77edo: Difference between revisions

← Older edit

VisualWikitext

Latest revision as of 09:38, 2 July 2025

← 76edo

77edo

78edo →

Prime factorization

7 × 11

Step size

15.5844 ¢

Fifth

45\77 (701.299 ¢)

Semitones (A1:m2)

7:6 (109.1 ¢ : 93.51 ¢)

Consistency limit

9

Distinct consistency limit

9

77 equal divisions of the octave (abbreviated 77edo or 77ed2), also called 77-tone equal temperament (77tet) or 77 equal temperament (77et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 77 equal parts of about 15.6 ¢ each. Each step represents a frequency ratio of 2^1/77, or the 77th root of 2.

Theory

With harmonic 3 less than a cent flat, harmonic 5 a bit over three cents sharp and 7 less flat than that, 77edo represents an excellent tuning choice for both valentine (hence also Carlos Alpha), the 31 & 46 temperament, and starling, the rank-3 temperament tempering out 126/125, giving the optimal patent val for 11-limit valentine and its 13-limit extension valentino, as well as 11-limit starling and oxpecker temperaments. For desirers of purer/more convincing harmonies of 19, it's also a great choice for nestoria (the extension of schismic to prime 19) so that ~16:19:24 can be heard to concord in isolation. It also gives the optimal patent val for grackle and various members of the unicorn family, with a generator of 4\77 instead of the 5\77 (which gives valentine); it is a very good choice for full-subgroup unicorn. These are 7-limit alicorn and 11- and 13-limit camahueto.

77et tempers out the schisma (32805/32768) in the 5-limit; 126/125, 1029/1024, and 6144/6125 in the 7-limit; 121/120, 176/175, 385/384, and 441/440 in the 11-limit; and 196/195, 351/350, 352/351, 676/675 and 729/728 in the 13-limit.

The 17 and 19 are tuned fairly well, making it consistent to the no-11 21-odd-limit. The equal temperament tempers out 256/255 in the 17-limit; and 171/170, 361/360, 513/512, and 1216/1215 in the 19-limit.

It also does surprisingly well (for its size) in a large range of very high odd-limits (41 to 125 range).

Prime harmonics

Approximation of prime harmonics in 77edo
Harmonic		2	3	5	7	11	13	17	19	23
Error	Absolute (¢)	+0.00	-0.66	+3.30	-2.59	-5.86	+1.03	+4.14	-1.41	-4.90
Error	Relative (%)	+0.0	-4.2	+21.2	-16.6	-37.6	+6.6	+26.5	-9.0	-31.4
Steps (reduced)		77 (0)	122 (45)	179 (25)	216 (62)	266 (35)	285 (54)	315 (7)	327 (19)	348 (40)

Approximation of prime harmonics in 77edo (continued)
Harmonic		29	31	37	41	43	47	53	59	61	67
Error	Absolute (¢)	-1.01	-7.37	-1.99	+7.30	+2.77	+4.62	-0.78	+0.57	+5.19	-1.38
Error	Relative (%)	-6.5	-47.3	-12.8	+46.8	+17.8	+29.7	-5.0	+3.6	+33.3	-8.9
Steps (reduced)		374 (66)	381 (73)	401 (16)	413 (28)	418 (33)	428 (43)	441 (56)	453 (68)	457 (72)	467 (5)

Subsets and supersets

Since 77 factors into primes as 7 × 11, 77edo contains 7edo and 11edo as subset edos.

