94edo: Difference between revisions

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== Theory ==

94edo is a remarkable all-around utility tuning system, good from low [[prime limit]] to very high prime limit situations. It is the first edo to be [[consistent]] through the [[23-odd-limit]], and no other edo is so consistent until [[282edo|282]] and [[311edo|311]] make their appearance.

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. It can also be thought of as the "sum" of [[41edo]] and [[53edo]] (41 + 53 = 94), both of which are known for their approximation of [[Pythagorean tuning]]. Therefore 94edo's fifth is the [[mediant]] of these two tunings' fifths; it is slightly sharp of just and less accurate than 53edo's fifth, but more accurate than 41edo's.

The list of 23-limit commas it tempers out is huge, but it is worth noting that it tempers out [[32805/32768]] and is thus a [[schismatic]] system, that it tempers out [[225/224]] and [[385/384]] and so is a [[marvel]] system, and that it also tempers out [[3125/3087]], [[4000/3969]], [[5120/5103]] and [[540/539]]. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[275/273]], and for a number of other temperaments, such as [[isis]].

94edo is an excellent edo for [[Carlos Beta]] scale, since the difference between ~~5 steps of 94edo and~~ 1 step of Carlos Beta is only -0.00314534 cents.

94edo is an excellent edo for [[Carlos Beta]] scale, since the difference between 1 step of Carlos Beta and 5 steps of 94edo is only 0.00314534 cents.

=== Prime harmonics ===

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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.

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.

The regular major second divisible into 16 equal parts can be helpful for realising some of the subtle tunings of Ancient Greek [[tetrachord]]al theory, [[Indian]] raga and Turkish [[maqam]], though it has not been used historically as a division in those musical cultures.

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.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

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== Scales ==

* [[Garibaldi5]]

* [[Garibaldi7]]

* [[Garibaldi12]]

* [[Garibaldi17]]

Since 94edo has a step of 12.766 cents, it also allows one to use its mos scales as circulating temperaments and is the first edo to allows one to use a ~~nohajira~~, pajara or miracle mos scale a as circulating temperament{{clarify}}.

Since 94edo has a step of 12.766 cents, it also allows one to use its [[mos]] scales as circulating temperaments and is the first edo to allows one to use a mohajira, pajara or miracle mos scale a as circulating temperament{{clarify}}.

{| class="wikitable mw-collapsible mw-collapsed"

|+ style=white-space:nowrap | Circulating temperaments in 94edo

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|}

== Music ==

; [[Cam Taylor]]

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

* [https://archive.org/details/4194EDOBosanquetAxis18thAug20181FeelingSadButWarmingUp Feeling Sad But Warming Up (in 2 parts)]

* [https://archive.org/details/4191edoPlayingWithThe13Limit Playing with the 13-limit]

[[Category:94edo| ]]

[[Category:Garibaldi]]

[[Category:Marvel]]

@@ Line 4: / Line 4: @@
 == Theory ==
 edo is a remarkable all-around utility tuning system, good from low [[prime limit]] to very high prime limit situations. It is the first edo to be [[consistent]] through the [[23-odd-limit]], and no other edo is so consistent until [[282edo|282]] and [[311edo|311]] make their appearance.
+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. It can also be thought of as the "sum" of [[41edo]] and [[53edo]] (41 + 53 = 94), both of which are known for their approximation of [[Pythagorean tuning]]. Therefore 94edo's fifth is the [[mediant]] of these two tunings' fifths; it is slightly sharp of just and less accurate than 53edo's fifth, but more accurate than 41edo's.
 The list of 23-limit commas it tempers out is huge, but it is worth noting that it tempers out [[32805/32768]] and is thus a [[schismatic]] system, that it tempers out [[225/224]] and [[385/384]] and so is a [[marvel]] system, and that it also tempers out [[3125/3087]], [[4000/3969]], [[5120/5103]] and [[540/539]]. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[275/273]], and for a number of other temperaments, such as [[isis]].
-edo is an excellent edo for [[Carlos Beta]] scale, since the difference between 5 steps of 94edo and 1 step of Carlos Beta is only -0.00314534 cents.
+edo is an excellent edo for [[Carlos Beta]] scale, since the difference between 1 step of Carlos Beta and 5 steps of 94edo is only 0.00314534 cents.
 === Prime harmonics ===
@@ Line 778: / Line 780: @@
 |
 |}
+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.
+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.
+The regular major second divisible into 16 equal parts can be helpful for realising some of the subtle tunings of Ancient Greek [[tetrachord]]al theory, [[Indian]] raga and Turkish [[maqam]], though it has not been used historically as a division in those musical cultures.
+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.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
@@ Line 943: / Line 956: @@
 == Scales ==
+* [[Garibaldi5]]
+* [[Garibaldi7]]
 * [[Garibaldi12]]
 * [[Garibaldi17]]
-Since 94edo has a step of 12.766 cents, it also allows one to use its mos scales as circulating temperaments and is the first edo to allows one to use a nohajira, pajara or miracle mos scale a as circulating temperament{{clarify}}.
+Since 94edo has a step of 12.766 cents, it also allows one to use its [[mos]] scales as circulating temperaments and is the first edo to allows one to use a mohajira, pajara or miracle mos scale a as circulating temperament{{clarify}}.
 {| class="wikitable mw-collapsible mw-collapsed"
 |+ style=white-space:nowrap | Circulating temperaments in 94edo
@@ Line 1,182: / Line 1,197: @@
 |}
+== Music ==
+; [[Cam Taylor]]
+* [https://archive.org/details/41-94edo09sept2017 4 Improvisations Saturday 9th September 2017]
+* [https://archive.org/details/4194EDOBosanquetAxis18thAug20181FeelingSadButWarmingUp Feeling Sad But Warming Up (in 2 parts)]
+* [https://archive.org/details/4191edoPlayingWithThe13Limit Playing with the 13-limit]
+[[Category:94edo| ]] <!-- main article -->
 [[Category:Garibaldi]]
 [[Category:Marvel]]