11edt: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
CompactStar (talk | contribs)
5,020 edits
No edit summary
+see also
 
(11 intermediate revisions by 4 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo|7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.
{{ED intro}}


From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.
== Theory ==
11edt can be seen as a very [[stretched and compressed tuning|stretched]] version of [[7edo]], with octaves sharpened by 10.3 cents. The octave stretching makes the [[3/2]] perfect fifth in better tune, while preserving a just [[3/1]] tritave.


From a no-2 point of view, 11edt has a [[5/3]] major sixth that is 19.8 cents flat. However, 11edt has an extremely inaccurate seventh harmonic [[7/1]], which is off by almost half a step (or about a semitone), which causes it to temper out [[49/45]] in the 7-limit. 11edt is at the extreme end of [[arcturus]] temperament, defined by tempering out [[15625/15309]] in the 3.5.7 subgroup. It gives an equalized interpretation for the [[9L 2s (3/1-equivalent)|sub-arcturus]] [[mos scale]].
The 11th harmonic, [[11/1]], only 1.6 cents flat, is very close to just. By exploiting the badly tuned seventh harmonic, 11edt tempers out [[35/33]] and [[77/75]] in the 11-limit. In the 3.5.11 subgroup, it tempers out [[125/121]].
=== Harmonics ===
{{Harmonics in equal|11|3|1|intervals=integer|columns=11}}
{{Harmonics in equal|11|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11edt (continued)}}
=== Subsets and supersets ===
11edt is the fifth [[prime equal division|prime edt]], following [[7edt]] and before [[13edt]], so it does not contain any nontrivial subset edts.


== Intervals ==
== Intervals ==
 
{| class="wikitable center-1 right-2 right-3"
{| class="wikitable center-all"
|-
! #
! #
! Cents
! [[Cent]]s
! Hekts
! [[Hekt]]s
! Approximate ratios
! Approximate ratios*
! [[Arcturus]] nonatonic notation
! [[Arcturus]]<br>enneatonic notation (J = 1/1)
|-
|-
| colspan = "3" | 0
| 0
| 0.0
| 0.0
| [[1/1]]
| [[1/1]]
| H
| J
|-
|-
| 1
| 1
| 172.9
| 172.9
| 118.1
| 118.1
| [[11/10]], [[10/9]]
| [[10/9]], [[11/10]]
| H#, Ib
| J#, Kb
|-
|-
| 2
| 2
| 345.8
| 345.8
| 236.2
| 236.2
| [[11/9]]
| [[5/4]], [[6/5]], [[11/9]], [[27/22]]
| I
| K
|-
|-
| 3
| 3
| 518.7
| 518.7
| 354.3
| 354.3
| [[4/3]], [[27/20]]
| [[4/3]], [[15/11]]
| J
| L
|-
|-
| 4
| 4
| 691.6
| 691.6
| 472.4
| 472.4
| [[3/2]], [[40/27]]
| [[3/2]]
| K
| M
|-
|-
| 5
| 5
| 864.5
| 864.5
| 590.5
| 590.5
| [[5/3]], [[28/17]], [[105/64]]
| [[5/3]], [[18/11]], [[33/20]]
| L
| N
|-
|-
| 6
| 6
| 1037.4
| 1037.4
| 708.6
| 708.6
| [[29/16]], [[20/11], [[64/35]]
| [[9/5]], [[11/6]], [[20/11]]
| L#, Mb
| N#, Ob
|-
|-
| 7
| 7
Line 58: Line 71:
| 826.7
| 826.7
| [[2/1]]
| [[2/1]]
| M
| O
|-
|-
| 8
| 8
| 1383.2
| 1383.2
| 944.8
| 944.8
|
| [[9/4]], [[11/5]]
| N
| P
|-
|-
| 9
| 9
| 1556.1
| 1556.1
| 1062.9
| 1062.9
|  
| [[5/2]], [[12/5]], [[22/9]], [[27/11]]
| O
| Q
|-
|-
| 10
| 10
| 1729
| 1729.0
| 1181
| 1181.0
|
| [[8/3]], [[11/4]]
| P
| R
|-
|-
| 11
| 11
| 1902
| 1902.0
| 1300
| 1300.0
|
| [[3/1]]
| H
| J
|}
|}
<nowiki/>* As a 2.3.5.11-subgroup temperament


== Scala tuning ==
== Music ==
Tuning in scala format is as follows:
=== Modern renderings ===
 
; {{W|Wolfgang Amadeus Mozart}}
<pre>
* [https://web.archive.org/web/20201127012444/http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 ''Piano Sonata No. 11'' in A major, K. 331] – using a 11 → 12 key mapping so octaves become tritaves
! E:\cakewalk\scales\11_of_tritave.scl
 
!
 
11 in tritave
 
!
 
