76edo: Difference between revisions

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The ''76 equal division'' 5-limit [[patent val]] is [[contorted|contorted]] in the 5-limit, reflecting the fact that 76 = 4 * 19. In the 7-limit it tempers out 2401/2400 as well as 81/80, and so supports [[Meantone_family#Squares|squares temperament]]. In the 11-limit, it tempers out 245/242 and 385/384, and supports the 24&26 temperament. In the 13-limit, it tempers out 105/104, 144/143, 351/350 and 364/363. While the 44\76 = 11\19 fifth is already flat, the 43\76 fifth, even flatter, is an almost perfect approximation to the [[Pelogic_family|hornbostel temperament]] POTE fifth, whereas its sharp fifth, 45\76, makes for an excellent [[Archytas_clan#Superpyth|superpyth]] fifth. Hence you can do hornbostel/mavila, squares/meantone, and superpyth all with the same equal division.

Using ~~non-patent vals~~, 76edo provides an excellent tuning for [[~~Teff|~~teff]] temperament, a low complexity, medium accuracy, and high limit (17 or 19) temperament.

== Theory ==

76edo's [[patent val]] is [[contorted]] in the 5-limit, reflecting the fact that {{nowrap|76 {{=}} 4 × 19}}. In the 7-limit it [[tempering out|tempers out]] [[2401/2400]] in addition to [[81/80]], and so [[support]]s the [[squares]] temperament. In the 11-limit, it tempers out [[245/242]] and [[385/384]], and supports [[pombe]], the {{nowrap|24 & 26}} temperament. In the 13-limit, it tempers out 105/104, 144/143, 351/350 and 364/363. While the {{nowrap|44\76 {{=}} 11\19}} fifth is already flat, the 43\76 fifth, even flatter, is an almost perfect approximation to the [[hornbostel]] temperament's [[POTE]] fifth, whereas its sharp fifth, 45\76, makes for an excellent [[superpyth]] fifth. Hence you can do hornbostel/mavila, squares/meantone, and superpyth all with the same equal division.

Using the 76dgh val, 76edo provides an excellent tuning for [[teff]] temperament, a low-complexity, medium-accuracy, and high-limit (17- or 19-limit) temperament.

=== Odd harmonics ===

=== Subsets and supersets ===

Since 76 factors into {{factorization|76}}, 76edo has subset edos {{EDOs| 2, 4, 19, and 38 }}. [[152edo]], which doubles it, is a [[zeta peak edo]].

== Intervals ==

== Notation ==

=== Ups and downs notation ===

76edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.

Using [[Helmholtz–Ellis]] accidentals, 76edo can also be notated using [[ups and downs notation]] along with Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals:

Here, a sharp raises by four steps, and a flat lowers by four steps, so arrows can be used to fill in the gap.

=== Sagittal notation ===

This notation uses the same sagittal sequence as EDOs [[62edo#Sagittal notation|62]] and [[69edo#Sagittal notation|69]], and is a superset of the notations for EDOs [[38edo#Sagittal notation|38]] and [[19edo#Sagittal notation|19]].

==== Evo flavor ====

File:76-EDO_Evo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 575 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 170 106 [[1053/1024]]

rect 170 80 330 106 [[567/550]]

default [[File:76-EDO_Evo_Sagittal.svg]]

</imagemap>

==== Revo flavor ====

File:76-EDO_Revo_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 604 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 170 106 [[1053/1024]]

rect 170 80 330 106 [[567/550]]

default [[File:76-EDO_Revo_Sagittal.svg]]

</imagemap>

==== Evo-SZ flavor ====

File:76-EDO_Evo-SZ_Sagittal.svg

desc none

rect 80 0 300 50 [[Sagittal_notation]]

rect 300 0 553 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 170 106 [[1053/1024]]

rect 170 80 330 106 [[567/550]]

default [[File:76-EDO_Evo-SZ_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 ==

=== Skip fretting ===

'''[[Skip fretting]] system 76 2 17''' optimizes the subminor chord, 0-17-44 of 76edo.

7/6 can be found on adjacent strings, as each string is tuned 17\76 apart.

If you’re playing a 6-string with the lowest string at an arbitrary root within 76edo, the strings will be as follows, from thickest to thinnest:

0 17 34 51 68 9

Note that the thinnest string is in the next octave.

Each fret is 2\76, or 1\38. This puts the perfect 5th at the 3rd string (34\76), 5th fret (10\76). That gives us 44\76. If you want to add more notes to your chord, 59\76 (12/7) can be found on the 4th string, 4th fret. The 8ve is found on the 5th string, 4th fret. The subminor 10th is found on the 6th string, 4th fret.

A regular minor chord, with a 19\76 3rd, is also easily playable. 19\76 is found on the 2nd string, 1st fret; as well as on the 6th string, 5th fret.

=== Keyboards ===

[[Lumatone mapping for 76edo|Lumatone mappings for 76edo]] are available.

