293edo: Difference between revisions

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== Theory ==

293edo does not approximate prime ~~harmonics~~ well all the way into the 41st, ~~unless~~ 30% ~~errors are considered "well", in which case it equally represents all of them~~. The first harmonic that it approximates ~~within 1 standard deviation of one step~~ is 43rd, which is 10% flat compared to the just intonated interval.

293edo does not approximate [[prime harmonic]]s well all the way into the 41st, with none approximated within 30% error. The first harmonic that it approximates well is the 43rd, which is 10% flat compared to the just intonated interval. As such, it is only [[consistent]] to the [[5-odd-limit]].

~~Nonetheless, it is~~ [[~~consistent~~]] in the ~~5-limit, where it~~ tempers out the [[parakleisma]] and {{monzo| -40 15 7 }}. ~~293edo~~ also tempers out the [[marvel comma]] ~~on the patent val~~ in the 7-limit.

Using the [[patent val]], the equal temperament [[tempering out|tempers out]] the [[parakleisma]] and {{monzo| -40 15 7 }}. It also tempers out the [[marvel comma]] in the 7-limit.

293bb val, with 170\293 fifth, is ~~an optimal~~ tuning for the meantone temperament.

The 293bb val, with 170\293 fifth, is a good tuning for the meantone temperament.

When it comes to the intervals that are not octave-reduced prime harmonics, some which are well-approximated are [[6/5]], [[11/7]], [[17/11]], [[19/17]], [[24/23]], [[25/17]], [[25/19]], and respectively their octave inversions. [[21/16]], which is a composite octave-reduced harmonic, is also well represented. These numbers are related to poor approximation of prime harmonics by cancelling out of the errors. For example, 19th and 17th ~~harmoincs~~ have +36 and +37 error respectively, which together cancels out to 1.

When it comes to the intervals that are not octave-reduced prime harmonics, some which are well-approximated are [[6/5]], [[11/7]], [[17/11]], [[19/17]], [[24/23]], [[25/17]], [[25/19]], and respectively their octave inversions. [[21/16]], which is a composite octave-reduced harmonic, is also well represented. These numbers are related to poor approximation of prime harmonics by cancelling out of the errors. For example, 19th and 17th harmonics have +36 and +37 error respectively, which together cancels out to 1.

One step of 293edo is at the edge of human pitch perception of 3.5 cents. When combined with low harmonicity, this opens 293edo to a wide range of interpretations.

One step of 293edo is at the edge of human pitch perception of 3.5 cents. When combined with low harmonicity, this opens 293edo to a wide range of interpretations. For example, 293edo also can be interpreted as a dual-interval tuning, with ''two notes'' instead of one assigned to a particular interval.

~~Likewise~~, 293edo also can be interpreted as a dual-interval tuning, with ''two notes'' instead of one assigned to a particular interval.

=== Odd harmonics ===

~~=== Relation to a calendar reform ===~~

The 33L 19s [[Maximal evenness|maximally even]] scale of 293edo has a real life application – it is a leap year pattern of a proposed calendar. It employs 62\293 as a generator, described as "accumulator" by the creator of the calendar himself. Likewise, a 71-note cycle with 260\293 generator can be constructed by analogy.

~~The corresponding rank two temperament is therefore called [[Symmetry454]].~~

=== ~~Miscellaneous properties~~ ===

=== Subsets and supersets ===

293edo is the 62nd [[prime edo]].

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293edo tempers out the 2.43 {{monzo| 1590 293 }} comma in the [[patent val]], equating a stack of 293 43rd harmonics with 1590 octaves.

. Using the patent val, it tempers out 225/224, 2500000/2470629, and 344373768/341796875 in the 7-limit; 6250/6237, 8019/8000, 14700/14641, and 16896/16807 in the 11-limit; 351/350, 625/624, 1625/1617, and 13122/13013 in the 13-limit; 715/714, 850/847, 1089/1088, 1377/1375, 2058/2057, and 2880/2873 in the 17-limit.

Using the patent val, it tempers out 225/224, 2500000/2470629, and 344373768/341796875 in the 7-limit; 6250/6237, 8019/8000, 14700/14641, and 16896/16807 in the 11-limit; 351/350, 625/624, 1625/1617, and 13122/13013 in the 13-limit; 715/714, 850/847, 1089/1088, 1377/1375, 2058/2057, and 2880/2873 in the 17-limit.

