Aberrismic theory: Difference between revisions

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[[groundfault]]'s '''aberrismic theory''' is a xen theoretical paradigm using '''aberrismas'''~~{{idiosyncratic}}~~, a type of scale step which can be added to a scale pattern to turn it into a scale of one rank higher. The aberrisma is a new category of melodic steps that are smaller than the steps in the original scale, which prototypically are categorical "seconds" such as whole tones and semitones. The typical range for an aberrisma is 20 to 60 [[cents]]; groundfault holds the optimal melodic size for an aberrisma to be approximately 40 cents. Examples of ternary patterns that can be made by adding aberrismas to the 5L2s diatonic MOS are:

[[groundfault]]'s '''aberrismic theory''' is a xen theoretical paradigm using '''aberrismas''', a type of scale step which can be added to a scale pattern to turn it into a scale of one rank higher. The aberrisma is a new category of melodic steps that are smaller than the steps in the original scale, which prototypically are categorical "seconds" such as whole tones and semitones. The typical range for an aberrisma is 25 to 55 [[cents]]; groundfault holds the optimal melodic size for an aberrisma to be approximately 40 cents. Examples of ternary patterns that can be made by adding aberrismas to the 5L2s diatonic MOS are:

* [[pinedye]] (5L2m1s or 1s)

* [[diasem]] (5L2m2s or 2s)

* [[blackdye]] (5L2m3s or 3s)

* [[~~diamech~~]] (5L2m4s or 4s)

* [[diaslen]] (5L2m4s or 4s)

* [[diachrome]] (5L2m5s or 5s)

== Edos with quasi-diatonic aberrismic scales ==

All edos 20 and above have an aberrismic scale of the form 5L2mks. If ''n'' = 5''p'' + 2''q'' where ''p'' > ''q'' > 1 (''n''-edo is a diatonic edo with step ratio ''p'':''q''), then (''n'' + ''k'')edo has a 5L2m''k''s scale with step ratio ''p'':''q'':1 for 1 ≤ ''k'' < ''q'', and (''n'' + ''rk'')edo has a 5L2m''k''s scale with step ratio ''p'':''q'':''r'' if 1 ≤ ''r'' ≤ ''rk'' < ''q''.

All edos 20 and above have an aberrismic scale of the form 5L2mks. If {{nowrap|''n'' {{=}} 5''p'' + 2''q''}} where {{nowrap|''p'' > ''q'' > 1}} (''n''-edo is a diatonic edo with step ratio ''p'':''q''), then ({{nowrap|''n'' + ''k''}})edo has a 5L2m''k''s scale with step ratio ''p'':''q'':1 for {{nowrap|1 ≤ ''k'' < ''q''}}, and ({{nowrap|''n'' + ''rk''}})edo has a 5L2m''k''s scale with step ratio ''p'':''q'':''r'' if {{nowrap|1 ≤ ''r'' ≤ ''rk'' < ''q''}}.

== Aberrismic theory and RTT ==

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Mathematically, this difference corresponds to choices of <math>\mathbb{Z}</math>-linear maps

<math>\alpha : \mathbb{Z}^3\langle\mathbf{L}, \mathbf{m}, \mathbf{s}\rangle \to \mathrm{JI}( 2, p_1, ..., p_s )/\mathrm{ker}(T)</math> (here <math>T</math> is a temperament map defined on the 2.p1....ps subgroup), determined by differing choices of <math>\alpha(\mathbf{L}), \alpha(\mathbf{m}), \alpha(\mathbf{s})</math> and subject to the constraint that <math>a\alpha(\mathbf{L}) + b\alpha(\mathbf{m}) + c\alpha(\mathbf{s}) = T(2)</math> for the ternary scale ''a'''''L'''''b'''''m'''''c'''''s'''. Thus there are two choices involved in interpreting a given ternary scale, namely the choice of temperament and the choice of where to map the scale steps. The assignment of scale steps to tempered intervals is chosen to improve coverage of important LCJI intervals.

=== Example: ~~blackdye~~ ===

=== Example: Blackdye ===

The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.

* ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists a tuning ~~with no deviation from JI~~, or that the temperament before applying the tuning map has the same rank as the JI subgroup. This also implies that there is only one JI ratio for each interval under such interpretations, unlike in temperaments that temper out commas.

* ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists an exact JI tuning, or that the temperament before applying the tuning map has the same rank as the JI subgroup. This also implies that there is only one JI ratio for each interval under such interpretations, unlike in temperaments that temper out commas.

* [[Ultrapyth]], 2.3.5.7.11.13[32 & 37], is a diatonic temperament generated by a fifth even sharper than in Superpyth. [[37edo]] provides a nearly optimal tuning. Note that we chose to regard the 3-step 2L + s as a 14/11 rather than as a 5/4, lest the interpretation merely be an extension of the untempered 2.3.5 one. groundfault terms the tuning of blackdye that makes aberrisma-altered Pyth thirds 13/11 and 14/11 ''Flutterpyth blackdye'', since [[Flutterpyth]] temperament is restricted to (maximally) 2.3.7.11.13.19 and does not include 5-limit thirds.

{| class="wikitable right-2 right-3 right-4 right-5"

|+ Blackdye intervals in two temperaments

|+ style="font-size: 105%;" | Blackdye intervals in two temperaments

|-

! | Interval class

! Interval class

! Sizes

! Untempered 2.3.5

! 2.3.7.11.13 Flutterpyth (extended to 13-limit Ultrapyth)

|-

!|[[TAMNAMS|1-steps]]

! [[TAMNAMS|1-steps]]

| s m L

| 81/80 16/15 10/9

| 143/140 22/21 160/143

|-

!|[[TAMNAMS|2-steps]]

! [[TAMNAMS|2-steps]]

| L + s L + m

| 9/8 32/27

| 8/7, 9/8 7/6

|-

!|[[TAMNAMS|3-steps]]

! [[TAMNAMS|3-steps]]

| L + 2s L + m + s 2L + s 2L + m

| 729/640 6/5 5/4 320/243

| 7/6 13/11 14/11 13/10

|-

!|[[TAMNAMS|4-steps]]

! [[TAMNAMS|4-steps]]

| 2L + 2s 2L + m + s

| 81/64 4/3

| 13/10 4/3

|-

!|[[TAMNAMS|5-steps]]

! [[TAMNAMS|5-steps]]

| 2L + m + 2s 2L + 2m + s 3L + 2s 3L + m + s

| 27/20 64/45 45/32 40/27

| 66/49 11/8 16/11 49/33

|-

!|[[TAMNAMS|6-steps]]

! [[TAMNAMS|6-steps]]

| 3L + m + 2s 3L + 2m + s

| 3/2 128/81

| 3/2 20/13

|-

!|[[TAMNAMS|7-steps]]

! [[TAMNAMS|7-steps]]

| 3L + m + 3s 3L + 2m + 2s 4L + m + 2s 4L + 2m + s

| 243/160 8/5 5/3 1280/729

| 20/13 11/7 22/13 12/7

|-

!|[[TAMNAMS|8-steps]]

! [[TAMNAMS|8-steps]]

| 4L + m + 3s 4L + 2m + 2s

| 27/16 16/9

| 12/7 7/4, 16/9

|-

!|[[TAMNAMS|9-steps]]

! [[TAMNAMS|9-steps]]

| 5L + 2m + s 5L + m + 2s 4L + 2m + 2s

| 9/5 15/8 160/81

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|}

== ~~Code~~ ==

== See also ==

~~Haskell function for edo tunings of aberrismic scales:~~

* [[Pseudo-MOS scale]]: A related but different concept of [[CompactStar]]

~~<syntaxhighlight lang="haskell">~~

~~import Data.List~~

-~~- `k`\`edo` in cents.~~

~~stepsOfEdoInCents~~ :~~: Int -> Int -> Double~~

~~stepsOfEdoInCents k edo = 1200*fromIntegral k/fromIntegral edo~~

~~{- Return a list of (edo, step ratio) tuples for the `(countL)L(countM)M(countS)s` aberrismic scale where `edo <= edoBound`,~~

~~where the tuning's s step satisfies the bound `aberLower` <= s <= `aberUpper`.~~

~~Non-coprime step ratios are reduced. -}~~

~~boundedEdosWithTernaryAberrismicScale :: Int -> Double -> Double -> Int -> Int -> Int -> [(Int, (Int, Int, Int))]~~

~~boundedEdosWithTernaryAberrismicScale edoBound aberLower aberUpper countL countM countS =~~

