9edf: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
CompactStar (talk | contribs)
autopatrolled
5,084 edits
No edit summary
Overthink (talk | contribs)
autopatrolled
2,697 edits
→Theory: mention difference with Carlos Alpha
 
(38 intermediate revisions by 8 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
9edf is the equal division of the just perfect fifth into 9 parts of 78 cents each, corresponding to 15.391524edo. It is nearly identical to [[Carlos Alpha]].  
{{ED intro}}
 
== Theory ==
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 in equal|9|3|2|columns=11}}
{{Harmonics in equal|9|3|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 9edf (continued)}}


== Intervals ==
== Intervals ==
{|class="wikitable"
{|class="wikitable center-1 right-2"
|-
! #
! Cents
! Approximate ratios
! [[1L 3s (fifth-equivalent)|Neptunian]] notation<br>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
!Cents
| 468.0
!Approximate ratios
| [[21/16]]
![[1L 3s (fifth-equivalent)|Neptunian]] notation
| E#, Fb
|-
|-
|0
| 7
|0.0
| 546.0
|[[1/1]]
| [[11/8]], [[15/11]]
|C
| F
|-
|-
|1
| 8
|78.0
| 624.0
|[[25/24]], [[21/20]]
| [[10/7]], [[36/25]]
|C#
| F#, Cb
|-
|-
|2
| 9
|156.0
| 702.0
|[[12/11]], [[11/10]]
| [[3/2]]
|Db
| C
|}
 
== Scales ==
{{Idiosyncratic terms}}
 
{| class="wikitable sortable"
|+
! rowspan="2" | Name
! rowspan="2" | Step pattern within 1 period
! colspan="3" | Interval of repetition
! rowspan="2" | First described by
|-
|-
|3
! (In 9edf steps)
|234.0
! (In cents)
|[[8/7]]
! (Nearby JI)
|D
|-
|-
|4
| Molten [[slendro]]
|312.0
| 3
|[[6/5]]
| 3
|D#, Eb
| 234.0
| [[8/7]]
| ''[[Budjarn Lambeth]] (2023)''
|-
|-
|5
| Pylon
|390.0
| 6 3
|[[5/4]]
| 9
|E
| 702.0
| [[3/2]]
| ''Budjarn Lambeth (2023)''
|-
|-
|6
| Quest
|468.0
| 7 2
|[[21/16]]
| 9
|E#, Fb
| 702.0
| [[3/2]]
| ''Budjarn Lambeth (2023)''
|-
|-
|7
| Snowcone
|546.0
| 3 6
|[[15/11]], [[11/8]]
| 9
|F
| 702.0
| [[3/2]]
| ''Budjarn Lambeth (2023)''
|-
|-
|8
| Molten [[pelog]]
|624.0
| 2 2 5 2 5
|[[10/7]], [[36/25]]
| 16
|F#, Cb
| 1247.9
| [[33/16]]
| ''Budjarn Lambeth (2023)''
|-
|-
|9
| Swan
|702.0
| 4 5 4 5 4 5 4
|[[3/2]]
| 31
|C
| 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 ==
* [[Alpha, beta, and gamma family of equal divisions]]


[[Category:Edf]]
[[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

Retrieved from "https://en.xen.wiki/index.php?title=9edf&oldid=226530"