== Music ==

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/IUqoN9YhTtQ ''microtonal improvisation in 76edo''] (2025)

@@ Line 1: / Line 1: @@
-The ''76 equal division'' 5-limit [[patent val]] is [[contorted|contorted]] in the 5-limit, reflecting the fact that 76 = 4 * 19. In the 7-limit it tempers out 2401/2400 as well as 81/80, and so supports [[Meantone_family#Squares|squares temperament]]. In the 11-limit, it tempers out 245/242 and 385/384, and supports the 24&amp;26 temperament. In the 13-limit, it tempers out 105/104, 144/143, 351/350 and 364/363. While the 44\76 = 11\19 fifth is already flat, the 43\76 fifth, even flatter, is an almost perfect approximation to the [[Pelogic_family|hornbostel temperament]] POTE fifth, whereas its sharp fifth, 45\76, makes for an excellent [[Archytas_clan#Superpyth|superpyth]] fifth. Hence you can do hornbostel/mavila, squares/meantone, and superpyth all with the same equal division.
+{{Infobox ET}}
+{{ED intro}}
-Using non-patent vals, 76edo provides an excellent tuning for [[Teff|teff]] temperament, a low complexity, medium accuracy, and high limit (17 or 19) temperament.
+== Theory ==
+edo's [[patent val]] is [[contorted]] in the 5-limit, reflecting the fact that {{nowrap|76 {{=}} 4 × 19}}. In the 7-limit it [[tempering out|tempers out]] [[2401/2400]] in addition to [[81/80]], and so [[support]]s the [[squares]] temperament. In the 11-limit, it tempers out [[245/242]] and [[385/384]], and supports [[pombe]], the {{nowrap|24 &amp; 26}} temperament. In the 13-limit, it tempers out 105/104, 144/143, 351/350 and 364/363. While the {{nowrap|44\76 {{=}} 11\19}} fifth is already flat, the 43\76 fifth, even flatter, is an almost perfect approximation to the [[hornbostel]] temperament's [[POTE]] fifth, whereas its sharp fifth, 45\76, makes for an excellent [[superpyth]] fifth. Hence you can do hornbostel/mavila, squares/meantone, and superpyth all with the same equal division.
+Using the 76dgh val, 76edo provides an excellent tuning for [[teff]] temperament, a low-complexity, medium-accuracy, and high-limit (17- or 19-limit) temperament.
+=== Odd harmonics ===
+{{Harmonics in equal|76}}
+=== Subsets and supersets ===
+Since 76 factors into {{factorization|76}}, 76edo has subset edos {{EDOs| 2, 4, 19, and 38 }}. [[152edo]], which doubles it, is a [[zeta peak edo]].
+== Intervals ==
+{{Interval table}}
+== Notation ==
+=== Ups and downs notation ===
+edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
+{{Sharpness-sharp4a}}
+Using [[Helmholtz–Ellis]] accidentals, 76edo can also be notated using [[ups and downs notation]] along with Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals:
+{{Sharpness-sharp4}}
+Here, a sharp raises by four steps, and a flat lowers by four steps, so arrows can be used to fill in the gap.
+=== Sagittal notation ===
+This notation uses the same sagittal sequence as EDOs [[62edo#Sagittal notation|62]] and [[69edo#Sagittal notation|69]], and is a superset of the notations for EDOs [[38edo#Sagittal notation|38]] and [[19edo#Sagittal notation|19]].
+==== Evo flavor ====
+<imagemap>
+File:76-EDO_Evo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 575 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 170 106 [[1053/1024]]
+rect 170 80 330 106 [[567/550]]
+default [[File:76-EDO_Evo_Sagittal.svg]]
+</imagemap>
+==== Revo flavor ====
+<imagemap>
+File:76-EDO_Revo_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 604 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 170 106 [[1053/1024]]
+rect 170 80 330 106 [[567/550]]
+default [[File:76-EDO_Revo_Sagittal.svg]]
+</imagemap>
+==== Evo-SZ flavor ====
+<imagemap>
+File:76-EDO_Evo-SZ_Sagittal.svg
+desc none
+rect 80 0 300 50 [[Sagittal_notation]]
+rect 300 0 553 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
+rect 20 80 170 106 [[1053/1024]]
+rect 170 80 330 106 [[567/550]]
+default [[File:76-EDO_Evo-SZ_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 ==
+=== Skip fretting ===
+'''[[Skip fretting]] system 76 2 17''' optimizes the subminor chord, 0-17-44 of 76edo.
+/6 can be found on adjacent strings, as each string is tuned 17\76 apart.
+If you’re playing a 6-string with the lowest string at an arbitrary root within 76edo, the strings will be as follows, from thickest to thinnest:
+17 34 51 68 9
+Note that the thinnest string is in the next octave.
+Each fret is 2\76, or 1\38. This puts the perfect 5th at the 3rd string (34\76), 5th fret (10\76). That gives us 44\76. If you want to add more notes to your chord, 59\76 (12/7) can be found on the 4th string, 4th fret. The 8ve is found on the 5th string, 4th fret. The subminor 10th is found on the 6th string, 4th fret.
+A regular minor chord, with a 19\76 3rd, is also easily playable. 19\76 is found on the 2nd string, 1st fret; as well as on the 6th string, 5th fret.
+=== Keyboards ===
+[[Lumatone mapping for 76edo|Lumatone mappings for 76edo]] are available.
+== Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/IUqoN9YhTtQ ''microtonal improvisation in 76edo''] (2025)