66edo: Difference between revisions

← Older edit

VisualWikitext

Latest revision as of 05:57, 26 July 2025

← 65edo

66edo

67edo →

Prime factorization

2 × 3 × 11

Step size

18.1818 ¢

Fifth

39\66 (709.091 ¢) (→ 13\22)

Semitones (A1:m2)

9:3 (163.6 ¢ : 54.55 ¢)

Dual sharp fifth

39\66 (709.091 ¢) (→ 13\22)

Dual flat fifth

38\66 (690.909 ¢) (→ 19\33)

Dual major 2nd

11\66 (200 ¢) (→ 1\6)

Consistency limit

3

Distinct consistency limit

3

66 equal divisions of the octave (abbreviated 66edo or 66ed2), also called 66-tone equal temperament (66tet) or 66 equal temperament (66et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 66 equal parts of about 18.2 ¢ each. Each step represents a frequency ratio of 2^1/66, or the 66th root of 2.

Theory

The patent val of 66edo is contorted in the 5-limit, tempering out the same commas (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 ¢ of error.

Odd harmonics

Approximation of odd harmonics in 66edo
Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+7.14	-4.50	-5.19	-3.91	-5.86	-4.16	+2.64	+4.14	-6.60	+1.95	+8.09
Error	Relative (%)	+39.2	-24.7	-28.5	-21.5	-32.2	-22.9	+14.5	+22.7	-36.3	+10.7	+44.5
Steps (reduced)		105 (39)	153 (21)	185 (53)	209 (11)	228 (30)	244 (46)	258 (60)	270 (6)	280 (16)	290 (26)	299 (35)

Subsets and supersets

Since 66 factors into 2 × 3 × 11, 66edo has subset edos 2, 3, 6, 11, 22, and 33. 198edo, which triples it, corrects its approximation to many of the lower harmonics.

