94edo: Difference between revisions

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Its step size is close to that of [[144/143]], which is consistently represented in this tuning system.

94edo can be thought of as two sets of [[47edo]] offset by one step of 94edo. It inherits from 47edo's good approximations of primes 5, 7, 13, and 17, while it dramatically improves on prime 3, as well as primes 11, 19, and 23 to a lesser degree.

=== As a tuning of other temperaments ===

=== ~~Significance~~ of ~~cassandra~~ ===

94edo can also be thought of as the "sum" of [[41edo]] and [[53edo]] {{nowrap|(41 + 53 {{=}} 94)}}, both of which are not only known for their approximation of [[Pythagorean tuning]], but also support a variety of [[Schismatic family|schismatic temperament]] known as [[Schismatic family#Cassandra|cassandra]] (which is itself a variety of [[Schismatic family#Garibaldi|garibaldi]]), tempering out [[32805/32768]], [[225/224]], and [[385/384]]. Therefore, 94edo's fifth is the [[mediant]] of these two edos' fifths; it is slightly sharp of just and less accurate than 53edo's fifth, but more accurate than 41edo's, and acts as a generator for a highly optimized and high-prime-limit form of cassandra. Few, if any, edos that support schismatic by [[Val|patent val]] have at least as high of a consistency limit as 94edo while also having a fifth that can stack to reach any interval in it.

~~=== Other temperament properties ===~~

The list of 23-limit commas it tempers out is huge, and in lower prime limits, it also tempers out [[3125/3087]], [[4000/3969]], [[5120/5103]] and [[540/539]]. It provides the [[optimal patent val]] for gassormic, the rank-5 temperament tempering out [[275/273]] (despite one edostep being very close in size to this comma), and for a number of other temperaments, such as [[isis]].

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=== Prime harmonics ===

=== Subsets and supersets ===

Since 94 factors into primes as {{nowrap| 2 × 47 }}, 94edo contains [[2edo]] and [[47edo]] as subset edos. It can be thought of as two sets of 47edo offset by one step of 94edo. It inherits from 47edo's good approximations of primes 5, 7, 13, and 17, while dramatically improving on prime 3, as well as primes 11, 19, and 23 to a lesser degree.

== Intervals ==

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There are perhaps nine functional minor thirds varying between 242.553 cents and 344.681 cents, and one can even go beyond those boundaries under the right conditions, so musicians playing in 94edo have a lot more flexibility in terms of the particular interval shadings they might use depending on context.

The perfect fifth has three, or perhaps even five, functional options, each differing by one step. ~~Although in most timbres only the central perfect fifth at 702.128 cents sounds consonant and stable, the~~ lower and higher variants provide a change in interval quality, and can be helpful in creating subsets which mimic other edos, and close the circle of fifths in different numbers of pitches. For example, a close approximation to 41edo can be made using a chain of forty 702.128 cent fifths and one wide fifth at 714.894 cents, with an improvement on the tuning of most simple consonances in close keys, but a 1-step variation in interval quality as one modulates to more distant keys.

The perfect fifth has three, or perhaps even five, functional options, each differing by one step. The lower and higher variants provide a change in interval quality, and can be helpful in creating subsets which mimic other edos, and close the circle of fifths in different numbers of pitches. For example, a close approximation to 41edo can be made using a chain of forty 702.128 cent fifths and one wide fifth at 714.894 cents, with an improvement on the tuning of most simple consonances in close keys, but a 1-step variation in interval quality as one modulates to more distant keys.

Every odd-numbered interval can generate the entire tuning of 94edo except for the 600-cent [[tritone]] (47\94), which divides the octave exactly in half.

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While having the whole gamut of 94 intervals available on a keyboard or other instrument would be quite a feat, one can get a lot out of a 41-tone chain of fifths (with the odd fifth one degree wide) or a 53-tone chain of fifths (with the odd fifth one degree narrow), where the subset behaves much like a well-temperament, arguably usable in all keys but with some interval size variation between closer and more distant keys.

== ~~Approximation to JI~~ ==

== Notation ==

==~~= Zeta peak index ===~~

94edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal_notation#Athenian_extension_single-shaft|Athenian extension]], with the apotome equating to 9 edosteps and the limma to 7 edosteps.

