 17
| 17
| | 2490.0018
| 2490.0018
| | [[21/20|21/5]]
| [[21/20|21/5]]
| |  
|  
|-
|-
| | 18
| 18
| | 2636.4724
| 2636.4724
| | [[8/7|32/7]]
| [[8/7|32/7]]
| |  
|  
|-
|-
| | 19
| 19
| | 2782.9431
| 2782.9431
| | [[5/1]]
| [[5/1]]
| |  
|  
|-
|-
| | 20
| 20
| | 2929.4138
| 2929.4138
| | [[49/36|49/9]]
| [[49/36|49/9]]
| |  
|  
|-
|-
| | 21
| 21
| | 3075.8845
| 3075.8845
| | 77/13
| 77/13
| |  
|  
|-
|-
| | 22
| 22
| | 3222.3552
| 3222.3552
| | 45/7
| 45/7
| |  
|  
|-
|-
| | 23
| 23
| | 3368.8259
| 3368.8259
| | '''exact [[7/1]]'''
| '''Exact [[7/1]]'''
| | [[7/4|harmonic seventh]] plus two octaves
| [[7/4|harmonic seventh]] plus two octaves
|}
|}
== Harmonics ==
{{Harmonics in equal|23|7|1|intervals=prime}}
{{Harmonics in equal|23|7|1|intervals=prime|collapsed=1|start=12}}


== See also ==
== See also ==
Line 134: Line 140:
* [[19ed5|19ED5]]: relative ED5
* [[19ed5|19ED5]]: relative ED5


[[Category:Ed7]]
 
[[Category:Edonoi]]
{{stub}}
[[Category:Bohlen-Pierce]]
[[Category:Bohlen–Pierce]]
[[Category:Bohpier]]
[[Category:Bohpier]]

Latest revision as of 19:20, 1 August 2025

← 22ed7 23ed7 24ed7 →
Prime factorization 23 (prime)
Step size 146.471 ¢ 
Octave 8\23ed7 (1171.77 ¢)
Twelfth 13\23ed7 (1904.12 ¢)
(convergent)
Consistency limit 7
Distinct consistency limit 4

23 equal divisions of the 7th harmonic (abbreviated 23ed7) is a nonoctave tuning system that divides the interval of 7/1 into 23 equal parts of about 146 ¢ each. Each step represents a frequency ratio of 71/23, or the 23rd root of 7. 23ED7 is related to the Bohlen–Pierce scale, but with the 7/1 rather than the 3/1 being just. The step size is about 146.4707 ¢, corresponding to 8.1928 EDO. It is almost identical to POTE generator for 7-limit bohpier temperament.

Intervals

Degree Cents Corresponding
JI intervals 		Comments
0 0.0000 exact 1/1
1 146.4707 49/45
2 292.9414 13/11
3 439.4121 9/7
4 585.8828 7/5
5 732.3535 49/32
6 878.8241 5/3
7 1025.2948 9/5
8 1171.7655 63/32
9 1318.2362 15/7
10 1464.7069 7/3
11 1611.1776 28/11
12 1757.6483 11/4
13 1904.1190 3/1
14 2050.5897 49/15
15 2197.0604 32/9
16 2343.5311 35/9
17 2490.0018 21/5
18 2636.4724 32/7
19 2782.9431 5/1
20 2929.4138 49/9
21 3075.8845 77/13
22 3222.3552 45/7
23 3368.8259 Exact 7/1 harmonic seventh plus two octaves

Harmonics

Approximation of prime harmonics in 23ed7
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -28.2 +2.2 -3.4 +0.0 -50.1 -46.4 -71.4 +29.0 -8.9 +29.3 +60.3
Relative (%) -19.3 +1.5 -2.3 +0.0 -34.2 -31.7 -48.8 +19.8 -6.0 +20.0 +41.1
Steps
(reduced) 		8
(8) 		13
(13) 		19
(19) 		23
(0) 		28
(5) 		30
(7) 		33
(10) 		35
(12) 		37
(14) 		40
(17) 		41
(18)
Approximation of prime harmonics in 23ed7
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) +46.9 +15.6 -66.8 +72.1 +10.6 -28.6 +60.2 +44.2 -56.2 +42.2 +51.9
Relative (%) +32.0 +10.7 -45.6 +49.3 +7.2 -19.5 +41.1 +30.2 -38.3 +28.8 +35.5
Steps
(reduced) 		43
(20) 		44
(21) 		44
(21) 		46
(0) 		47
(1) 		48
(2) 		49
(3) 		50
(4) 		50
(4) 		51
(5) 		52
(6)

See also

  • 13ED3: relative ED3 (Bohlen-Pierce scale)
  • 19ED5: relative ED5


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