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. ~~Nonetheless~~, it is [[consistent]] in the 5-limit.

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]].

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

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.

~~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~~.

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

~~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~~.

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.

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

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.

=== Odd harmonics ===

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

=== Subsets and supersets ===

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~~ ===

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.

~~293edo tempers out {{monzo| 8 14 -13 }} ([[parakleisma]]) and {{monzo| -40 15 7 }} in the 5-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 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 ==

; [[Eliora]]

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

* [https://www.youtube.com/watch?v=KYcS2hSd93Y ''Whiplash''] (2022) – 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}}
+{{ED 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. Nonetheless, it is [[consistent]] in the 5-limit.
+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]].
-bb val, with 170\293 fifth, is an optimal tuning for the meantone temperament.
+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.
-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.
+The 293bb val, with 170\293 fifth, is a good tuning for the meantone temperament.
-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.
+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.
-Likewise, 293edo also can be interpreted as a dual-interval tuning, with ''two notes'' instead of one assigned to a particular interval.
+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.
 === Odd harmonics ===
-{{Harmonics in equal|293|columns=10}}
+{{Harmonics in equal|293}}
-=== Relation to a calendar reform ===
+=== Subsets and supersets ===
-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 ===
 edo is the 62nd [[prime edo]].
@@ Line 28: / 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.
-edo tempers out {{monzo| 8 14 -13 }} ([[parakleisma]]) and {{monzo| -40 15 7 }} in the 5-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 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 42: / 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 60: / 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 ==
+; [[Eliora]]
-* [https://www.youtube.com/watch?v=KYcS2hSd93Y Whiplash] by [[Cinnamon Mavka]] – using the Symmetry454[52] scale.
+* [https://www.youtube.com/watch?v=KYcS2hSd93Y ''Whiplash''] (2022) – 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]]