172.90500
 
345.81000
 
518.71500
 
691.62000
 
864.52500
 
1037.43000
 
1210.33500
 
1383.24000
 
1556.14500
 
1729.05000
 
3/1
</pre>
 
== Pieces ==
 
Mozart's [http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 sonata #11 in A Major K331 in 11 EDT] (using a 11 =&gt; 12 key mapping so octaves become tritaves)


[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 Frozen Time Occupies Wall Street] by [http://www.chrisvaisvil.com Chris Vaisvil] =&gt;[http://chrisvaisvil.com/?p=1392  information about the piece]
=== 21st century ===
; [[Chris Vaisvil]]
* ''Frozen Time Occupies Wall Street'' (2011) – [https://www.chrisvaisvil.com/frozen-time-occupies-wall-street/ blog] | [https://web.archive.org/web/20220911143825/http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 play]  
* ''Molly's Playground'' (2011) – [https://www.chrisvaisvil.com/mollys-playground/ blog] | [https://web.archive.org/web/20201127013949/http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 play]


[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 Molly's Playground] by [http://www.chrisvaisvil.com Chris Vaisvil] =&gt; [http://chrisvaisvil.com/?p=1716  information about the piece]
== See also ==
[[category:macrotonal]]
* [[7edo]] – relative edo
* [[18ed6]] – relative ed6

Latest revision as of 12:48, 26 May 2025

← 10edt 11edt 12edt →
Prime factorization 11 (prime)
Step size 172.905 ¢ 
Octave 7\11edt (1210.34 ¢)
(semiconvergent)
Consistency limit 6
Distinct consistency limit 5

11 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 11edt or 11ed3), is a nonoctave tuning system that divides the interval of 3/1 into 11 equal parts of about 173 ¢ each. Each step represents a frequency ratio of 31/11, or the 11th root of 3.

Theory

11edt can be seen as a very stretched version of 7edo, with octaves sharpened by 10.3 cents. The octave stretching makes the 3/2 perfect fifth in better tune, while preserving a just 3/1 tritave.

From a no-2 point of view, 11edt has a 5/3 major sixth that is 19.8 cents flat. However, 11edt has an extremely inaccurate seventh harmonic 7/1, which is off by almost half a step (or about a semitone), which causes it to temper out 49/45 in the 7-limit. 11edt is at the extreme end of arcturus temperament, defined by tempering out 15625/15309 in the 3.5.7 subgroup. It gives an equalized interpretation for the sub-arcturus mos scale.

The 11th harmonic, 11/1, only 1.6 cents flat, is very close to just. By exploiting the badly tuned seventh harmonic, 11edt tempers out 35/33 and 77/75 in the 11-limit. In the 3.5.11 subgroup, it tempers out 125/121.

Harmonics

Approximation of harmonics in 11edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +10.3 +0.0 +20.7 -19.8 +10.3 -83.6 +31.0 +0.0 -9.5 -1.6 +20.7
Relative (%) +6.0 +0.0 +12.0 -11.5 +6.0 -48.4 +17.9 +0.0 -5.5 -0.9 +12.0
Steps
(reduced) 		7
(7) 		11
(0) 		14
(3) 		16
(5) 		18
(7) 		19
(8) 		21
(10) 		22
(0) 		23
(1) 		24
(2) 		25
(3)
Approximation of harmonics in 11edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +55.0 -73.3 -19.8 +41.3 -63.6 +10.3 -83.3 +0.8 -83.6 +8.7 -68.2 +31.0
Relative (%) +31.8 -42.4 -11.5 +23.9 -36.8 +6.0 -48.2 +0.5 -48.4 +5.1 -39.5 +17.9
Steps
(reduced) 		26
(4) 		26
(4) 		27
(5) 		28
(6) 		28
(6) 		29
(7) 		29
(7) 		30
(8) 		30
(8) 		31
(9) 		31
(9) 		32
(10)

Subsets and supersets

11edt is the fifth prime edt, following 7edt and before 13edt, so it does not contain any nontrivial subset edts.

Intervals

# Cents Hekts Approximate ratios* Arcturus
enneatonic notation (J = 1/1)
0 0.0 0.0 1/1 J
1 172.9 118.1 10/9, 11/10 J#, Kb
2 345.8 236.2 5/4, 6/5, 11/9, 27/22 K
3 518.7 354.3 4/3, 15/11 L
4 691.6 472.4 3/2 M
5 864.5 590.5 5/3, 18/11, 33/20 N
6 1037.4 708.6 9/5, 11/6, 20/11 N#, Ob
7 1210.3 826.7 2/1 O
8 1383.2 944.8 9/4, 11/5 P
9 1556.1 1062.9 5/2, 12/5, 22/9, 27/11 Q
10 1729.0 1181.0 8/3, 11/4 R
11 1902.0 1300.0 3/1 J

* As a 2.3.5.11-subgroup temperament

Music

Modern renderings

Wolfgang Amadeus Mozart

21st century

Chris Vaisvil
  • Frozen Time Occupies Wall Street (2011) – blog | play
  • Molly's Playground (2011) – blog | play

See also

Retrieved from "https://en.xen.wiki/index.php?title=11edt&oldid=197763"