Using the 293b val, it tempers out 16875/16807, 20000/19683, and 65625/65536 in the 7-limit; 896/891, 6875/6804, 9375/9317, and 12005/11979 in the 11-limit; 352/351, 364/363, 1716/1715, and 8125/8019 in the 13-limit.

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=== Rank-2 temperaments ===

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

! Periods per ~~Octave~~

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

|1

| 1

|11\293

| 11\293

|45.06

| 45.06

|36/35

| 36/35

|[[Quartonic]] (293bcd)

| [[Quartonic]] (293bcd)

|-

| 1

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| [[Symmetry454]]

|-

|1

| 1

|118\293

| 118\293

|483.28

| 483.28

|320/243

| 320/243

|[[Hemiseven]] (293de)

| [[Hemiseven]] (293de)

|-

|1

| 1

|143\293

| 143\293

|585.66

| 585.66

|7/5

| 7/5

|[[Merman]] (293ef)

| [[Merman]] (293ef)

|-

|1

| 1

|170\293

| 170\293

|696.25

| 696.25

|3/2

| 3/2

|[[Meantone]] (293bb)

| [[Meantone]] (293bb)

|}

=== Scales ===

The 33L 19s [[Maximal evenness|maximally even]] scale of 293edo is a leap year pattern of a proposed calendar. It employs 62\293 as a generator, described as "accumulator" by the creator of the calendar himself. Likewise, a 71-note cycle with 260\293 generator can be constructed by analogy.

The corresponding rank two temperament is therefore called [[Symmetry454]].

== Music ==

; [[Cinnamon Mavka]]

* [https://www.youtube.com/watch?v=KYcS2hSd93Y Whiplash~~] by [[Cinnamon Mavka]~~] – using the Symmetry454[52] scale.

* [https://www.youtube.com/watch?v=KYcS2hSd93Y ''Whiplash''] – using the Symmetry454[52] scale.

== External links ==

* [https://individual.utoronto.ca/kalendis/leap/52-293-sym454-leap-years.htm 52\293 Symmetry454 Leap Years]

~~[[Category:Equal divisions of the octave|###]] ~~

~~[[Category:Prime EDO]]~~

[[Category:Listen]]