Interval table

Steps	Cents	Approximate ratios	Ups and downs notation (Dual flat fifth 38\66)	Ups and downs notation (Dual sharp fifth 39\66)
0	0	1/1	D	D
1	18.2		^D, vE♭♭♭♭	^D, vvE♭
2	36.4		D♯, E♭♭♭♭	^^D, vE♭
3	54.5	31/30, 32/31, 33/32, 34/33	^D♯, vE♭♭♭	^³D, E♭
4	72.7	24/23	D𝄪, E♭♭♭	^⁴D, ^E♭
5	90.9	20/19	^D𝄪, vE♭♭	v⁴D♯, ^^E♭
6	109.1	16/15, 33/31	D♯𝄪, E♭♭	v³D♯, ^³E♭
7	127.3	14/13	^D♯𝄪, vE♭	vvD♯, ^⁴E♭
8	145.5		D𝄪𝄪, E♭	vD♯, v⁴E
9	163.6	11/10, 34/31	^D𝄪𝄪, vE	D♯, v³E
10	181.8		E	^D♯, vvE
11	200	28/25	^E, vF♭♭♭	^^D♯, vE
12	218.2	17/15, 25/22	E♯, F♭♭♭	E
13	236.4		^E♯, vF♭♭	^E, vvF
14	254.5	22/19	E𝄪, F♭♭	^^E, vF
15	272.7	34/29	^E𝄪, vF♭	F
16	290.9	13/11	E♯𝄪, F♭	^F, vvG♭
17	309.1		^E♯𝄪, vF	^^F, vG♭
18	327.3	29/24	F	^³F, G♭
19	345.5		^F, vG♭♭♭♭	^⁴F, ^G♭
20	363.6	21/17	F♯, G♭♭♭♭	v⁴F♯, ^^G♭
21	381.8		^F♯, vG♭♭♭	v³F♯, ^³G♭
22	400	29/23	F𝄪, G♭♭♭	vvF♯, ^⁴G♭
23	418.2	14/11	^F𝄪, vG♭♭	vF♯, v⁴G
24	436.4		F♯𝄪, G♭♭	F♯, v³G
25	454.5	13/10	^F♯𝄪, vG♭	^F♯, vvG
26	472.7	21/16, 25/19	F𝄪𝄪, G♭	^^F♯, vG
27	490.9		^F𝄪𝄪, vG	G
28	509.1		G	^G, vvA♭
29	527.3	19/14, 23/17	^G, vA♭♭♭♭	^^G, vA♭
30	545.5	26/19	G♯, A♭♭♭♭	^³G, A♭
31	563.6		^G♯, vA♭♭♭	^⁴G, ^A♭
32	581.8	7/5	G𝄪, A♭♭♭	v⁴G♯, ^^A♭
33	600	17/12, 24/17	^G𝄪, vA♭♭	v³G♯, ^³A♭
34	618.2	10/7	G♯𝄪, A♭♭	vvG♯, ^⁴A♭
35	636.4		^G♯𝄪, vA♭	vG♯, v⁴A
36	654.5	19/13	G𝄪𝄪, A♭	G♯, v³A
37	672.7	28/19, 31/21, 34/23	^G𝄪𝄪, vA	^G♯, vvA
38	690.9		A	^^G♯, vA
39	709.1		^A, vB♭♭♭♭	A
40	727.3	32/21	A♯, B♭♭♭♭	^A, vvB♭
41	745.5	20/13	^A♯, vB♭♭♭	^^A, vB♭
42	763.6		A𝄪, B♭♭♭	^³A, B♭
43	781.8	11/7	^A𝄪, vB♭♭	^⁴A, ^B♭
44	800	35/22	A♯𝄪, B♭♭	v⁴A♯, ^^B♭
45	818.2		^A♯𝄪, vB♭	v³A♯, ^³B♭
46	836.4	34/21	A𝄪𝄪, B♭	vvA♯, ^⁴B♭
47	854.5		^A𝄪𝄪, vB	vA♯, v⁴B
48	872.7		B	A♯, v³B
49	890.9		^B, vC♭♭♭	^A♯, vvB
50	909.1	22/13	B♯, C♭♭♭	^^A♯, vB
51	927.3	29/17	^B♯, vC♭♭	B
52	945.5	19/11	B𝄪, C♭♭	^B, vvC
53	963.6		^B𝄪, vC♭	^^B, vC
54	981.8	30/17	B♯𝄪, C♭	C
55	1000	25/14	^B♯𝄪, vC	^C, vvD♭
56	1018.2		C	^^C, vD♭
57	1036.4	20/11, 31/17	^C, vD♭♭♭♭	^³C, D♭
58	1054.5	35/19	C♯, D♭♭♭♭	^⁴C, ^D♭
59	1072.7	13/7	^C♯, vD♭♭♭	v⁴C♯, ^^D♭
60	1090.9	15/8	C𝄪, D♭♭♭	v³C♯, ^³D♭
61	1109.1	19/10	^C𝄪, vD♭♭	vvC♯, ^⁴D♭
62	1127.3	23/12	C♯𝄪, D♭♭	vC♯, v⁴D
63	1145.5	31/16, 33/17	^C♯𝄪, vD♭	C♯, v³D
64	1163.6		C𝄪𝄪, D♭	^C♯, vvD
65	1181.8		^C𝄪𝄪, vD	^^C♯, vD
66	1200	2/1	D	D

Notation

Sagittal notation

This notation uses the same sagittal sequence as 59-EDO, and is a superset of the notations for EDOs 22 and 11.

Evo flavor

Revo flavor

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's 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

31/16: string 2 fret 9

Music

Bryan Deister

@@ Line 1: / Line 1: @@
-The <b>66 equal division</b> or <b>66-EDO</b> divides the octave into 66 equal parts of 18.182 cents each. 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]].
+{{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.
+== 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)

66edo: Difference between revisions

Latest revision as of 05:57, 26 July 2025

Contents

Theory

Odd harmonics

Subsets and supersets

Interval table

Notation

Sagittal notation

Evo flavor

Revo flavor

Instruments

Lumatone

Skip fretting

Music

Navigation menu

66edo: Difference between revisions

Latest revision as of 05:57, 26 July 2025

Theory

Odd harmonics

Subsets and supersets

Interval table

Notation

Sagittal notation

Evo flavor

Revo flavor

Instruments

Lumatone

Skip fretting

Music

Navigation menu

Search