~~let~~

~~sizesOfS = [1..edoBound] -- smallest s possible in n-edo is 1\n~~

~~sizesOfM = [2..edoBound] -- smallest m possible in n-edo is 2\n~~

~~sizesOfL = [3..edoBound] -- smallest L possible in n-edo is 3\n~~

~~in sortBy (\x y -> compare (fst x) (fst y)) -- sort the list, which is finite, by the edo~~

~~$ filter (\x -> (fst x) <= edoBound) -- filter edos that exceed `edoBound`~~

~~[ (edo, (x, y, z)) -- divide step sizes by gcd~~

~~| x <- sizesOfL, y <- sizesOfM, z <- sizesOfS,~~

~~let edo = countL*x + countM*y + countS*z,~~

~~let aberSize = stepsOfEdoInCents z edo, -- compute aberrisma size in given tuning~~

~~x > y && y > z && aberLower <= aberSize && aberSize <= aberUpper ]~~

{-

~~`boundedEdosWithTernaryAberrismicScale 53 20.0 60.0 5 2 3` returns:~~

`[(22,(3,2,1)),(27,(4,2,1)),(29,(4,3,1)),(32,(5,2,1)),(34,(5,3,1)),(36,(5,4,1)),(37,(6,2,1)),(39,(6,3,1)),(41,(6,4,1)),(42,(6,3,2)),(42,(7,2,1)),(43,(6,5,1)),(44,(6,4,2)),(44,(7,3,1)),(46,(6,5,2)),(46,(7,4,1)),(47,(7,3,2)),(47,(8,2,1)),(48,(7,5,1)),(49,(7,4,2)),(49,(8,3,1)),(50,(7,6,1)),(51,(7,5,2)),(51,(8,4,1)),(52,(8,3,2)),(52,(9,2,1)),(53,(7,6,2)),(53,(8,5,1))]`

-}

~~</syntaxhighlight>~~

[[Category:Terms]]

[[Category:Aberrismic theory|*]]

[[Category:Aberrismic theory| ]]