~~[[Category:Marvel]]~~

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|293}}
+{{EDO intro}}
 == Theory ==
-edo does not approximate prime harmonics well all the way into the 41st, unless 30% errors are considered "well", in which case it equally represents all of them. The first harmonic that it approximates within 1 standard deviation of one step is 43rd, which is 10% flat compared to the just intonated interval.
+edo does not approximate [[prime harmonic]]s well all the way into the 41st, with none approximated within 30% error. The first harmonic that it approximates well is the 43rd, which is 10% flat compared to the just intonated interval. As such, it is only [[consistent]] to the [[5-odd-limit]].
-Nonetheless, it is [[consistent]] in the 5-limit, where it tempers out the [[parakleisma]] and {{monzo| -40 15 7 }}. 293edo also tempers out the [[marvel comma]] on the patent val in the 7-limit.
+Using the [[patent val]], the equal temperament [[tempering out|tempers out]] the [[parakleisma]] and {{monzo| -40 15 7 }}. It also tempers out the [[marvel comma]] in the 7-limit.
-bb val, with 170\293 fifth, is an optimal tuning for the meantone temperament.
+The 293bb val, with 170\293 fifth, is a good tuning for the meantone temperament.
-When it comes to the intervals that are not octave-reduced prime harmonics, some which are well-approximated are [[6/5]], [[11/7]], [[17/11]], [[19/17]], [[24/23]], [[25/17]], [[25/19]], and respectively their octave inversions. [[21/16]], which is a composite octave-reduced harmonic, is also well represented. These numbers are related to poor approximation of prime harmonics by cancelling out of the errors. For example, 19th and 17th harmoincs have +36 and +37 error respectively, which together cancels out to 1.
+When it comes to the intervals that are not octave-reduced prime harmonics, some which are well-approximated are [[6/5]], [[11/7]], [[17/11]], [[19/17]], [[24/23]], [[25/17]], [[25/19]], and respectively their octave inversions. [[21/16]], which is a composite octave-reduced harmonic, is also well represented. These numbers are related to poor approximation of prime harmonics by cancelling out of the errors. For example, 19th and 17th harmonics have +36 and +37 error respectively, which together cancels out to 1.
-One step of 293edo is at the edge of human pitch perception of 3.5 cents. When combined with low harmonicity, this opens 293edo to a wide range of interpretations.
+One step of 293edo is at the edge of human pitch perception of 3.5 cents. When combined with low harmonicity, this opens 293edo to a wide range of interpretations. For example, 293edo also can be interpreted as a dual-interval tuning, with ''two notes'' instead of one assigned to a particular interval.
-Likewise, 293edo also can be interpreted as a dual-interval tuning, with ''two notes'' instead of one assigned to a particular interval.
 === Odd harmonics ===
-{{Harmonics in equal|293|columns=10}}
+{{Harmonics in equal|293}}
-=== Relation to a calendar reform ===
-The 33L 19s [[Maximal evenness|maximally even]] scale of 293edo has a real life application – it is a leap year pattern of a proposed calendar. It employs 62\293 as a generator, described as "accumulator" by the creator of the calendar himself. Likewise, a 71-note cycle with 260\293 generator can be constructed by analogy.
-The corresponding rank two temperament is therefore called [[Symmetry454]].
-=== Miscellaneous properties ===
+=== Subsets and supersets ===
 edo is the 62nd [[prime edo]].
@@ Line 30: / Line 23: @@
 edo tempers out the 2.43 {{monzo| 1590 293 }} comma in the [[patent val]], equating a stack of 293 43rd harmonics with 1590 octaves.
-. Using the patent val, it tempers out 225/224, 2500000/2470629, and 344373768/341796875 in the 7-limit; 6250/6237, 8019/8000, 14700/14641, and 16896/16807 in the 11-limit; 351/350, 625/624, 1625/1617, and 13122/13013 in the 13-limit; 715/714, 850/847, 1089/1088, 1377/1375, 2058/2057, and 2880/2873 in the 17-limit.
+Using the patent val, it tempers out 225/224, 2500000/2470629, and 344373768/341796875 in the 7-limit; 6250/6237, 8019/8000, 14700/14641, and 16896/16807 in the 11-limit; 351/350, 625/624, 1625/1617, and 13122/13013 in the 13-limit; 715/714, 850/847, 1089/1088, 1377/1375, 2058/2057, and 2880/2873 in the 17-limit.
 Using the 293b val, it tempers out 16875/16807, 20000/19683, and 65625/65536 in the 7-limit; 896/891, 6875/6804, 9375/9317, and 12005/11979 in the 11-limit; 352/351, 364/363, 1716/1715, and 8125/8019 in the 13-limit.
@@ Line 44: / Line 37: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-! Periods<br>per Octave
+! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
-|1
+| 1
-|11\293
+| 11\293
-|45.06
+| 45.06
-|36/35
+| 36/35
-|[[Quartonic]] (293bcd)
+| [[Quartonic]] (293bcd)
 |-
 | 1
@@ Line 62: / Line 55: @@
 | [[Symmetry454]]
 |-
-|1
+| 1
-|118\293
+| 118\293
-|483.28
+| 483.28
-|320/243
+| 320/243
-|[[Hemiseven]] (293de)
+| [[Hemiseven]] (293de)
 |-
-|1
+| 1
-|143\293
+| 143\293
-|585.66
+| 585.66
-|7/5
+| 7/5
-|[[Merman]] (293ef)
+| [[Merman]] (293ef)
 |-
-|1
+| 1
-|170\293
+| 170\293
-|696.25
+| 696.25
-|3/2
+| 3/2
-|[[Meantone]] (293bb)
+| [[Meantone]] (293bb)
 |}
+=== Scales ===
+The 33L 19s [[Maximal evenness|maximally even]] scale of 293edo is a leap year pattern of a proposed calendar. It employs 62\293 as a generator, described as "accumulator" by the creator of the calendar himself. Likewise, a 71-note cycle with 260\293 generator can be constructed by analogy.
+The corresponding rank two temperament is therefore called [[Symmetry454]].
 == Music ==
+; [[Cinnamon Mavka]]
-* [https://www.youtube.com/watch?v=KYcS2hSd93Y Whiplash] by [[Cinnamon Mavka]] – using the Symmetry454[52] scale.
+* [https://www.youtube.com/watch?v=KYcS2hSd93Y ''Whiplash''] – using the Symmetry454[52] scale.
 == External links ==
 * [https://individual.utoronto.ca/kalendis/leap/52-293-sym454-leap-years.htm 52\293 Symmetry454 Leap Years]
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
-[[Category:Prime EDO]]
 [[Category:Listen]]
-[[Category:Marvel]]