{{~~ZPI~~

{| class="wikitable" style="text-align: center;"

| ~~zpi~~ = ~~532~~

!Degree

| ~~steps~~ = ~~93.9836761074943~~

!−9

| ~~step size~~ = ~~12.7681747480009~~

!−8

| ~~tempered height~~ = ~~8.806201~~

!−7

| ~~pure height~~ = ~~8.585151~~

!−6

| ~~integral~~ = ~~1.394050~~

!−5

| ~~gap~~ = ~~17.832744~~

!−4

| ~~octave~~ = ~~1200.20842631208~~

!−3

| ~~consistent~~ = 24

!−2

| ~~distinct = 15~~

!−1

}}

!0

!+1

!+2

!+3

!+4

!+5

!+6

!+7

!+8

!+9

|-

!Evo

|{{sagittal|b}}

|{{sagittal|b}}{{sagittal|~|(}}

|{{sagittal|b}}{{sagittal|/|}}

|{{sagittal|b}}{{sagittal|(|(}}

|{{sagittal|b}}{{sagittal|/|\}}

| rowspan="2" |{{sagittal|\!/}}

| rowspan="2" |{{sagittal|(!(}}

| rowspan="2" |{{sagittal|\!}}

| rowspan="2" |{{sagittal|~!(}}

| rowspan="2" |{{sagittal||//|}}

| rowspan="2" |{{sagittal|~|(}}

| rowspan="2" |{{sagittal|/|}}

| rowspan="2" |{{sagittal|(|(}}

| rowspan="2" |{{sagittal|/|\}}

|{{sagittal|#}}{{sagittal|\!/}}

|{{sagittal|#}}{{sagittal|(!(}}

|{{sagittal|#}}{{sagittal|\!}}

|{{sagittal|#}}{{sagittal|~!(}}

|{{sagittal|#}}

|-

!Revo

|{{sagittal|\!!/}}

|{{sagittal|(!!(}}

|{{sagittal|!!/}}

|{{sagittal|~!!(}}

|{{sagittal|(!)}}

|{{sagittal|(|)}}

|{{sagittal|~||(}}

|{{sagittal|||\}}

|{{sagittal|(||(}}

|{{sagittal|/||\}}

|}

== Regular temperament properties ==

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Below are some 23-limit temperaments supported by 94et. It might be noted that 94, a very good tuning for [[garibaldi temperament]], shows us how to extend it to the 23-limit.

* {{nowrap|46 & 94~~}} {{multival| 8 30 -18 -4 -28 8 -24 2 …~~ }}

* {{nowrap|46 & 94}}

* {{nowrap|68 & 94~~}} {{multival| 20 28 2 -10 24 20 34 52 …~~ }}

* {{nowrap|68 & 94}}

* {{nowrap|53 & 94}} ~~{{multival| 1 -8 -14 23 20 -46 -3 -35 … }}~~ (one garibaldi)

* {{nowrap|53 & 94}} (one garibaldi)

* {{nowrap|41 & 94}} ~~{{multival| 1 -8 -14 23 20 48 -3 -35 … }}~~ (another garibaldi, only differing in the mappings of 17 and 23)

* {{nowrap|41 & 94}} (another garibaldi, only differing in the mappings of 17 and 23)

* {{nowrap|135 & 94}} ~~{{multival| 1 -8 -14 23 20 48 -3 59 … }}~~ (another garibaldi)

* {{nowrap|135 & 94}} (another garibaldi)

* {{nowrap|130 & 94}} ~~{{multival| 6 -48 10 -50 26 6 -18 -22 … }}~~ (a pogo extension)

* {{nowrap|130 & 94}} (a pogo extension)

* {{nowrap|58 & 94}} ~~{{multival| 6 46 10 44 26 6 -18 -22 … }}~~ (a supers extension)

* {{nowrap|58 & 94}} (a supers extension)

* {{nowrap|50 & 94~~}} {{multival| 24 -4 40 -12 10 24 22 6 …~~ }}

* {{nowrap|50 & 94}}

* {{nowrap|72 & 94}} ~~{{multival| 12 -2 20 -6 52 12 -36 -44 … }}~~ (a gizzard extension)

* {{nowrap|72 & 94}} (a gizzard extension)

* {{nowrap|80 & 94~~}} {{multival| 18 44 30 38 -16 18 40 28 …~~ }}

* {{nowrap|80 & 94}}

* 94 solo ~~{{multival| 12 -2 20 -6 -42 12 -36 -44 … }}~~ (a rank one temperament!)

* 94 solo (a rank one temperament!)

Temperaments to which 94et can be detempered:

* [[Satin]] ({{nowrap|94 & 311}}) ~~{{multival| 3 70 -42 69 -34 50 85 83 … }}~~

* [[Satin]] ({{nowrap|94 & 311}})

* {{nowrap|94 & 422~~}} {{multival| 8 124 -18 90 -28 102 164 96 …~~ }}

* {{nowrap|94 & 422}}

== Scales ==

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== Music ==

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/Zx4xbJhXmgc ''microtonal improvisation in 94edo''] (2025)

; [[Cam Taylor]]

* [https://archive.org/details/41-94edo09sept2017 4 Improvisations Saturday 9th September 2017]

