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== Theory ==
== Theory ==
9edf corresponds to 15.391524edo. It is closely related to [[Carlos Alpha]], and can be used as a temperament of the 3/2.5/4.7/8.11/8 subgroup. Carlos Alpha can be seen as 9edf with an independent dimension for 2.  
9edf corresponds to 15.385602edo. It is closely related to [[Carlos Alpha]] (though the Carlos Alpha generator is 0.03{{C}} narrower at 77.965{{C}}), and can be used as a temperament of the 3/2.5/4.8/7.11/8 subgroup. [[Valentine]] can be seen as 9edf with an independent dimension for 2.  


=== Harmonics ===
=== Harmonics ===
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|-
|-
| 9
| 9
| 701.9
| 702.0
| [[3/2]]
| [[3/2]]
| C
| C
Line 69: Line 69:


== Scales ==
== Scales ==
{{Idiosyncratic terms}}
{| class="wikitable sortable"
{| class="wikitable sortable"
|+
|+
! rowspan="2" |Name
! rowspan="2" | Name
! rowspan="2" |Step pattern within 1 period
! rowspan="2" | Step pattern within 1 period
! colspan="2" |Interval of repetition
! colspan="3" | Interval of repetition
! rowspan="2" |First described by
! rowspan="2" | First described by
|-
|-
!(in 9edf steps)
! (In 9edf steps)
!(in cents)
! (In cents)
! (Nearby JI)
|-
|-
|Molten [[slendro]]
| Molten [[slendro]]
|3
| 3
|3
| 3
|234.0
| 234.0
|''Budjarn Lambeth (2023)''
| [[8/7]]
| ''[[Budjarn Lambeth]] (2023)''
|-
|-
|Pylon
| Pylon
|6 3
| 6 3
|9
| 9
|702.0
| 702.0
|''Budjarn Lambeth (2023)''
| [[3/2]]
| ''Budjarn Lambeth (2023)''
|-
|-
|Quest
| Quest
|7 2
| 7 2
|9
| 9
|702.0
| 702.0
|''Budjarn Lambeth (2023)''
| [[3/2]]
| ''Budjarn Lambeth (2023)''
|-
|-
|Snowcone
| Snowcone
|3 6
| 3 6
|9
| 9
|702.0
| 702.0
|''Budjarn Lambeth (2023)''
| [[3/2]]
| ''Budjarn Lambeth (2023)''
|-
|-
|Molten [[pelog]]
| Molten [[pelog]]
|2 2 5 2 5
| 2 2 5 2 5
|16
| 16
|1247.9
| 1247.9
|''Budjarn Lambeth (2023)''
| [[33/16]]
| ''Budjarn Lambeth (2023)''
|-
|-
|Swan
| Swan
|4 5 4 5 4 5 4
| 4 5 4 5 4 5 4
|31
| 31
|2417.8
| 2417.8
|''Budjarn Lambeth (2023)''
| [[4/1]]
| ''Budjarn Lambeth (2023)''
|-
|-
|Livewire
| Livewire
|1 8 1 8 1 8 1 8 1 8 1
| 1 8 1 8 1 8 1 8 1 8 1
|46
| 46
|3587.8
| 3587.8
|''[[Budjarn Lambeth]] (2023)''
| [[8/1]]
| ''Budjarn Lambeth (2023)''
|-
|-
|Purgatory
| Purgatory
|8 1 8 1 8 1 8 1 8 1 8 1 8
| 8 1 8 1 8 1 8 1 8 1 8 1 8
|62
| 62
|4835.7
| 4835.7
|''Budjarn Lambeth (2023)''
| [[16/1]]
| ''Budjarn Lambeth (2023)''
|-
|-
|Cloudscape
| Cloudscape
|5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5
| 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5
|72
| 72
|5615.6
| 5615.6
|''Budjarn Lambeth (2023)''
| [[128/5]]
| ''Budjarn Lambeth (2023)''
|-
|-
|Corrugated
| Corrugated
|2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2
| 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2
|135
| 135
|10529.3
| 10529.3
|''Budjarn Lambeth (2023)''
| [[437/1]]
| ''Budjarn Lambeth (2023)''
|}
|}
None of these scales are approximated from another tuning unless specified otherwise. For exact [[cents]] values see [[User:BudjarnLambeth/Longer versions of scales#9edf]].
None of these scales are approximated from another tuning unless specified otherwise. For exact [[cents]] values see [[User:BudjarnLambeth/Longer versions of scales#9edf]].
Line 143: Line 156:
== Music ==
== Music ==
See [[Carlos Alpha #Music]]. Many are tuned to exactly 9edf.  
See [[Carlos Alpha #Music]]. Many are tuned to exactly 9edf.  
== See also ==
* [[Alpha, beta, and gamma family of equal divisions]]