Intervals

#	Cents	Approximate ratios*	Ups and downs notation
0	0.0	1/1	D
1	15.6	81/80, 91/90, 99/98, 105/104	^D, ^^E♭♭
2	31.2	49/48, 55/54, 64/63, 65/64, 100/99	^^D, ^³E♭♭
3	46.8	33/32, 36/35, 40/39, 45/44, 50/49	^³D, v³E♭
4	62.3	26/25, 27/26, 28/27	v³D♯, vvE♭
5	77.9	21/20, 22/21, 25/24	vvD♯, vE♭
6	93.5	18/17, 19/18, 20/19	vD♯, E♭
7	109.1	16/15, 17/16	D♯, ^E♭
8	124.7	14/13, 15/14	^D♯, ^^E♭
9	140.3	13/12	^^D♯, ^³E♭
10	155.8	11/10, 12/11	^³D♯, v³E
11	171.4	21/19	v³D𝄪, vvE
12	187.0	10/9	vvD𝄪, vE
13	202.6	9/8	E
14	218.2	17/15	^E, ^^F♭
15	233.8	8/7	^^E, ^³F♭
16	249.4	15/13, 22/19	^³E, v³F
17	264.9	7/6	v³E♯, vvF
18	280.5	20/17	vvE♯, vF
19	296.1	13/11, 19/16, 32/27	F
20	311.7	6/5	^F, ^^G♭♭
21	327.3	98/81	^^F, ^³G♭♭
22	342.9	11/9, 17/14	^³F, v³G♭
23	358.4	16/13, 21/17	v³F♯, vvG♭
24	374.0	26/21, 56/45	vvF♯, vG♭
25	389.6	5/4	vF♯, G♭
26	405.2	19/15, 24/19, 33/26	F♯, ^G♭
27	420.8	14/11, 32/25	^F♯, ^^G♭
28	436.4	9/7	^^F♯, ^³G♭
29	451.9	13/10	^³F♯, v³G
30	467.5	17/13, 21/16	v³F𝄪, vvG
31	483.1	120/91	vvF𝄪, vG
32	498.7	4/3	G
33	514.3	27/20	^G, ^^A♭♭
34	529.9	19/14	^^G, ^³A♭♭
35	545.5	11/8, 15/11, 26/19	^³G, v³A♭
36	561.0	18/13	v³G♯, vvA♭
37	576.6	7/5	vvG♯, vA♭
38	592.2	24/17, 38/27, 45/32	vG♯, A♭
…	…	…

* As a 19-limit temperament

Notation

Ups and downs notation

77edo can be notated using ups and downs. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
Sharp symbol
Flat symbol

Alternatively, sharps and flats with arrows borrowed from Helmholtz–Ellis notation can be used:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
Sharp symbol
Flat symbol

Sagittal notation

Evo flavor

Revo flavor

Approximation to JI

Zeta peak index

Tuning			Strength				Octave (cents)		Integer limit
ZPI	Steps per 8ve	Step size (cents)	Height		Integral	Gap	Size	Stretch	Consistent	Distinct
ZPI	Steps per 8ve	Step size (cents)	Tempered	Pure	Integral	Gap	Size	Stretch	Consistent	Distinct
414zpi	76.991854	15.586065	8.194847	8.145298	1.311364	17.029289	1200.126969	0.126969	10	10

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[-122 77⟩	[⟨77 122]]	+0.207	0.207	1.33
2.3.5	32805/32768, 1594323/1562500	[⟨77 122 179]]	−0.336	0.785	5.04
2.3.5.7	126/125, 1029/1024, 10976/10935	[⟨77 122 179 216]]	−0.021	0.872	5.59
2.3.5.7.11	121/120, 126/125, 176/175, 10976/10935	[⟨77 122 179 216 266]]	+0.322	1.039	6.66
2.3.5.7.11.13	121/120, 126/125, 176/175, 196/195, 676/675	[⟨77 122 179 216 266 285]]	+0.222	0.974	6.25

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperament
1	4\77	62.3	28/27	Unicorn / alicorn (77e) / camahueto (77) / qilin (77)
1	5\77	77.9	21/20	Valentine
1	9\77	140.3	13/12	Tsaharuk
1	15\77	233.8	8/7	Guiron
1	16\77	249.4	15/13	Hemischis (77e)
1	20\77	311.7	6/5	Oolong
1	23\77	358.4	16/13	Restles
1	31\77	483.1	45/34	Hemiseven
1	32\77	498.7	4/3	Grackle
1	34\77	529.9	512/375	Tuskaloosa / muscogee
1	36\77	561.0	18/13	Demivalentine
7	32\77 (1\77)	498.7 (15.6)	4/3 (81/80)	Absurdity
11	32\77 (3\77)	498.7 (46.8)	4/3 (36/35)	Hendecatonic

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Instruments

Skip fretting

Skip fretting system 77 9 11 is a skip fretting system that tunes strings 11\77 apart, with frets placed at intervals of 9\77, or 8.555...-edo. All examples on this page are for 7-string guitar.

