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{{Infobox ET}}
{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].
{{ED intro}}


Lookalikes: [[17edo]], [[27edt]]
== Theory ==
10edf is related to [[17edo]], but with the [[3/2|perfect fifth]] rather than the [[2/1|octave]] being just. The octave is compressed by about 6.68{{c}}, a small but significant deviation. 10edf is [[consistent]] to the [[integer limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the 4-integer-limit. This makes 10edf a suitable tuning perhaps in the [[5-limit]], but overcompressed in any other limits, as well as the no-5 13-limit, where 17edo is best at.


==Intervals==
=== Harmonics ===
{| class="wikitable"
{{Harmonics in equal|10|3|2|intervals=integer|columns=11}}
!degree
{{Harmonics in equal|10|3|2|intervals=integer|columns=12|start=12|collapsed=true|Approximation of harmonics in 10edf (continued)}}
!
 
![[1L 3s (fifth-equivalent)|Neptunian]] notation using 8\10edf
=== Subsets and supersets ===
![[Ed9/4|Neapolitan]] notation using 3/10edf
Since 10 factors into primes as {{nowrap| 2 × 5 }}, 10edf contains [[2edf]] and [[5edf]] as subset edfs.
 
== Intervals ==
{| class="wikitable center-all right-2"
|-
|-
! colspan="2" |0
! #
|C
! Cents
|F
! [[1L 3s (fifth-equivalent)|Neptunian]] notation<br>using 8\10edf
! [[Ed9/4|Neapolitan]] notation<br>using 3/10edf
|-
| 0
| 0.0
| C
| F
|-
|-
| 1
| 1
|70.1955
| 70.2
|^C, vDb
| ^C, vDb
|F^, Gb
| F^, Gb
|-
|-
|2
| 2
|140.391
| 140.4
|C#, Db
| C#, Db
|F#, Gd
| F#, Gd
|-
|-
|3
| 3
|210.5865
| 210.6
|vD
| vD
|G
| G
|-
|-
|4
| 4
|280.782
| 280.8
|D
| D
|G^, Ab
| G^, Ab
|-
|-
|5
| 5
|350.9775
| 351.0
|^D, vE
| ^D, vE
|G#, Ad
| G#, Ad
|-
|-
|6
| 6
|421.173
| 421.2
|E
| E
|A
| A
|-
|-
|7
| 7
|491.3685
| 491.4
|^E, vF
| ^E, vF
|A^, Hb
| A^, Hb
|-
|-
| 8
| 8
|561.564
| 561.6
|F
| F
|A#, Hd
| A#, Hd
|-
|-
|9
| 9
|631.7595
| 631.8
|^F, vC
| ^F, vC
|H
| H
|-
|-
|10
| 10
|701.955
| 702.0
|C
| C
|B
| B
|-
|-
|11
| 11
|772.1505
| 772.2
|^C, vDb
| ^C, vDb
|B^, Cb
| B^, Cb
|-
|-
|12
| 12
|842.346
| 842.3
|C#, Db
| C#, Db
|B#, Cd
| B#, Cd
|-
|-
|13
| 13
|912.5415
| 912.5
|vD
| vD
|C
| C
|-
|-
|14
| 14
|982.737
| 982.7
|D
| D
|C^, Db
| C^, Db
|-
|-
|15
| 15
|1052.9325
| 1052.9
|^D, vE
| ^D, vE
|C#, Dd
| C#, Dd
|-
|-
|16
| 16
|1123.128
| 1123.1
|E
| E
|D
| D
|-
|-
|17
| 17
|1193.3235
| 1193.3
|^E, vF
| ^E, vF
|D^, Eb
| D^, Eb
|-
|-
|18
| 18
|1263.519
| 1263.5
|F
| F
|D#, Eb
| D#, Eb
|-
|-
|19
| 19
|1333.7145
| 1333.7
|^F, vC
| ^F, vC
|E
| E
|-
|-
|20
| 20
|1403.91
| 1403.9
|C
| C
|F
| F
|}
|}
==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.
== Music ==
 
