9ed5/2: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{ED intro}}
{{ED intro}}
== Theory ==
9ed5/2 can function as a generator chain for the [[tetracot]] temperament.
=== Harmonics ===
{{Harmonics in equal|9|5|2|columns=11}}
{{Harmonics in equal|9|5|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 9ed5/2 (continued)}}


== Intervals ==
== Intervals ==
{| class="wikitable mw-collapsible"
{| class="wikitable center-1 right-2"
|+
! #
!Step
! Cents
!Interval (¢)
! Approximated ratios
!JI approximated
|-
!Simplified ratios
| 0
| 0
| 1/1
|-
|-
|1
| 1
|176.26
| 176
|21/19
| 10/9, 11/10, 21/19
|
|-
|-
|2
| 2
|352.51
| 353
|32/26
| 11/9, 16/13
|16/13
|-
|-
|3
| 3
|528.77
| 529
|19/14
| 15/11, 19/14, 27/20
|
|-
|-
|4
| 4
|705.03
| 705
|21/14
| 3/2
|3/2
|-
|-
|5
| 5
|881.29
| 881
|35/21, 43/26, 48/29
| 5/3
|5/3
|-
|-
|6
| 6
|1057.54
| 1058
|26/14, 35/19, 48/26
| 11/6, 24/13
|13/7, 24/13
|-
|-
|7
| 7
|1233.80
| 1234
|39/19, 43/21
| 45/22
|
|-
|-
|8
| 8
|1410.06
| 1410
|43/19
| 9/4
|
|-
|-
|9
| 9
|1586.31
| 1586
|5/2
| 5/2
|
|}
|}
The subgroup interpretation used is 5/2.14.19.21.26.29.32.35.39.43.48. Other interpretations are possible. Don't forget that fractions can multiply.
== Harmonics ==
{{Harmonics in equal
| steps = 9
| num = 5
| denom = 2
}}
{{Harmonics in equal
| steps = 9
| num = 5
| denom = 2
| start = 12
| collapsed = 1
}}
{{stub}}

Latest revision as of 14:34, 27 May 2026

← 8ed5/2 9ed5/2 10ed5/2 →
Prime factorization 32
Step size 176.257 ¢ 
Octave 7\9ed5/2 (1233.8 ¢)
(semiconvergent)
Twelfth 11\9ed5/2 (1938.83 ¢)
Consistency limit 6
Distinct consistency limit 5

9 equal divisions of 5/2 (abbreviated 9ed5/2) is a nonoctave tuning system that divides the interval of 5/2 into 9 equal parts of about 176 ¢ each. Each step represents a frequency ratio of (5/2)1/9, or the 9th root of 5/2.

Theory

9ed5/2 can function as a generator chain for the tetracot temperament.

Harmonics

Approximation of harmonics in 9ed5/2
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +33.8 +36.9 +67.6 +33.8 +70.7 -19.9 -74.9 +73.7 +67.6 +78.9 -71.8
Relative (%) +19.2 +20.9 +38.4 +19.2 +40.1 -11.3 -42.5 +41.8 +38.4 +44.7 -40.7
Steps
(reduced) 		7
(7) 		11
(2) 		14
(5) 		16
(7) 		18
(0) 		19
(1) 		20
(2) 		22
(4) 		23
(5) 		24
(6) 		24
(6)
Approximation of harmonics in 9ed5/2 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -34.1 +13.9 +70.7 -41.1 +30.2 -68.7 +13.9 -74.9 +16.9 -63.6 +35.7 -38.0
Relative (%) -19.3 +7.9 +40.1 -23.3 +17.2 -39.0 +7.9 -42.5 +9.6 -36.1 +20.3 -21.6
Steps
(reduced) 		25
(7) 		26
(8) 		27
(0) 		27
(0) 		28
(1) 		28
(1) 		29
(2) 		29
(2) 		30
(3) 		30
(3) 		31
(4) 		31
(4)

Intervals

# Cents Approximated ratios
0 0 1/1
1 176 10/9, 11/10, 21/19
2 353 11/9, 16/13
3 529 15/11, 19/14, 27/20
4 705 3/2
5 881 5/3
6 1058 11/6, 24/13
7 1234 45/22
8 1410 9/4
9 1586 5/2
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