@@ Line 7: / Line 7: @@
 Its step size is close to that of [[144/143]], which is consistently represented in this tuning system.
-edo can be thought of as two sets of [[47edo]] offset by one step of 94edo. It inherits from 47edo's good approximations of primes 5, 7, 13, and 17, while it dramatically improves on prime 3, as well as primes 11, 19, and 23 to a lesser degree.
+=== As a tuning of other temperaments ===
-=== Significance of cassandra ===
 edo can also be thought of as the "sum" of [[41edo]] and [[53edo]] {{nowrap|(41 + 53 {{=}} 94)}}, both of which are not only known for their approximation of [[Pythagorean tuning]], but also support a variety of [[Schismatic family|schismatic temperament]] known as [[Schismatic family#Cassandra|cassandra]] (which is itself a variety of [[Schismatic family#Garibaldi|garibaldi]]), tempering out [[32805/32768]], [[225/224]], and [[385/384]]. Therefore, 94edo's fifth is the [[mediant]] of these two edos' fifths; it is slightly sharp of just and less accurate than 53edo's fifth, but more accurate than 41edo's, and acts as a generator for a highly optimized and high-prime-limit form of cassandra. Few, if any, edos that support schismatic by [[Val|patent val]] have at least as high of a consistency limit as 94edo while also having a fifth that can stack to reach any interval in it.
-=== Other temperament properties ===
 The list of 23-limit commas it tempers out is huge, and in lower prime limits, it also tempers out [[3125/3087]], [[4000/3969]], [[5120/5103]] and [[540/539]]. It provides the [[optimal patent val]] for gassormic, the rank-5 temperament tempering out [[275/273]] (despite one edostep being very close in size to this comma), and for a number of other temperaments, such as [[isis]].
@@ Line 19: / Line 16: @@
 === Prime harmonics ===
 {{Harmonics in equal|94|columns=11}}
+=== Subsets and supersets ===
+Since 94 factors into primes as {{nowrap| 2 × 47 }}, 94edo contains [[2edo]] and [[47edo]] as subset edos. It can be thought of as two sets of 47edo offset by one step of 94edo. It inherits from 47edo's good approximations of primes 5, 7, 13, and 17, while dramatically improving on prime 3, as well as primes 11, 19, and 23 to a lesser degree.
 == Intervals ==
@@ Line 894: / Line 894: @@
 There are perhaps nine functional minor thirds varying between 242.553 cents and 344.681 cents, and one can even go beyond those boundaries under the right conditions, so musicians playing in 94edo have a lot more flexibility in terms of the particular interval shadings they might use depending on context.
-The perfect fifth has three, or perhaps even five, functional options, each differing by one step. Although in most timbres only the central perfect fifth at 702.128 cents sounds consonant and stable, the lower and higher variants provide a change in interval quality, and can be helpful in creating subsets which mimic other edos, and close the circle of fifths in different numbers of pitches. For example, a close approximation to 41edo can be made using a chain of forty 702.128 cent fifths and one wide fifth at 714.894 cents, with an improvement on the tuning of most simple consonances in close keys, but a 1-step variation in interval quality as one modulates to more distant keys.
+The perfect fifth has three, or perhaps even five, functional options, each differing by one step. The lower and higher variants provide a change in interval quality, and can be helpful in creating subsets which mimic other edos, and close the circle of fifths in different numbers of pitches. For example, a close approximation to 41edo can be made using a chain of forty 702.128 cent fifths and one wide fifth at 714.894 cents, with an improvement on the tuning of most simple consonances in close keys, but a 1-step variation in interval quality as one modulates to more distant keys.
 Every odd-numbered interval can generate the entire tuning of 94edo except for the 600-cent [[tritone]] (47\94), which divides the octave exactly in half.
@@ Line 902: / Line 902: @@
