20edf: Difference between revisions

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Carlos Gamma as we know today is non-octave, even tho it was originally intended to include the octaves
 
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{{Infobox ET}}
{{Infobox ET}}
==Intervals==
{{ED intro}}
 
== Theory ==
20edf corresponds to 34.1902edo. It is closely related to [[Carlos Gamma]] and the [[gammic]] temperament, which adds an independent dimension for the [[2/1|octave]] (although strictly speaking, the "canonical" optimized Carlos Gamma tuning is not exactly 20edf, with its fifth stretched by the microscopic amount of 0.016{{c}}). It very accurately represents the intervals [[5/4]], with 11 steps, and [[17/16]], with 3 steps.
 
=== Harmonics ===
{{Harmonics in equal|20|3|2|columns=11}}
{{Harmonics in equal|20|3|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 20edf (continued)}}
 
== Intervals ==
The first steps up to two just perfect fifths should give a feeling of the granularity of this system…
The first steps up to two just perfect fifths should give a feeling of the granularity of this system…
{| class="wikitable"
{| class="wikitable mw-collapsible"
|+ Intervals of 20edf
|-
!Degrees
!Degrees
!
!3/2.5/4.17/16 interpretation
!Cents
|-
|-
|1
|1
|51/50
|35.1
|35.1
|-
|-
| |2
|2
| |70.2
|25/24
|70.2
|-
|-
| |3
|3
| |105.29
|17/16
|105.29
|-
|-
| |4
|4
| |140.39
|625/576, 867/800
|140.39
|-
|-
| |5
|5
| |175.49
|320/289, 425/384
|175.49
|-
|-
| |6
|6
| |210.59
|96/85
|210.59
|-
|-
| |7
|7
| |245.68
|144/125
|245.68
|-
|-
| |8
|8
| |280.78
|20/17
|280.78
|-
|-
| |9
|9
| |315.88
|6/5
|315.88
|-
|-
| |10
|10
| |350.98
|153/125, 125/102
|350.98
|-
|-
| |11
|11
| |386.075
|5/4
|386.075
|-
|-
| |12
|12
| |421.17
|51/40
|421.17
|-
|-
| |13
|13
| |456.27
|125/96
|456.27
|-
|-
| |14
|14
| |491.37
|85/64
|491.37
|-
|-
| |15
|15
| |526.47
|576/425, 867/640
|526.47
|-
|-
| |16
|16
| |561.56
|400/289, 864/625
|561.56
|-
|-
| |17
|17
| |596.66
|24/17
|596.66
|-
|-
| |18
|18
| |631.76
|36/25
|631.76
|-
|-
| |19
|19
| |666.86
|25/17
|666.86
|-
|-
| |20
|'''20'''
| |701.955
|'''3/2'''
|'''701.955'''
|-
|-
|21
|21
|153/100
|737.05
|737.05
|-
|-
|22
|22
|25/16
|772.15
|772.15
|-
|-
|23
|23
|51/32
|807.25
|807.25
|-
|-
|24
|24
|625/384
|842.35
|842.35
|-
|-
|25
|25
|425/256, 480/289
|877.44
|877.44
|-
|-
|26
|26
|144/85
|912.54
|912.54
|-
|-
|27
|27
|216/125
|947.64
|947.64
|-
|-
|28
|28
|30/17
|982.74
|982.74
|-
|-
|29
|29
|9/5
|1017.835
|1017.835
|-
|-
|30
|30
|125/68
|1052.93
|1052.93
|-
|-
|31
|31
|15/8
|1088.03
|1088.03
|-
|-
|32
|32
|153/80
|1123.13
|1123.13
|-
|-
|33
|33
|125/64
|1158.23
|1158.23
|-
|-
|34
|34
|255/128
|1193.32
|1193.32
|-
|-
|35
|35
|864/425
|1228.42
|1228.42
|-
|-
|36
|36
|600/289
|1263.52
|1263.52
|-
|-
|37
|37
|36/17
|1298.62
|1298.62
|-
|-
|38
|38
|54/25
|1333.715
|1333.715
|-
|-
|39
|39
|75/34
|1368.81
|1368.81
|-
|-
|40
|'''40'''
|1403.91
|'''9/4'''
|'''1403.91'''
|}
|}
==Scale tree==
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: 34.28571 cents (4\7/20 = 1\35) to 36 cents (3\5/20 = 3/100)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || || || || ||34.2857||
|-
| || || || || || ||27\47||34.4681||
|-
| || || || || ||23\40|| ||34.5000||
|-
| || || || || || ||42\73||34.52055||
|-
| || || || ||19\33|| || ||34.{{Overline|54}}||
|-
| || || || || || ||53\92||34.5652||
|-
| || || || || ||34\59|| ||34.5763||
|-
| || || || || || ||49\85||34.5882||
|-
| || || ||15\26|| || || ||34.6153||
|-
| || || || || || ||56\97||34.6392||
|-
| || || || || ||41\71|| ||34.6479||
|-
| || || || || || ||67\116||34.6551||
|-
| || || || ||26\45|| || ||34.{{Overline|6}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||34.6789||
|-
| || || || || ||37\64|| ||34.6875||
|-
| || || || || || ||48\83||34.6988||
|-
| || ||11\19|| || || || ||34.7368||The generator closest to a just [[11/9]] for EDOs less than 2000
|-
| || || || || || ||51\88||34.7{{Overline|72}}||
|-
| || || || || ||40\69|| ||34.7826||
|-
| || || || || || ||69\119||34.7899||
|-
| || || || ||29\50|| || ||34.8000||
|-
| || || || || || ||76\131||34.8092||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||34.{{Overline|814}}||
|-
| || || || || || ||65\112||34.8214||
|-
| || || ||18\31|| || || ||34.8387||[[Meantone]] is in this region
|-
| || || || || || ||61\105||34.8571||
|-
| || || || || ||43\74|| ||34.{{Overline|864}}||
|-
| || || || || || ||68\117||34.8718||
|-
| || || || ||25\43|| || ||34.8837||
|-
| || || || || || ||57\98||34.8980||
|-
| || || || || ||32\55|| ||34.{{Overline|90}}||
|-
| || || || || || ||39\67||34.9254||
|-
| ||7\12|| || || || || ||35.0000||
|-
| || || || || || ||38\65||35.0769||
|-
| || || || || ||31\53|| ||35.0843||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||35.1064||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||35.12195||
|-
| || || || || || ||65\111||35.{{Overline|135}}||
|-
| || || || || ||41\70|| ||35.1429||
|-
| || || || || || ||58\99||35.{{Overline|15}}||
|-
| || || ||17\29|| || || ||35.1724||
|-
| || || || || || ||61\104||35.1923||
|-
| || || || || ||44\75|| ||35.2000||
|-
| || || || || || ||71\121||35.2066||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||35.2174||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||35.2294||
|-
| || || || || ||37\63|| ||35.2381||
|-
| || || || || || ||47\80||35.2500||
|-
| || ||10\17|| || || || ||35.2941||
|-
| || || || || || ||43\73||35.3425||
|-
| || || || || ||33\56|| ||35.3571||
|-
| || || || || || ||56\95||35.3684||
|-
| || || || ||23\39|| || ||35.3846||
|-
| || || || || || ||59\100||35.4000||
|-
| || || || || ||36\61|| ||35.4098||
|-
| || || || || || ||49\83||35.4217||
|-
| || || ||13\22|| || || ||35.{{Overline|45}}||[[Archy]] is in this region
|-
| || || || || || ||42\71||35.4930||
|-
| || || || || ||29\49|| ||35.5102||
|-
| || || || || || ||45\76||35.5263||
|-
| || || || ||16\27|| || ||35.{{Overline|5}}||
|-
| || || || || || ||35\59||35.5932||
|-
| || || || || ||19\32|| ||35.6250||
|-
| || || || || || ||22\37||35.{{Overline|675}}||
|-
|3\5|| || || || || || ||36.0000||
|}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.
{{stub}}
[[Category:edf]]
[[Category:Todo:expand]]