Intervals

0\77=1/1: string 2 open

77\77=2/1: string 7 fret 11

45\77=3/2: string 2 fret 5

25\77=5/4: string 1 fret 4

62\77=7/4: string 6 fret 2

35\77=11/8: string 4 fret 10

54\77=13/8: string 2 fret 6

7\77=17/16: string 1 fret 2

19\77=19/16: string 5 fret 7

40\77=23/16: string 4 fret 2

Chords

x00030x: Neutral 9th (saj6, v5)

Keyboards

Lumatone mappings for 77edo are available.

Music

[[Bryan Deister]

microtonal improvisation in 77edo (2025)

Jake Freivald

A Seed Planted^{[dead link]}, in an organ version of Claudi Meneghin.

Joel Grant Taylor

Star 1-GrimaldiA+Bmod

Chris Vaisvil

77et Star

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro}}
+{{ED intro}}
 == Theory ==
-With [[3/1|harmonic 3]] less than a cent flat, [[5/1|harmonic 5]] a bit over three cents sharp and [[7/1|7]]'s less flat than that, 77edo represents an excellent tuning choice for both [[valentine]], the {{nowrap|31 &amp; 46}} temperament, and [[starling]], the [[126/125]] [[planar temperament]], giving the [[optimal patent val]] for [[11-limit]] valentine and its [[13-limit]] extensions dwynwen and valentino, as well as 11-limit starling and [[oxpecker]] temperaments. It also gives the optimal patent val for [[grackle]] and various members of the [[unicorn family]], with a [[generator]] of 4\77 instead of the 5\77 (which gives valentine). These are 7-limit [[Unicorn family #Alicorn|alicorn]] and 11- and 13-limit [[Unicorn family #Camahueto|camahueto]].
+With [[3/1|harmonic 3]] less than a cent flat, [[5/1|harmonic 5]] a bit over three cents sharp and [[7/1|7]] less flat than that, 77edo represents an excellent tuning choice for both [[valentine]] (hence also [[Carlos Alpha]]), the {{nowrap|31 & 46}} temperament, and [[starling]], the [[rank-3 temperament]] [[tempering out]] [[126/125]], giving the [[optimal patent val]] for [[11-limit]] valentine and its [[13-limit]] extension [[valentino]], as well as 11-limit starling and [[oxpecker]] temperaments. For desirers of purer/more convincing harmonies of 19, it's also a great choice for [[nestoria]] (the extension of schismic to prime 19) so that ~16:19:24 can be heard to concord in isolation. It also gives the optimal patent val for [[grackle]] and various members of the [[unicorn family]], with a [[generator]] of 4\77 instead of the 5\77 (which gives valentine); it is a very good choice for full-subgroup [[unicorn]]. These are 7-limit [[unicorn family #Alicorn|alicorn]] and 11- and 13-limit [[unicorn family #Camahueto|camahueto]].
-et tempers out [[32805/32768]] in the [[5-limit]], [[126/125]], [[1029/1024]], and [[6144/6125]] in the 7-limit, [[121/120]], [[176/175]], [[385/384]], and [[441/440]] in the 11-limit, and [[196/195]], [[351/350]], [[352/351]], [[676/675]] and [[729/728]] in the 13-limit.
+et tempers out the [[schisma]] (32805/32768) in the [[5-limit]]; [[126/125]], [[1029/1024]], and [[6144/6125]] in the 7-limit; [[121/120]], [[176/175]], [[385/384]], and [[441/440]] in the 11-limit; and [[196/195]], [[351/350]], [[352/351]], [[676/675]] and [[729/728]] in the 13-limit.
 The [[17/1|17]] and [[19/1|19]] are tuned fairly well, making it [[consistent]] to the no-11 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]], and [[1216/1215]] in the 19-limit.