 While having the whole gamut of 94 intervals available on a keyboard or other instrument would be quite a feat, one can get a lot out of a 41-tone chain of fifths (with the odd fifth one degree wide) or a 53-tone chain of fifths (with the odd fifth one degree narrow), where the subset behaves much like a well-temperament, arguably usable in all keys but with some interval size variation between closer and more distant keys.
-== Approximation to JI ==
+== Notation ==
-=== Zeta peak index ===
+edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal_notation#Athenian_extension_single-shaft|Athenian extension]], with the apotome equating to 9 edosteps and the limma to 7 edosteps.
-{{ZPI
+{| class="wikitable" style="text-align: center;"
-| zpi = 532
+!Degree
-| steps = 93.9836761074943
+!−9
-| step size = 12.7681747480009
+!−8
-| tempered height = 8.806201
+!−7
-| pure height = 8.585151
+!−6
-| integral = 1.394050
+!−5
-| gap = 17.832744
+!−4
-| octave = 1200.20842631208
+!−3
-| consistent = 24
+!−2
-| distinct = 15
+!−1
-}}
+!0
+!+1
+!+2
+!+3
+!+4
+!+5
+!+6
+!+7
+!+8
+!+9
+|-
+!Evo
+|{{sagittal|b}}
+|{{sagittal|b}}{{sagittal|~|(}}
+|{{sagittal|b}}{{sagittal|/|}}
+|{{sagittal|b}}{{sagittal|(|(}}
+|{{sagittal|b}}{{sagittal|/|\}}
+| rowspan="2" |{{sagittal|\!/}}
+| rowspan="2" |{{sagittal|(!(}}
+| rowspan="2" |{{sagittal|\!}}
+| rowspan="2" |{{sagittal|~!(}}
+| rowspan="2" |{{sagittal||//|}}
+| rowspan="2" |{{sagittal|~|(}}
+| rowspan="2" |{{sagittal|/|}}
+| rowspan="2" |{{sagittal|(|(}}
+| rowspan="2" |{{sagittal|/|\}}
+|{{sagittal|#}}{{sagittal|\!/}}
+|{{sagittal|#}}{{sagittal|(!(}}
+|{{sagittal|#}}{{sagittal|\!}}
+|{{sagittal|#}}{{sagittal|~!(}}
+|{{sagittal|#}}
+|-
+!Revo
+|{{sagittal|\!!/}}
+|{{sagittal|(!!(}}
+|{{sagittal|!!/}}
+|{{sagittal|~!!(}}
+|{{sagittal|(!)}}
+|{{sagittal|(|)}}
+|{{sagittal|~||(}}
+|{{sagittal|||\}}
+|{{sagittal|(||(}}
+|{{sagittal|/||\}}
+|}
 == Regular temperament properties ==
@@ Line 1,067: / Line 1,110: @@
 Below are some 23-limit temperaments supported by 94et. It might be noted that 94, a very good tuning for [[garibaldi temperament]], shows us how to extend it to the 23-limit.
-* {{nowrap|46 &amp; 94}} {{multival| 8 30 -18 -4 -28 8 -24 2 … }}
+* {{nowrap|46 &amp; 94}}
-* {{nowrap|68 &amp; 94}} {{multival| 20 28 2 -10 24 20 34 52 … }}
+* {{nowrap|68 &amp; 94}}
-* {{nowrap|53 &amp; 94}} {{multival| 1 -8 -14 23 20 -46 -3 -35 … }} (one garibaldi)
+* {{nowrap|53 &amp; 94}}  (one garibaldi)
-* {{nowrap|41 &amp; 94}} {{multival| 1 -8 -14 23 20 48 -3 -35 … }} (another garibaldi, only differing in the mappings of 17 and 23)
+* {{nowrap|41 &amp; 94}}  (another garibaldi, only differing in the mappings of 17 and 23)
-* {{nowrap|135 &amp; 94}} {{multival| 1 -8 -14 23 20 48 -3 59 … }} (another garibaldi)
+* {{nowrap|135 &amp; 94}}  (another garibaldi)
-* {{nowrap|130 &amp; 94}} {{multival| 6 -48 10 -50 26 6 -18 -22 … }} (a pogo extension)
+* {{nowrap|130 &amp; 94}}  (a pogo extension)
-* {{nowrap|58 &amp; 94}} {{multival| 6 46 10 44 26 6 -18 -22 … }} (a supers extension)
+* {{nowrap|58 &amp; 94}}  (a supers extension)
-* {{nowrap|50 &amp; 94}} {{multival| 24 -4 40 -12 10 24 22 6 … }}
+* {{nowrap|50 &amp; 94}}
-* {{nowrap|72 &amp; 94}} {{multival| 12 -2 20 -6 52 12 -36 -44 … }} (a gizzard extension)
+* {{nowrap|72 &amp; 94}}  (a gizzard extension)
-* {{nowrap|80 &amp; 94}} {{multival| 18 44 30 38 -16 18 40 28 … }}
+* {{nowrap|80 &amp; 94}}
-* 94 solo {{multival| 12 -2 20 -6 -42 12 -36 -44 … }} (a rank one temperament!)
+* 94 solo  (a rank one temperament!)
 Temperaments to which 94et can be detempered:
-* [[Satin]] ({{nowrap|94 & 311}}) {{multival| 3 70 -42 69 -34 50 85 83 … }}
+* [[Satin]] ({{nowrap|94 & 311}})
-* {{nowrap|94 & 422}} {{multival| 8 124 -18 90 -28 102 164 96 … }}
+* {{nowrap|94 & 422}}
 == Scales ==
@@ Line 1,098: / Line 1,141: @@
 == Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/Zx4xbJhXmgc ''microtonal improvisation in 94edo''] (2025)
 ; [[Cam Taylor]]
 * [https://archive.org/details/41-94edo09sept2017 4 Improvisations Saturday 9th September 2017]