66edo: Difference between revisions

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The 66 equal division divides the octave into 66 equal parts of 18.182 cents each. The patent 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.

== Theory ==

[[~~Category~~:~~ammonite~~]]

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 ===

=== 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.

== Octave stretch or compression ==

66edo tunes multiple-of-3 [[harmonic]]s quite sharp, but most other simple harmonics quite flat.

Slight [[octave stretching]] turns 66edo into a [[dual-fifth]] tuning, sharing error near-equally between its two [[mapping]]s of 3, while improving most other simple harmonics. Slightly stretched tunings of 66edo include [[zpi|340zpi]], [[equal tuning|219ed10]] & [[equal tuning|209ed9]].

Substantial octave stretching will flip the mapping of many harmonics including 3, but will reduce the error of most of them. Though it will introduce more mapping inconsistencies. Such tunings of 66edo include [[zpi|338zpi]] and [[104edt]].

Moderate ''[[octave shrinking]]'' will instead improve upon 66edo's best mapping of prime 3 dramatically. It reduce the error on most other simple harmonics. It flips the mapping of many of them, but introduces less internal mapping inconsistencies than stretching by the same amount. Moderately octave-compressed tunings for 66edo include [[105edt]] and [[zpi|342zpi]].

== 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

31/16: string 2 fret 9

== Music ==

; [[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: @@
-The 66 equal division divides the octave into 66 equal parts of 18.182 cents each. The patent 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]].
+{{Infobox ET}}
+{{ED intro}}
-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.
+== Theory ==
-[[Category:ammonite]]
+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}}
+=== 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.
+== Octave stretch or compression ==
+edo tunes multiple-of-3 [[harmonic]]s quite sharp, but most other simple harmonics quite flat.
+Slight [[octave stretching]] turns 66edo into a [[dual-fifth]] tuning, sharing error near-equally between its two [[mapping]]s of 3, while improving most other simple harmonics. Slightly stretched tunings of 66edo include [[zpi|340zpi]], [[equal tuning|219ed10]] & [[equal tuning|209ed9]].
+Substantial octave stretching will flip the mapping of many harmonics including 3, but will reduce the error of most of them. Though it will introduce more mapping inconsistencies. Such tunings of 66edo include [[zpi|338zpi]] and [[104edt]].
+Moderate ''[[octave shrinking]]'' will instead improve upon 66edo's best mapping of prime 3 dramatically. It reduce the error on most other simple harmonics. It flips the mapping of many of them, but introduces less internal mapping inconsistencies than stretching by the same amount. Moderately octave-compressed tunings for 66edo include  [[105edt]] and [[zpi|342zpi]].
+== 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
+/16: string 2 fret 9
+== Music ==
+; [[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)