; [[Peter Kosmorsky]]
Generator range: 68.57143 cents (4\7/10 = 2\35) to 72 cents (3\5/10 = 3\50)
* [https://www.archive.org/details/10Edf ''10 edf''] (archived 2011)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || || || || ||68.5714||
|-
| || || || || || ||27\47||68.9362||
|-
| || || || || ||23\40|| ||69.000||
|-
| || || || || || ||42\73||69.0411||
|-
| || || || ||19\33|| || ||69.0909||
|-
| || || || || || ||53\92||69.1304||
|-
| || || || || ||34\59|| ||69.1525||
|-
| || || || || || ||49\85||69.1765||
|-
| || || ||15\26|| || || ||69.2308||
|-
| || || || || || ||56\97||69.2784||
|-
| || || || || ||41\71|| ||69.2958||
|-
| || || || || || ||67\116||69.3103||
|-
| || || || ||26\45|| || ||69.3333||[[Flattone]] is in this region
|-
| || || || || || ||63\109||69.3578||
|-
| || || || || ||37\64|| ||69.3750||
|-
| || || || || || ||48\83||69.3976||
|-
| || ||11\19|| || || || ||69.4737||
|-
| || || || || || ||51\88||69.5455||
|-
| || || || || ||40\69|| ||69.5652||
|-
| || || || || || ||69\119||69.5798||
|-
| || || || ||29\50|| || ||69.6000||
|-
| || || || || || ||66\131||69.6183||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||69.6296||
|-
| || || || || || ||65\112||69.6429||
|-
| || || ||18\31|| || || ||69.6774||[[Meantone]] is in this region
|-
| || || || || || ||61\105||69.7143||
|-
| || || || || ||43\74|| ||69.7297||
|-
| || || || || || ||68\117||69.7436||
|-
| || || || ||25\43|| || ||69.7674||
|-
| || || || || || ||57\98||69.7959||
|-
| || || || || ||32\55|| ||69.8182||
|-
| || || || || || ||39\67||69.8507||
|-
| ||7\12|| || || || || ||70.000||
|-
| || || || || || ||38\65||70.1539||
|-
| || || || || ||31\53|| ||70.1887||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||70.2128||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||70.2409||
|-
| || || || || || ||65\111||70.2703||
|-
| || || || || ||41\70|| ||70.2857||
|-
| || || || || || ||58\99||70.3030||
|-
| || || ||17\29|| || || ||70.3448||
|-
| || || || || || ||61\104||70.3846||
|-
| || || || || ||44\75|| ||70.4000||
|-
| || || || || || ||71\121||70.4132||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||70.4348||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||70.5487||
|-
| || || || || ||37\63|| ||70.4762||
|-
| || || || || || ||47\80||70.5000||
|-
| || ||10\17|| || || || ||70.5882||The generator closest to a just [[25/24]] for EDOs less than 200
|-
| || || || || || ||43\73||70.6849||
|-
| || || || || ||33\56|| ||70.7143||
|-
| || || || || || ||56\95||70.7368||
|-
| || || || ||23\39|| || ||70.7692||
|-
| || || || || || ||59\100||70.8000||
|-
| || || || || ||36\61|| ||70.8197||
|-
| || || || || || ||49\83||70.8434||
|-
| || || ||13\22|| || || ||70.9091||[[Archy]] is in this region
|-
| || || || || || ||42\71||70.9859||
|-
| || || || || ||29\49|| ||71.0204||
|-
| || || || || || ||45\76||71.0526||
|-
| || || || ||16\27|| || ||71.1111||
|-
| || || || || || ||35\59||71.1864||
|-
| || || || || ||19\32|| ||71.2500||
|-
| || || || || || ||22\37||71.3514||
|-
|3\5|| || || || || || ||72.000||
|}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.
== See also ==
==Music==
* [[17edo]] – relative edo
*http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]
* [[27edt]] – relative edt
* [[44ed6]] – relative ed6


[[Category:Edf]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:todo:expand]]
[[Category:todo:improve synopsis]]

Latest revision as of 15:29, 19 June 2025

← 9edf 10edf 11edf →
Prime factorization 2 × 5
Step size 70.1955 ¢ 
Octave 17\10edf (1193.32 ¢)
(semiconvergent)
Twelfth 27\10edf (1895.28 ¢)
(semiconvergent)
Consistency limit 7
Distinct consistency limit 6

10 equal divisions of the perfect fifth (abbreviated 10edf or 10ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 10 equal parts of about 70.2 ¢ each. Each step represents a frequency ratio of (3/2)1/10, or the 10th root of 3/2.

Theory

10edf is related to 17edo, but with the perfect fifth rather than the octave being just. The octave is compressed by about 6.68 ¢, a small but significant deviation. 10edf is consistent to the 7-integer-limit, but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the 4-integer-limit. This makes 10edf a suitable tuning perhaps in the 5-limit, but overcompressed in any other limits, as well as the no-5 13-limit, where 17edo is best at.

Harmonics

Approximation of harmonics in 10edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -6.7 -6.7 -13.4 +21.5 -13.4 +0.6 -20.0 -13.4 +14.8 -9.8 -20.0
Relative (%) -9.5 -9.5 -19.0 +30.6 -19.0 +0.8 -28.5 -19.0 +21.1 -13.9 -28.5
Steps
(reduced) 		17
(7) 		27
(7) 		34
(4) 		40
(0) 		44
(4) 		48
(8) 		51
(1) 		54
(4) 		57
(7) 		59
(9) 		61
(1)
Approximation of harmonics in 10edf
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -18.2 -6.1 +14.8 -26.7 +8.7 -20.0 +26.8 +8.2 -6.1 -16.5 -23.2 -26.7
Relative (%) -25.9 -8.7 +21.1 -38.0 +12.4 -28.5 +38.1 +11.6 -8.7 -23.4 -33.1 -38.0
Steps
(reduced) 		63
(3) 		65
(5) 		67
(7) 		68
(8) 		70
(0) 		71
(1) 		73
(3) 		74
(4) 		75
(5) 		76
(6) 		77
(7) 		78
(8)

Subsets and supersets

Since 10 factors into primes as 2 × 5, 10edf contains 2edf and 5edf as subset edfs.

Intervals

# Cents Neptunian notation
using 8\10edf 		Neapolitan notation
using 3/10edf
0 0.0 C F
1 70.2 ^C, vDb F^, Gb
2 140.4 C#, Db F#, Gd
3 210.6 vD G
4 280.8 D G^, Ab
5 351.0 ^D, vE G#, Ad
6 421.2 E A
7 491.4 ^E, vF A^, Hb
8 561.6 F A#, Hd
9 631.8 ^F, vC H
10 702.0 C B
11 772.2 ^C, vDb B^, Cb
12 842.3 C#, Db B#, Cd
13 912.5 vD C
14 982.7 D C^, Db
15 1052.9 ^D, vE C#, Dd
16 1123.1 E D
17 1193.3 ^E, vF D^, Eb
18 1263.5 F D#, Eb
19 1333.7 ^F, vC E
20 1403.9 C F

Music

Peter Kosmorsky

See also

Retrieved from "https://en.xen.wiki/index.php?title=10edf&oldid=200023"