66edo: Difference between revisions

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== Theory ==

The [[patent val]] of 66edo is [[contorted]] in the 5-limit, [[tempering out]] the same [[comma]]s ([[250/243]], [[2048/2025]], [[3125/3072]], etc.) as [[22edo]]. In the 7-limit it tempers out [[686/675]] and [[1029/1024]], in the 11-limit [[55/54]], [[100/99]] and [[121/120]], in the 13-limit [[91/90]], [[169/168]], [[196/195]] and in the 17-limit [[136/135]] and [[256/255]]. It provides the [[optimal patent val]] for the 11- and 13-limit [[ammonite]] temperament. Otherwise, 66edo is not exceptional when it comes to approximating prime harmonics; however, it contains a quite accurate approximation to the 5:7:9:11:13 chord and can therefore be used for various [[primodal]] [[over-5]] scales.

The 66b val tempers out [[16875/16384]] in the 5-limit, [[126/125]], [[1728/1715]] and [[2401/2400]] in the 7-limit, [[99/98]] and [[385/384]] in the 11-limit, and [[105/104]], [[144/143]] and [[847/845]] in the 13-limit.

109 steps of 66edo is extremely close to the [[acoustic pi]] with only +0.023{{c}} of error.

=== Odd harmonics ===

The [[patent val]] is [[contorted]] in the 5-limit, tempering out the same commas 250/243, 2048/2025 and 3125/3072 as [[22edo|22edo]]. In the 7-limit it tempers out 686/675 and 1029/1024, in the 11-limit 55/54, 100/99 and 121/120, in the 13-limit 91/90, 169/168, 196/195 and in the 17-limit 136/135 and 256/255. It provides the [[Optimal_patent_val|optimal patent val]] for 11- and 13-limit [[Porcupine_family#Ammonite|ammonite temperament]].

~~The 66b val tempers out 16875/16384 in~~ the 5-~~limit~~, ~~126~~/~~125, 1728~~/~~1715 and 2401~~/~~2400 in the 7~~-~~limit, 99~~/~~98 and 385~~/~~384 in the 11~~-~~limit, and 105~~/~~104,~~ 144/143 ~~and 847~~/~~845~~ in the 13-~~limit~~.

=== Subsets and supersets ===

Since 66 factors into {{factorization|66}}, 66edo has subset edos {{EDOs| 2, 3, 6, 11, 22, and 33 }}. [[198edo]], which triples it, corrects its approximation to many of the lower harmonics.

== Interval table ==

== Notation ==

=== Sagittal notation ===

This notation uses the same sagittal sequence as [[59edo#Sagittal notation|59-EDO]], and is a superset of the notations for EDOs [[22edo#Sagittal notation|22]] and [[11edo#Sagittal notation|11]].

==== Evo flavor ====

File:66-EDO_Evo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 748 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 190 106 [[513/512]]

rect 190 80 320 106 [[144/143]]

rect 320 80 430 106 [[81/80]]

rect 430 80 570 106 [[1053/1024]]

default [[File:66-EDO_Evo_Sagittal.svg]]

</imagemap>

==== Revo flavor ====

File:66-EDO_Revo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 694 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 190 106 [[513/512]]

rect 190 80 320 106 [[144/143]]

rect 320 80 430 106 [[81/80]]

rect 430 80 570 106 [[1053/1024]]

default [[File:66-EDO_Revo_Sagittal.svg]]

</imagemap>

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

== Instruments ==

=== Lumatone ===

[[Lumatone mapping for 66edo]]

=== Skip fretting ===

'''Skip fretting system 66 7 11''' is a [[skip fretting]] system for [[66edo]]. All examples on this page are for 7-string [[guitar]].

; Prime harmonics

1/1: string 2 open

2/1: string 1 fret 11 and string 7 fret 11

3/2: string 3 fret 4

5/4: string 2 fret 3

7/4: string 3 fret 6

11/8: string 5 fret 9

13/8: string 3 fret 5

17/16: string 6 fret 4

19/16: string 5 fret 7

23/16: string 2 fret 5

29/16: string 4 fret 5

~~109 steps of 66edo is extremely close to [[Pitave|Pi]] with only +0.023 cents of error.~~

31/16: string 2 fret 9

[[~~Category:Equal divisions of the octave|##~~]] <!-~~- 2-digit number -->~~

== Music ==

[~~[Category~~:~~Ammonite]~~]

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/jHwtQg0aR14 ''Spring Yard Zone (microtonal version in 66edo) - Sonic The Hedgehog''] (2022)

* [https://www.youtube.com/shorts/Cn28RJSSgQ0 ''microtonal improvisation in 66edo''] (2025)

