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{{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}}
{{todo|improve synopsis|text=Make it longer, describe more about the tuning.}}
 
== 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.  
 
=== 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)}}


== Approximation of harmonics ==
{{Harmonics in equal
| steps = 9
| num = 3
| denom = 2
| columns = 18
}}
== Intervals ==
== Intervals ==
{|class="wikitable"
{|class="wikitable center-1 right-2"
|-
|-
!#
! #
!Cents
! Cents
!Approximate ratios
! Approximate ratios
![[1L 3s (fifth-equivalent)|Neptunian]] notation using 7\9edf
! [[1L 3s (fifth-equivalent)|Neptunian]] notation<br>using 7\9edf
|-
|-
|0
| 0
|0.0
| 0.0
|[[1/1]]
| [[1/1]]
|C
| C
|-
|-
|1
| 1
|78.0
| 78.0
|[[25/24]], [[21/20]]
| [[21/20]], [[22/21]], [[25/24]]
|C#
| C#
|-
|-
|2
| 2
|156.0
| 156.0
|[[12/11]], [[11/10]]
| [[11/10]], [[12/11]]
|Db
| Db
|-
|-
|3
| 3
|234.0
| 234.0
|[[8/7]]
| [[8/7]]
|D
| D
|-
|-
|4
| 4
|312.0
| 312.0
|[[6/5]]
| [[6/5]]
|D#, Eb
| D#, Eb
|-
|-
|5
| 5
|390.0
| 390.0
|[[5/4]]
| [[5/4]]
|E
| E
|-
|-
|6
| 6
|468.0
| 468.0
|[[21/16]]
| [[21/16]]
|E#, Fb
| E#, Fb
|-
|-
|7
| 7
|546.0
| 546.0
|[[15/11]], [[11/8]]
| [[11/8]], [[15/11]]
|F
| F
|-
|-
|8
| 8
|624.0
| 624.0
|[[10/7]], [[36/25]]
| [[10/7]], [[36/25]]
|F#, Cb
| F#, Cb
|-
|-
|9
| 9
|702.0
| 702.0
|[[3/2]]
| [[3/2]]
|C
| C
|}
|}


== Scales within 9edf ==
== Scales ==
Livewire (this is the original/default tuning)
Livewire (this is the original/default tuning)


Line 264: Line 263:


== Music ==
== Music ==
See [[Carlos Alpha #Music]]. Many are in tuned to exactly 9edf.


* ’HUH?’ by Mandrake (2022): https://m.youtube.com/watch?v=J5gntG5LrOk
[[Category:Edf]]
[[Category:Listen]]
[[Category:Listen]]

Revision as of 08:58, 26 January 2025

← 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 15.391524edo. 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.

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

Livewire (this is the original/default tuning)

  • 77.995
  • 701.955
  • 779.950
  • 1403.905
  • 1481.905
  • 2105.865
  • 2183.860
  • 2807.820
  • 2885.815
  • 3509.775
  • 3587.770


Corrugated (this is the original/default tuning)

  • 155.990
  • 701.955
  • 857.945
  • 1403.910
  • 1559.900
  • 2105.865
  • 2261.855
  • 2807.820
  • 2963.810
  • 3509.775
  • 3665.765
  • 4211.730
  • 4367.720
  • 4913.685
  • 5069.675
  • 5615.640
  • 5771.630
  • 6317.595
  • 6473.585
  • 7019.550
  • 7175.540
  • 7721.505
  • 7877.495
  • 8423.460


Snowcone (this is the original/default tuning)

  • 233.985
  • 701.955
  • 935.940
  • 1403.910
  • 1637.895
  • 2105.865
  • 2339.850
  • 2807.820
  • 3041.805
  • 3509.775
  • 3743.760
  • 4211.730
  • 4445.715
  • 4913.685
  • 5147.670
  • 5615.640
  • 5849.625
  • 6317.595
  • 6551.580
  • 7019.550
  • 7253.535
  • 7721.505
  • 7955.490
  • 8423.460


Swan (this is the original/default tuning)

  • 311.980
  • 701.955
  • 1013.935
  • 1403.910
  • 1715.890
  • 2105.865
  • 2417.845


Cloudscape (this is the original/default tuning)

  • 389.975
  • 701.955
  • 1091.930
  • 1403.910
  • 1793.885
  • 2105.865
  • 2495.840
  • 2807.820
  • 3197.795
  • 3509.775
  • 3899.750
  • 4211.730
  • 4601.705
  • 4913.685
  • 5303.660
  • 5615.640
  • 6005.615


Pylon (this is the original/default tuning)

  • 467.970
  • 701.955
  • 1169.925
  • 1403.910
  • 1871.880
  • 2105.865
  • 2573.835
  • 2807.820
  • 3275.790
  • 3509.775
  • 3977.745
  • 4211.730
  • 4679.700
  • 4913.685
  • 5381.655
  • 5615.640
  • 6083.610
  • 6317.595
  • 6785.565
  • 7019.550
  • 7487.520
  • 7721.505
  • 8189.475
  • 8423.460


Quest (this is the original/default tuning)

  • 545.965
  • 701.955
  • 1247.920
  • 1403.910
  • 1949.875
  • 2105.865
  • 2651.830
  • 2807.820
  • 3353.785
  • 3509.775
  • 4055.740
  • 4211.730
  • 4757.695
  • 4913.685
  • 5459.650
  • 5615.640
  • 6161.605
  • 6317.595
  • 6863.560
  • 7019.550
  • 7565.515
  • 7721.505
  • 8267.470
  • 8423.460


Purgatory (this is the original/default tuning)

  • 623.960
  • 701.955
  • 1325.915
  • 1403.910
  • 2027.870
  • 2105.865
  • 2729.825
  • 2807.820
  • 3431.780
  • 3509.775
  • 4133.735
  • 4211.730
  • 4835.690


Molten Pelog (this is the original/default tuning)

  • 155.990
  • 311.980
  • 701.955
  • 857.945
  • 1247.920


Molten Slendro (exact tuning from 3edf)

  • 233.985
  • 467.970
  • 701.955
  • 935.940
  • 1169.925

Music

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

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