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{{todo|add source|research|cleanup|inline=1}}
{{Infobox ET|9ed4/3}}
{{Infobox ET|9ed4/3}}
9ed4/3, also known as '''Noleta''', is a tuning system based on the division of the [[perfect fourth]] (4/3) into 9 equal parts, each 55.3383 [[cent]]s in size. The term seems to be coined by [[Ron Sword]]: [http://www.nonoctave.com/forum/messages/9197.html?n=12 Nonoctaves.com: Messages: 9197]
'''9ed4/3''', also known as '''Noleta''', is a tuning system based on the division of the [[perfect fourth]] (4/3) into 9 equal parts, each 55.3383 [[cent]]s in size; this corresponds to 21.6848[[edo]], or approximately to every three steps of [[65edo]]. The name ‘Noleta’ seems to be coined by [[Ron Sword]]<ref>[http://www.nonoctave.com/forum/messages/9197.html?n=12 Nonoctave.com: Messages: 9197]{{dead link}}</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100326.html Some Mysterious Scales] on [[Yahoo Groups tuning lists]]</ref><ref>[https://www.facebook.com/groups/2229924481/?multi_permalinks=10150273357164482 Ron Sword's reply on Facebook (archived group)]</ref>.


Regular temperaments that divide 4/3 into 9 equal parts include:
Regular temperaments that divide 4/3 into 9 equal parts include:
Line 12: Line 14:


== Intervals ==
== Intervals ==
{{Interval table|9ed4/3}}
9ed4/3 has good approximations of [[5/4]] and [[11/10]] (the latter from [[3ed4/3]]) and their fourth-complements [[16/15]] and [[40/33]]. Treated as a 4/3.5/4.11/8 temperament, it tempers out the commas [[4000/3993]] and [[5632/5625]], a structure that is identical to [[Escapade family#2.3.5.11 subgroup|escapade]]. [[32/31]] can additionally be added as a representation of a single step, tempering out [[961/960|S31]] and [[1024/1023|S32]].
 
All ratios in the below table are in the 88-integer-limit and in the 4/3.5/4.11/8.31/24 [[subgroup]].
 
{| class="wikitable sortable center-1 right-2"
! &#35;
! Cents
! class="unsortable" | Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
|-
| 1
| 55.338
| 33/32, 32/31, 31/30
|-
| 2
| 110.677
| 16/15, 33/31
|-
| 3
| 166.015
| 11/10
|-
| 4
| 221.353
| 25/22
|-
| 5
| 276.692
| 75/64, 88/75
|-
| 6
| 332.030
| 40/33
|-
| 7
| 387.368
| 5/4
|-
| 8
| 442.707
| 31/24, 40/31
|-
| 9
| 498.045
| '''exact [[4/3]]'''
|}


== Harmonics ==
== Harmonics ==
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}}
}}


-----
== References ==
[http://launch.groups.yahoo.com/group/tuning/message/100326 This topic is under discussion...]     
<references/>
[[Category:edonoi]][[Category:nonoctave]]
 
[[Category:Nonoctave]]

Latest revision as of 06:41, 17 August 2025

Todo: add source , research, cleanup
← 8ed4/3 9ed4/3 10ed4/3 →
Prime factorization 32
Step size 55.3383 ¢ 
Octave 22\9ed4/3 (1217.44 ¢)
Twelfth 34\9ed4/3 (1881.5 ¢)
Consistency limit 2
Distinct consistency limit 2

9ed4/3, also known as Noleta, is a tuning system based on the division of the perfect fourth (4/3) into 9 equal parts, each 55.3383 cents in size; this corresponds to 21.6848edo, or approximately to every three steps of 65edo. The name ‘Noleta’ seems to be coined by Ron Sword[1][2][3].

Regular temperaments that divide 4/3 into 9 equal parts include:

Intervals

9ed4/3 has good approximations of 5/4 and 11/10 (the latter from 3ed4/3) and their fourth-complements 16/15 and 40/33. Treated as a 4/3.5/4.11/8 temperament, it tempers out the commas 4000/3993 and 5632/5625, a structure that is identical to escapade. 32/31 can additionally be added as a representation of a single step, tempering out S31 and S32.

All ratios in the below table are in the 88-integer-limit and in the 4/3.5/4.11/8.31/24 subgroup.

# Cents Approximate ratios
0 0.0 1/1
1 55.338 33/32, 32/31, 31/30
2 110.677 16/15, 33/31
3 166.015 11/10
4 221.353 25/22
5 276.692 75/64, 88/75
6 332.030 40/33
7 387.368 5/4
8 442.707 31/24, 40/31
9 498.045 exact 4/3

Harmonics

Approximation of harmonics in 9ed4/3
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +17.4 -20.5 -20.5 -19.4 -3.0 +6.8 -3.0 +14.4 -2.0 -0.9 +14.4
Relative (%) +31.5 -37.0 -37.0 -35.1 -5.4 +12.3 -5.4 +26.1 -3.5 -1.7 +26.1
Steps
(reduced) 		22
(4) 		34
(7) 		43
(7) 		50
(5) 		56
(2) 		61
(7) 		65
(2) 		69
(6) 		72
(0) 		75
(3) 		78
(6)
Approximation of harmonics in 9ed4/3
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -13.5 +24.3 +15.5 +14.4 +20.2 -23.5 -6.4 +15.5 -13.6 +16.5 -5.1
Relative (%) -24.3 +43.8 +28.0 +26.1 +36.4 -42.4 -11.5 +28.0 -24.6 +29.8 -9.2
Steps
(reduced) 		80
(8) 		83
(2) 		85
(4) 		87
(6) 		89
(8) 		90
(0) 		92
(2) 		94
(4) 		95
(5) 		97
(7) 		98
(8)

References

  1. ↑ Nonoctave.com: Messages: 9197[dead link]
  2. ↑ Some Mysterious Scales on Yahoo Groups tuning lists
  3. ↑ Ron Sword's reply on Facebook (archived group)
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