9edf: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
18,215 edits
Contribution (talk | contribs)
2,880 edits
 
(22 intermediate revisions by 5 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}}
== Approximation of harmonics ==
 
{{Harmonics in equal
== Theory ==
| steps = 9
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.  
| num = 3
 
| denom = 2
=== Harmonics ===
| columns = 18
{{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
! 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
| 701.9
|[[3/2]]
| [[3/2]]
|C
| C
|}
|}
==Scale tree==
EDF scales can be approximated in [[EDO]]s by subdividing diatonic fifths. If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.
If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.


Generator range: 76.1905 cents (4\7/9 = 4\63) to 80 cents (3\5/9 = 1\15)
== Scales ==
{| class="wikitable center-all"
{| class="wikitable sortable"
! colspan="7" |Fifth
|+
!Cents
! rowspan="2" |Name
!Comments
! rowspan="2" |Step pattern within 1 period
! colspan="3" |Interval of repetition
! rowspan="2" |First described by
|-
|-
|4\7|| || || || || || ||76.1905||
!(in 9edf steps)
!(in cents)
!(nearby JI)
|-
|-
| || || || || || ||27\47||76.596||
|Molten [[slendro]]
|3
|3
|234.0
|[[8/7]]
|''[[Budjarn Lambeth]] (2023)''
|-
|-
| || || || || ||23\40|| ||76.{{Overline|6}}||
|Pylon
|6 3
|9
|702.0
|[[3/2]]
|''Budjarn Lambeth (2023)''
|-
|-
| || || || || || ||42\73||76.712||
|Quest
|7 2
|9
|702.0
|[[3/2]]
|''Budjarn Lambeth (2023)''
|-
|-
| || || || ||19\33|| || ||76.{{Overline|76}}||
|Snowcone
|3 6
|9
|702.0
|[[3/2]]
|''Budjarn Lambeth (2023)''
|-
|-
| || || || || || ||53\92||76.812||
|Molten [[pelog]]
|2 2 5 2 5
|16
|1247.9
|[[33/16]]
|''Budjarn Lambeth (2023)''
|-
|-
| || || || || ||34\59|| ||76.836||
|Swan
|4 5 4 5 4 5 4
|31
|2417.8
|[[4/1]]
|''Budjarn Lambeth (2023)''
|-
|-
| || || || || || ||49\85||76.893||
|Livewire
|1 8 1 8 1 8 1 8 1 8 1
|46
|3587.8
|[[8/1]]
|''Budjarn Lambeth (2023)''
|-
|-
| || || ||15\26|| || || ||76.932||
|Purgatory
|8 1 8 1 8 1 8 1 8 1 8 1 8
|62
|4835.7
|[[16/1]]
|''Budjarn Lambeth (2023)''
|-
|-
| || || || || || ||56\97||76.976||
|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)''
|-
|-
| || || || || ||41\71|| ||76.995||
|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
| || || || || || ||67\116||77.0115||
|135
|-
|10529.3
| || || || ||26\45|| || ||77.{{Overline|037}}||[[Flattone]] is in this region
|[[437/1]]
|-
|''Budjarn Lambeth (2023)''
| || || || || || ||63\109||77.064||
|}
|-
None of these scales are approximated from another tuning unless specified otherwise. For exact [[cents]] values see [[User:BudjarnLambeth/Longer versions of scales#9edf]].
| || || || || ||37\64|| ||77.08{{Overline|3}}||
|-
| || || || || || ||48\83||77.108||
|-
| || ||11\19|| || || || ||77.193||
|-
| || || || || || ||51\88||77.{{Overline|27}}||
|-
| || || || || ||40\69|| ||77.295||
|-
| || || || || || ||69\119||77.311||
|-
| || || || ||29\50|| || ||77.{{Overline|3}}||
|-
| || || || || || ||76\131||77.354||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||77.366||
|-
| || || || || || ||65\112||77.381||
|-
| || || ||18\31|| || || ||77.419||[[Meantone]] is in this region
|-
| || || || || || ||61\105||77.460||
|-
| || || || || ||43\74|| ||77.4775||
|-
| || || || || || ||68\117||77.493||
|-
| || || || ||25\43|| || ||77.519||
|-
| || || || || || ||57\98||77.551||
|-
| || || || || ||32\55|| ||77.{{Overline|57}}||
|-
| || || || || || ||39\67||77.612||
|-
| ||7\12|| || || || || ||77.{{Overline|7}}||
|-
| || || || || || ||38\65||77.949||
|-
| || || || || ||31\53|| ||77.987||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||78.014||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||78.049||
|-
| || || || || || ||65\111||78.{{Overline|078}}||
|-
| || || || || ||41\70|| ||78.095||
|-
| || || || || || ||58\99||78.1145||
|-
| || || ||17\29|| || || ||78.161||
|-
| || || || || || ||61\104||78.205||
|-
| || || || || ||44\75|| ||78.{{Overline|2}}||
|-
| || || || || || ||71\121||78.237||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||78.261||[[Neogothic]] is in this region
The generator closest to a just [[11/7]] for EDOs less than 1800
|-
| || || || || || ||64\109||78.2875||
|-
| || || || || ||37\63|| ||78.307||
|-
| || || || || || ||47\80||78.{{Overline|3}}||
|-
| || ||10\17|| || || || ||78.431||
|-
| || || || || || ||43\73||78.539||
|-
| || || || || ||33\56|| ||78.571||
|-
| || || || || || ||56\95||78.5965||
|-
| || || || ||23\39|| || ||78.6325||
|-
| || || || || || ||59\100||78.{{Overline|6}}||
|-
| || || || || ||36\61|| ||78.6885||
|-
| || || || || || ||49\83||78.715||
|-
| || || ||13\22|| || || ||78.{{Overline|78}}||[[Archy]] is in this region
|-
| || || || || || ||42\71||78.873||
|-
| || || || || ||29\49|| ||78.912||
|-
| || || || || || ||45\76||78.947||
|-
| || || || ||16\27|| || ||79.012||
|-
| || || || || || ||35\59||79.096||
|-
| || || || || ||19\32|| ||79.1{{Overline|6}}||
|-
| || || || || || ||22\37||79.{{Overline|279}}||
|-
|3\5|| || || || || || ||80.000||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.
 
Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
 
== Scales within 9edf ==
Tightrope
 
* 77.995
* 701.955
* 779.950
* 1403.905
* 1481.905
* 2105.865
* 2183.860
* 2807.820
* 2885.815
* 3509.775
* 3587.770
 
 
Corrugated
 
* 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
 
* 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
 
* 311.980
* 701.955
* 1013.935
* 1403.910
* 1715.890
* 2105.865
* 2417.845
 
 
Cloudscape
 
* 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
 
* 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
 
* 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
 
* 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
 
* 155.990
* 545.965
* 701.955
* 857.945
* 1247.920
 
 
Molten Slendro
 
* 233.985
* 467.970
* 701.955
* 935.940
* 1169.925


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


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

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.

Retrieved from "https://en.xen.wiki/index.php?title=9edf&oldid=203659"