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|66}}
+{{ED intro}}
 == Theory ==
+The [[patent val]] of 66edo is [[contorted]] in the 5-limit, [[tempering out]] the same [[comma]]s ([[250/243]], [[2048/2025]], [[3125/3072]], etc.) as [[22edo]]. In the 7-limit it tempers out [[686/675]] and [[1029/1024]], in the 11-limit [[55/54]], [[100/99]] and [[121/120]], in the 13-limit [[91/90]], [[169/168]], [[196/195]] and in the 17-limit [[136/135]] and [[256/255]]. It provides the [[optimal patent val]] for the 11- and 13-limit [[ammonite]] temperament. Otherwise, 66edo is not exceptional when it comes to approximating prime harmonics; however, it contains a quite accurate approximation to the 5:7:9:11:13 chord and can therefore be used for various [[primodal]] [[over-5]] scales.
+The 66b val tempers out [[16875/16384]] in the 5-limit, [[126/125]], [[1728/1715]] and [[2401/2400]] in the 7-limit, [[99/98]] and [[385/384]] in the 11-limit, and [[105/104]], [[144/143]] and [[847/845]] in the 13-limit.
+steps of 66edo is extremely close to the [[acoustic pi]] with only +0.023{{c}} of error.
+=== Odd harmonics ===
 {{Harmonics in equal|66}}
-The [[patent val]] is [[contorted]] in the 5-limit, tempering out the same commas 250/243, 2048/2025 and 3125/3072 as [[22edo|22edo]]. In the 7-limit it tempers out 686/675 and 1029/1024, in the 11-limit 55/54, 100/99 and 121/120, in the 13-limit 91/90, 169/168, 196/195 and in the 17-limit 136/135 and 256/255. It provides the [[Optimal_patent_val|optimal patent val]] for 11- and 13-limit [[Porcupine_family#Ammonite|ammonite temperament]].
-The 66b val tempers out 16875/16384 in the 5-limit, 126/125, 1728/1715 and 2401/2400 in the 7-limit, 99/98 and 385/384 in the 11-limit, and 105/104, 144/143 and 847/845 in the 13-limit.
+=== Subsets and supersets ===
+Since 66 factors into {{factorization|66}}, 66edo has subset edos {{EDOs| 2, 3, 6, 11, 22, and 33 }}. [[198edo]], which triples it, corrects its approximation to many of the lower harmonics.
+== Interval table ==
+{{Interval table}}
+== Notation ==
+=== Sagittal notation ===
+This notation uses the same sagittal sequence as [[59edo#Sagittal notation|59-EDO]], and is a superset of the notations for EDOs [[22edo#Sagittal notation|22]] and [[11edo#Sagittal notation|11]].
+==== Evo flavor ====
+<imagemap>
+File:66-EDO_Evo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 748 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 190 106 [[513/512]]
+rect 190 80 320 106 [[144/143]]
+rect 320 80 430 106 [[81/80]]
+rect 430 80 570 106 [[1053/1024]]
+default [[File:66-EDO_Evo_Sagittal.svg]]
+</imagemap>
+==== Revo flavor ====
+<imagemap>
+File:66-EDO_Revo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 694 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 190 106 [[513/512]]
+rect 190 80 320 106 [[144/143]]
+rect 320 80 430 106 [[81/80]]
+rect 430 80 570 106 [[1053/1024]]
+default [[File:66-EDO_Revo_Sagittal.svg]]
+</imagemap>
+In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
+== Instruments ==
+=== Lumatone ===
+[[Lumatone mapping for 66edo]]
+=== Skip fretting ===
+'''Skip fretting system 66 7 11''' is a [[skip fretting]] system for [[66edo]]. All examples on this page are for 7-string [[guitar]].
+; Prime harmonics
+/1: string 2 open
+/1: string 1 fret 11 and string 7 fret 11
+/2: string 3 fret 4
+/4: string 2 fret 3
+/4: string 3 fret 6
+/8: string 5 fret 9
+/8: string 3 fret 5
+/16: string 6 fret 4
+/16: string 5 fret 7
+/16: string 2 fret 5
+/16: string 4 fret 5
-steps of 66edo is extremely close to [[Pitave|Pi]] with only +0.023 cents of error.
+/16: string 2 fret 9
-[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
+== Music ==
-[[Category:Ammonite]]
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/jHwtQg0aR14 ''Spring Yard Zone (microtonal version in 66edo) - Sonic The Hedgehog''] (2022)
+* [https://www.youtube.com/shorts/Cn28RJSSgQ0 ''microtonal improvisation in 66edo''] (2025)