9edf: Difference between revisions

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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]], and can be used as a temperament of the [[Half-prime subgroup|3/2.5/4.7/8.11/8 subgroup]]. Carlos Alpha can be seen as 9edf with an independent dimension for 2.  


=== Harmonics ===
=== Harmonics ===
Line 69: Line 69:


== Scales ==
== Scales ==
Livewire (this is the original/default tuning)
{| class="wikitable sortable"
 
|+
* 77.995
! rowspan="2" |Name
* 701.955
! rowspan="2" |Step pattern within 1 period
* 779.950
! colspan="3" |Interval of repetition
* 1403.905
! rowspan="2" |First described by
* 1481.905
|-
* 2105.865
!(in 9edf steps)
* 2183.860
!(in cents)
* 2807.820
!(nearby JI)
* 2885.815
|-
* 3509.775
|Molten [[slendro]]
* 3587.770
|3
 
|3
 
|234.0
Corrugated (this is the original/default tuning)
|[[8/7]]
 
|''[[Budjarn Lambeth]] (2023)''
* 155.990
|-
* 701.955
|Pylon
* 857.945
|6 3
* 1403.910
|9
* 1559.900
|702.0
* 2105.865
|[[3/2]]
* 2261.855
|''Budjarn Lambeth (2023)''
* 2807.820
|-
* 2963.810
|Quest
* 3509.775
|7 2
* 3665.765
|9
* 4211.730
|702.0
* 4367.720
|[[3/2]]
* 4913.685
|''Budjarn Lambeth (2023)''
* 5069.675
|-
* 5615.640
|Snowcone
* 5771.630
|3 6
* 6317.595
|9
* 6473.585
|702.0
* 7019.550
|[[3/2]]
* 7175.540
|''Budjarn Lambeth (2023)''
* 7721.505
|-
* 7877.495
|Molten [[pelog]]
* 8423.460
|2 2 5 2 5
 
|16
 
|1247.9
Snowcone (this is the original/default tuning)
|[[33/16]]
 
|''Budjarn Lambeth (2023)''
* 233.985
|-
* 701.955
|Swan
* 935.940
|4 5 4 5 4 5 4
* 1403.910
|31
* 1637.895
|2417.8
* 2105.865
|[[4/1]]
* 2339.850
|''Budjarn Lambeth (2023)''
* 2807.820
|-
* 3041.805
|Livewire
* 3509.775
|1 8 1 8 1 8 1 8 1 8 1
* 3743.760
|46
* 4211.730
|3587.8
* 4445.715
|[[8/1]]
* 4913.685
|''Budjarn Lambeth (2023)''
* 5147.670
|-
* 5615.640
|Purgatory
* 5849.625
|8 1 8 1 8 1 8 1 8 1 8 1 8
* 6317.595
|62
* 6551.580
|4835.7
* 7019.550
|[[16/1]]
* 7253.535
|''Budjarn Lambeth (2023)''
* 7721.505
|-
* 7955.490
|Cloudscape
* 8423.460
|5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5
 
|72
 
|5615.6
Swan (this is the original/default tuning)
|[[128/5]]
 
|''Budjarn Lambeth (2023)''
* 311.980
|-
* 701.955
|Corrugated
* 1013.935
|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
* 1403.910
|135
* 1715.890
|10529.3
* 2105.865
|[[437/1]]
* 2417.845
|''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]].
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 ==
== Music ==

Latest revision as of 20:28, 25 June 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 15.385602edo. 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.

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
9 701.9 3/2 C

Scales

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.

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