Latest revision as of 11:40, 6 August 2025

← 19edf 20edf 21edf →
Prime factorization 22 × 5
Step size 35.0978 ¢ 
Octave 34\20edf (1193.32 ¢) (→ 17\10edf)
Twelfth 54\20edf (1895.28 ¢) (→ 27\10edf)
Consistency limit 7
Distinct consistency limit 7

20 equal divisions of the perfect fifth (abbreviated 20edf or 20ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 20 equal parts of about 35.1 ¢ each. Each step represents a frequency ratio of (3/2)1/20, or the 20th root of 3/2.

Theory

20edf corresponds to 34.1902edo. It is closely related to Carlos Gamma and the gammic temperament, which adds an independent dimension for the octave (although strictly speaking, the "canonical" optimized Carlos Gamma tuning is not exactly 20edf, with its fifth stretched by the microscopic amount of 0.016 ¢). It very accurately represents the intervals 5/4, with 11 steps, and 17/16, with 3 steps.

Harmonics

Approximation of harmonics in 20edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -6.7 -6.7 -13.4 -13.6 -13.4 +0.6 +15.1 -13.4 +14.8 -9.8 +15.1
Relative (%) -19.0 -19.0 -38.0 -38.7 -38.0 +1.6 +42.9 -38.0 +42.3 -27.9 +42.9
Steps
(reduced) 		34
(14) 		54
(14) 		68
(8) 		79
(19) 		88
(8) 		96
(16) 		103
(3) 		108
(8) 		114
(14) 		118
(18) 		123
(3)
Approximation of harmonics in 20edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +16.9 -6.1 +14.8 +8.4 +8.7 +15.1 -8.3 +8.2 -6.1 -16.5 +11.9 +8.4
Relative (%) +48.1 -17.4 +42.3 +23.9 +24.9 +42.9 -23.8 +23.2 -17.4 -46.9 +33.8 +23.9
Steps
(reduced) 		127
(7) 		130
(10) 		134
(14) 		137
(17) 		140
(0) 		143
(3) 		145
(5) 		148
(8) 		150
(10) 		152
(12) 		155
(15) 		157
(17)

Intervals

The first steps up to two just perfect fifths should give a feeling of the granularity of this system…

Intervals of 20edf
Degrees 3/2.5/4.17/16 interpretation Cents
1 51/50 35.1
2 25/24 70.2
3 17/16 105.29
4 625/576, 867/800 140.39
5 320/289, 425/384 175.49
6 96/85 210.59
7 144/125 245.68
8 20/17 280.78
9 6/5 315.88
10 153/125, 125/102 350.98
11 5/4 386.075
12 51/40 421.17
13 125/96 456.27
14 85/64 491.37
15 576/425, 867/640 526.47
16 400/289, 864/625 561.56
17 24/17 596.66
18 36/25 631.76
19 25/17 666.86
20 3/2 701.955
21 153/100 737.05
22 25/16 772.15
23 51/32 807.25
24 625/384 842.35
25 425/256, 480/289 877.44
26 144/85 912.54
27 216/125 947.64
28 30/17 982.74
29 9/5 1017.835
30 125/68 1052.93
31 15/8 1088.03
32 153/80 1123.13
33 125/64 1158.23
34 255/128 1193.32
35 864/425 1228.42
36 600/289 1263.52
37 36/17 1298.62
38 54/25 1333.715
39 75/34 1368.81
40 9/4 1403.91
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