[[Category:Listen]]
[[Category:Listen]]

Latest revision as of 19:21, 20 March 2026

← 8edf 9edf 10edf →
Prime factorization 32
Step size 77.995 ¢ 
Octave 15\9edf (1169.93 ¢) (→ 5\3edf)
Twelfth 24\9edf (1871.88 ¢) (→ 8\3edf)
Consistency limit 3
Distinct consistency limit 3

9 equal divisions of the perfect fifth (abbreviated 9edf or 9ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 9 equal parts of about 78 ¢ each. Each step represents a frequency ratio of (3/2)1/9, or the 9th root of 3/2.

Theory

9edf corresponds to 15.385602edo. It is closely related to Carlos Alpha (though the Carlos Alpha generator is 0.03 ¢ narrower at 77.965 ¢), and can be used as a temperament of the 3/2.5/4.8/7.11/8 subgroup. Valentine can be seen as 9edf with an independent dimension for 2.

Harmonics

Approximation of harmonics in 9edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -30.1 -30.1 +17.8 +21.5 +17.8 -15.0 -12.2 +17.8 -8.6 -17.6 -12.2
Relative (%) -38.6 -38.6 +22.9 +27.6 +22.9 -19.3 -15.7 +22.9 -11.0 -22.5 -15.7
Steps
(reduced) 		15
(6) 		24
(6) 		31
(4) 		36
(0) 		40
(4) 		43
(7) 		46
(1) 		49
(4) 		51
(6) 		53
(8) 		55
(1)
Approximation of harmonics in 9edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +5.2 +32.9 -8.6 +35.7 +8.7 -12.2 -27.8 -38.6 +32.9 +30.3 +31.4 +35.7
Relative (%) +6.7 +42.2 -11.0 +45.8 +11.2 -15.7 -35.7 -49.5 +42.2 +38.9 +40.2 +45.8
Steps
(reduced) 		57
(3) 		59
(5) 		60
(6) 		62
(8) 		63
(0) 		64
(1) 		65
(2) 		66
(3) 		68
(5) 		69
(6) 		70
(7) 		71
(8)

Intervals

# Cents Approximate ratios Neptunian notation
using 7\9edf
0 0.0 1/1 C
1 78.0 21/20, 22/21, 25/24 C#
2 156.0 11/10, 12/11 Db
3 234.0 8/7 D
4 312.0 6/5 D#, Eb
5 390.0 5/4 E
6 468.0 21/16 E#, Fb
7 546.0 11/8, 15/11 F
8 624.0 10/7, 36/25 F#, Cb
9 702.0 3/2 C

Scales

This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community.
Name Step pattern within 1 period Interval of repetition First described by
(In 9edf steps) (In cents) (Nearby JI)
Molten slendro 3 3 234.0 8/7 Budjarn Lambeth (2023)
Pylon 6 3 9 702.0 3/2 Budjarn Lambeth (2023)
Quest 7 2 9 702.0 3/2 Budjarn Lambeth (2023)
Snowcone 3 6 9 702.0 3/2 Budjarn Lambeth (2023)
Molten pelog 2 2 5 2 5 16 1247.9 33/16 Budjarn Lambeth (2023)
Swan 4 5 4 5 4 5 4 31 2417.8 4/1 Budjarn Lambeth (2023)
Livewire 1 8 1 8 1 8 1 8 1 8 1 46 3587.8 8/1 Budjarn Lambeth (2023)
Purgatory 8 1 8 1 8 1 8 1 8 1 8 1 8 62 4835.7 16/1 Budjarn Lambeth (2023)
Cloudscape 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 72 5615.6 128/5 Budjarn Lambeth (2023)
Corrugated 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 7 2 135 10529.3 437/1 Budjarn Lambeth (2023)

None of these scales are approximated from another tuning unless specified otherwise. For exact cents values see User:BudjarnLambeth/Longer versions of scales#9edf.

Music

See Carlos Alpha #Music. Many are tuned to exactly 9edf.

See also

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