-edo is an excellent edo for [[Carlos Alpha]], since the difference between 5 steps of 77edo and 1 step of Carlos Alpha is only −0.042912 cents.
+It also does surprisingly well (for its size) in a large range of very high odd-limits (41 to 125 range).
 === Prime harmonics ===
 {{Harmonics in equal|77|columns=9}}
-{{Harmonics in equal|77|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 77edo (continued)}}
+{{Harmonics in equal|77|columns=10|start=10|collapsed=true|title=Approximation of prime harmonics in 77edo (continued)}}
 === Subsets and supersets ===
-Since 77 factors into primes as {{nowrap| 7 × 11 }}, 77edo contains [[7edo]] and [[11edo]] as subset edos.
+Since 77 factors into primes as {{nowrap|7 × 11}}, 77edo contains [[7edo]] and [[11edo]] as subset edos.
 == Intervals ==
@@ Line 228: / Line 228: @@
 == Notation ==
+=== Ups and downs notation ===
+edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
+{{Sharpness-sharp7a}}
+Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
+{{Sharpness-sharp7}}
 === Sagittal notation ===
 ==== Evo flavor ====
@@ Line 252: / Line 261: @@
 default [[File:77-EDO_Revo_Sagittal.svg]]
 </imagemap>
-=== Ups and downs notation ===
-Using [[Helmholtz–Ellis]] accidentals, 77edo can be notated using [[ups and downs notation]]:
-{{Sharpness-sharp7}}
 == Approximation to JI ==
 === Zeta peak index ===
-{| class="wikitable center-all"
+{{ZPI
-|-
+| zpi = 414
-! colspan="3" | Tuning
+| steps = 76.9918536925042
-! colspan="3" | Strength
+| step size = 15.5860645308353
-! colspan="2" | Closest edo
+| tempered height = 8.194847
-! colspan="2" | Integer limit
+| pure height = 8.145298
-|-
+| integral = 1.311364
-! ZPI
+| gap = 17.029289
-! Steps per octave
+| octave = 1200.12696887432
-! Step size (cents)
+| consistent = 10
-! Height
+| distinct = 10
-! Integral
+}}
-! Gap
-! Edo
-! Octave (cents)
-! Consistent
-! Distinct
-|-
-| [[414zpi]]
-| 76.9918536925042
-| 15.5860645308353
-| 8.194847
-| 1.311364
-| 17.029289
-| 77edo
-| 1200.12696887432
-| 10
-| 10
-|}
 == Regular temperament properties ==
@@ Line 427: / Line 414: @@
 |}
 <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+== Instruments ==
+=== Skip fretting ===
+'''Skip fretting system 77 9 11''' is a [[skip fretting]] system that tunes strings 11\77 apart, with frets placed at intervals of 9\77, or 8.555...-edo. All examples on this page are for 7-string [[guitar]].
+; Intervals
+\77=1/1: string 2 open
+\77=2/1: string 7 fret 11
+\77=3/2: string 2 fret 5
+\77=5/4: string 1 fret 4
+\77=7/4: string 6 fret 2
+\77=11/8: string 4 fret 10
+\77=13/8: string 2 fret 6
+\77=17/16: string 1 fret 2
+\77=19/16: string 5 fret 7
+\77=23/16: string 4 fret 2
+; Chords
+x00030x: Neutral 9th (saj6, v5)
+=== Keyboards ===
+[[Lumatone mapping for 77edo|Lumatone mappings for 77edo]] are available.
 == Music ==
+; [[Bryan Deister]
+* [https://www.youtube.com/shorts/wSZez2KgP2U ''microtonal improvisation in 77edo''] (2025)
 ; [[Jake Freivald]]
 * [http://soonlabel.com/xenharmonic/wp-content/uploads/2013/10/Freivald-J.-A-Seed-Planted-2nd-Version-77edo.mp3 ''A Seed Planted'']{{dead link}}, in an [https://web.archive.org/web/20190412162407/http://soonlabel.com/xenharmonic/archives/1391 organ version] of [[Claudi Meneghin]].

77edo: Difference between revisions

Latest revision as of 09:38, 2 July 2025

Contents