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== Theory ==
== Theory ==
From a no-two 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.
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]].
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]].


=== Relation to edos ===
=== Harmonics ===
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.
{{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)}}


=== Prime harmonics ===
== Intervals ==
{{Harmonics in equal|11|3|1}}
{| class="wikitable center-1 right-2 right-3"
{{Harmonics in equal|11|3|1|intervals=prime}}
 
=== Interval table ===
{| class="wikitable center-all"
|-
|-
! #
! #
Line 23: Line 22:
! [[Arcturus]] nonatonic notation (J = 1/1)
! [[Arcturus]] nonatonic notation (J = 1/1)
|-
|-
| colspan = "3" | 0
| 0
| 0.0
| 0.0
| [[1/1]]
| [[1/1]]
| J
| J
Line 82: Line 83:
|-
|-
| 10
| 10
| 1729
| 1729.0
| 1181
| 1181.0
|
|
| R
| R
|-
|-
| 11
| 11
| 1902
| 1902.0
| 1300
| 1300.0
|
|
| J
| J
|}
|}


== Pieces ==
== Music ==
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 => 12 key mapping so octaves become tritaves)
=== Modern renderings ===
 
; {{W|Wolfgang Amadeus Mozart}}
[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 Frozen Time Occupies Wall Street] by [http://www.chrisvaisvil.com Chris Vaisvil] =>[http://chrisvaisvil.com/?p=1392  information about the piece]
* [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


[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 Molly's Playground] by [http://www.chrisvaisvil.com Chris Vaisvil] => [http://chrisvaisvil.com/?p=1716  information about the piece]
=== 21st century ===
[[category:macrotonal]]
; [[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]

Revision as of 12:29, 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)

Intervals

# Cents Hekts Approximate ratios Arcturus nonatonic notation (J = 1/1)
0 0.0 0.0 1/1 J
1 172.9 118.1 11/10, 10/9 J#, Kb
2 345.8 236.2 11/9 K
3 518.7 354.3 4/3, 27/20 L
4 691.6 472.4 3/2, 40/27 M
5 864.5 590.5 5/3, 28/17, 105/64 N
6 1037.4 708.6 29/16, 20/11, 64/35 N#, Ob
7 1210.3 826.7 2/1 O
8 1383.2 944.8 P
9 1556.1 1062.9 Q
10 1729.0 1181.0 R
11 1902.0 1300.0 J

Music

Modern renderings

Wolfgang Amadeus Mozart

21st century

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