94edo: Difference between revisions

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The '''94 equal temperament''', often abbreviated '''94-tET''', '''94-[[EDO]]''', or '''94-ET''', is the scale derived by dividing the [[octave]] into 94 equally-sized steps, where each step ~~represents a frequency ratio of~~ 12.766 [[cent]]s. It is a remarkable all-around utility temperament, good from low prime limit to very high prime limit situations. It is the first equal temperament to be [[consistent]] though the [[23-limit]], and no other equal temperament is so consistent until 282 and 311 make their appearance.

The '''94 equal temperament''', often abbreviated '''94-tET''', '''94-[[EDO]]''', or '''94-ET''', is the scale derived by dividing the [[octave]] into 94 equally-sized steps, where each step is 12.766 [[cent|cents]].

~~The list of 23-limit commas it tempers out is huge, but it's 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 five temperament tempering out 275/273~~, and ~~for a number of~~ other ~~temperaments, such as~~ [[~~isis~~]].

== Theory ==

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

~~Also~~, ~~see~~ the [[~~Table~~ of ~~94edo intervals~~]].

The list of 23-limit commas it tempers out is huge, but it's 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 five temperament tempering out [[275/273]], and for a number of other temperaments, such as [[isis]].

== Approximations to prime harmonics==

:''See also: [[Table of 94edo intervals]]''

== Approximations to prime harmonics ==

{| class="wikitable center-all"

! colspan="2" |

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|-

! rowspan="2" |Error

!absolute (¢)

! absolute (¢)

| 0.0

| +0.17

Line 34:

Line 37:

| +3.90

|-

!relative (%)

! relative (%)

|0.0

| 0.0

| +1.4

| -26.1

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== Rank two temperaments ==

{| class="wikitable ~~right~~-all ~~center~~-4 left-5"

{| class="wikitable center-all right-3 left-5"

! Periods per ~~ ~~ octave

! Periods per octave

! Generator

! Cents

! Associated ratio

! Associated ratio

! Temperament

|-

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| 64/63

| [[Ketchup]]

|-

~~| 2~~

~~| 43\94~~

~~| 548.936~~

~~| 11/8~~

~~| [[Kleischismic]]~~

|-

| 2

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| 27/25

| [[Fifive]]

|-

| 2

| 30\94

| 382.979

| 5/4

| [[Wizard]] / [[gizzard]]

|-

| 2

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|-

| 2

| 30\94

| 43\94

| ~~382~~.~~979~~

| 548.936

| 5/4

| 11/8

| [[~~Wizard]] / [[gizzard~~]]

| [[Kleischismic]]

|}

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

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.

* 46&94 <<8 30 -18 -4 -28 8 -24 2 ... ||

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* 94 solo <<12 -2 20 -6 -42 12 -36 -44 ... || (a rank one temperament!)

Temperaments for which 94 is a [[~~MOS|~~MOS]]:

Temperaments for which 94 is a [[MOS]]:

* 311&94 <<3 70 -42 69 -34 50 85 83...||

@@ Line 1: / Line 1: @@
-The '''94 equal temperament''', often abbreviated '''94-tET''', '''94-[[EDO]]''', or '''94-ET''', is the scale derived by dividing the [[octave]] into 94 equally-sized steps, where each step represents a frequency ratio of 12.766 [[cent]]s. It is a remarkable all-around utility temperament, good from low prime limit to very high prime limit situations. It is the first equal temperament to be [[consistent]] though the [[23-limit]], and no other equal temperament is so consistent until 282 and 311 make their appearance.
+The '''94 equal temperament''', often abbreviated '''94-tET''', '''94-[[EDO]]''', or '''94-ET''', is the scale derived by dividing the [[octave]] into 94 equally-sized steps, where each step is 12.766 [[cent|cents]].
-The list of 23-limit commas it tempers out is huge, but it's 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 five temperament tempering out 275/273, and for a number of other temperaments, such as [[isis]].
+== Theory ==
+edo is a remarkable all-around utility temperament, good from low [[prime limit]] to very high prime limit situations. It is the first equal temperament to be [[consistent]] through the [[23-limit]], and no other equal temperament is so consistent until [[282edo|282]] and [[311edo|311]] make their appearance.
-Also, see the [[Table of 94edo intervals]].
+The list of 23-limit commas it tempers out is huge, but it's 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 five temperament tempering out [[275/273]], and for a number of other temperaments, such as [[isis]].
-== Approximations to prime harmonics==
+:''See also: [[Table of 94edo intervals]]''
+== Approximations to prime harmonics ==
 {| class="wikitable center-all"
 ! colspan="2" |
@@ Line 21: / Line 24: @@
 |-
 ! rowspan="2" |Error
-!absolute (¢)
+! absolute (¢)
 | 0.0
 | +0.17
@@ Line 34: / Line 37: @@
 | +3.90
 |-
-!relative (%)
+! relative (%)
-|0.0
+| 0.0
 | +1.4
 | -26.1
@@ Line 50: / Line 53: @@
 == Rank two temperaments ==
-{| class="wikitable right-all center-4 left-5"
+{| class="wikitable center-all right-3 left-5"
-! Periods <br> per <br> octave
+! Periods<br>per octave
 ! Generator
 ! Cents
-! Associated <br> ratio
+! Associated<br>ratio
 ! Temperament
 |-
@@ Line 92: / Line 95: @@
 | 64/63
 | [[Ketchup]]
-|-
-| 2
-| 43\94
-| 548.936
-| 11/8
-| [[Kleischismic]]
 |-
 | 2
@@ Line 104: / Line 101: @@
 | 27/25
 | [[Fifive]]
+|-
+| 2
+| 30\94
+| 382.979
+| 5/4
+| [[Wizard]] / [[gizzard]]
 |-
 | 2
@@ Line 112: / Line 115: @@
 |-
 | 2
-| 30\94
+| 43\94
-| 382.979
+| 548.936
-| 5/4
+| 11/8
-| [[Wizard]] / [[gizzard]]
+| [[Kleischismic]]
 |}
-Below are some 23-limit temperaments supported by 94et. It might be noted that 94, a very good tuning for [[Garibaldi_temperament|garibaldi temperament]], shows us how to extend it to the 23-limit.
+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.
 * 46&amp;94 &lt;&lt;8 30 -18 -4 -28 8 -24 2 ... ||
@@ Line 132: / Line 135: @@
 * 94 solo &lt;&lt;12 -2 20 -6 -42 12 -36 -44 ... || (a rank one temperament!)
-Temperaments for which 94 is a [[MOS|MOS]]:
+Temperaments for which 94 is a [[MOS]]:
 * 311&amp;94 &lt;&lt;3 70 -42 69 -34 50 85 83...||