@@ Line 1: / Line 1: @@
-[[groundfault]]'s '''aberrismic theory''' is a xen theoretical paradigm using '''aberrismas'''{{idiosyncratic}}, a type of scale step which can be added to a scale pattern to turn it into a scale of one rank higher. The aberrisma is a new category of melodic steps that are smaller than the steps in the original scale, which prototypically are categorical "seconds" such as whole tones and semitones. The typical range for an aberrisma is 20 to 60 [[cents]]; groundfault holds the optimal melodic size for an aberrisma to be approximately 40 cents. Examples of ternary patterns that can be made by adding aberrismas to the 5L2s diatonic MOS are:
+[[groundfault]]'s '''aberrismic theory''' is a xen theoretical paradigm using '''aberrismas''', a type of scale step which can be added to a scale pattern to turn it into a scale of one rank higher. The aberrisma is a new category of melodic steps that are smaller than the steps in the original scale, which prototypically are categorical "seconds" such as whole tones and semitones. The typical range for an aberrisma is 25 to 55 [[cents]]; groundfault holds the optimal melodic size for an aberrisma to be approximately 40 cents. Examples of ternary patterns that can be made by adding aberrismas to the 5L2s diatonic MOS are:
 * [[pinedye]] (5L2m1s or 1s)
 * [[diasem]] (5L2m2s or 2s)
 * [[blackdye]] (5L2m3s or 3s)
-* [[diamech]] (5L2m4s or 4s)
+* [[diaslen]] (5L2m4s or 4s)
 * [[diachrome]] (5L2m5s or 5s)
 {{User:Inthar/Template:Notation}}
 == Edos with quasi-diatonic aberrismic scales ==
-All edos 20 and above have an aberrismic scale of the form 5L2mks. If ''n'' = 5''p'' + 2''q'' where ''p'' > ''q'' > 1 (''n''-edo is a diatonic edo with step ratio ''p'':''q''), then (''n'' + ''k'')edo has a 5L2m''k''s scale with step ratio ''p'':''q'':1 for 1 &le; ''k'' < ''q'', and (''n'' + ''rk'')edo has a 5L2m''k''s scale with step ratio ''p'':''q'':''r'' if 1 &le; ''r'' &le; ''rk'' < ''q''.
+All edos 20 and above have an aberrismic scale of the form 5L2mks. If {{nowrap|''n'' {{=}} 5''p'' + 2''q''}} where {{nowrap|''p'' > ''q'' > 1}} (''n''-edo is a diatonic edo with step ratio ''p'':''q''), then ({{nowrap|''n'' + ''k''}})edo has a 5L2m''k''s scale with step ratio ''p'':''q'':1 for {{nowrap|1 &le; ''k'' < ''q''}}, and ({{nowrap|''n'' + ''rk''}})edo has a 5L2m''k''s scale with step ratio ''p'':''q'':''r'' if {{nowrap|1 &le; ''r'' &le; ''rk'' < ''q''}}.
 == Aberrismic theory and RTT ==
@@ Line 17: / Line 17: @@
 Mathematically, this difference corresponds to choices of <math>\mathbb{Z}</math>-linear maps
 <math>\alpha : \mathbb{Z}^3\langle\mathbf{L}, \mathbf{m}, \mathbf{s}\rangle \to \mathrm{JI}( 2, p_1, ..., p_s )/\mathrm{ker}(T)</math> (here <math>T</math> is a temperament map defined on the 2.p<sub>1</sub>....p<sub>s</sub> subgroup), determined by differing choices of <math>\alpha(\mathbf{L}), \alpha(\mathbf{m}), \alpha(\mathbf{s})</math> and subject to the constraint that <math>a\alpha(\mathbf{L}) + b\alpha(\mathbf{m}) + c\alpha(\mathbf{s}) = T(2)</math> for the ternary scale ''a'''''L'''''b'''''m'''''c'''''s'''. Thus there are two choices involved in interpreting a given ternary scale, namely the choice of temperament and the choice of where to map the scale steps. The assignment of scale steps to tempered intervals is chosen to improve coverage of important LCJI intervals.
-=== Example: blackdye ===
+=== Example: Blackdye ===
 The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.
-* ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists a tuning with no deviation from JI, or that the temperament before applying the tuning map has the same rank as the JI subgroup. This also implies that there is only one JI ratio for each interval under such interpretations, unlike in temperaments that temper out commas.
+* ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists an exact JI tuning, or that the temperament before applying the tuning map has the same rank as the JI subgroup. This also implies that there is only one JI ratio for each interval under such interpretations, unlike in temperaments that temper out commas.
 * [[Ultrapyth]], 2.3.5.7.11.13[32 & 37], is a diatonic temperament generated by a fifth even sharper than in Superpyth. [[37edo]] provides a nearly optimal tuning. Note that we chose to regard the 3-step 2L + s as a 14/11 rather than as a 5/4, lest the interpretation merely be an extension of the untempered 2.3.5 one. groundfault terms the tuning of blackdye that makes aberrisma-altered Pyth thirds 13/11 and 14/11 ''Flutterpyth blackdye'', since [[Flutterpyth]] temperament is restricted to (maximally) 2.3.7.11.13.19 and does not include 5-limit thirds.
 {| class="wikitable right-2 right-3 right-4 right-5"
-|+ Blackdye intervals in two temperaments
+|+ style="font-size: 105%;" | Blackdye intervals in two temperaments
 |-
-! | Interval class
+! Interval class
 ! Sizes
 ! Untempered 2.3.5
 ! 2.3.7.11.13 Flutterpyth (extended to 13-limit Ultrapyth)
 |-
-!|[[TAMNAMS|1-steps]]
+! [[TAMNAMS|1-steps]]
 | s<br/>m<br/>L
 | 81/80<br/>16/15<br/>10/9
 | 143/140<br/>22/21<br/>160/143
 |-
-!|[[TAMNAMS|2-steps]]
+! [[TAMNAMS|2-steps]]
 | L + s<br/>L + m
 | 9/8<br/>32/27
 | 8/7, 9/8<br/>7/6
 |-
-!|[[TAMNAMS|3-steps]]
+! [[TAMNAMS|3-steps]]
 | L + 2s<br/>L + m + s<br/>2L + s<br/>2L + m
 | 729/640<br/>6/5<br/>5/4<br/>320/243
 | 7/6<br/>13/11<br/>14/11<br/>13/10
 |-
-!|[[TAMNAMS|4-steps]]
+! [[TAMNAMS|4-steps]]
 | 2L + 2s<br/>2L + m + s
 | 81/64<br/>4/3
 | 13/10<br/>4/3
 |-
-!|[[TAMNAMS|5-steps]]
+! [[TAMNAMS|5-steps]]
 | 2L + m + 2s<br/>2L + 2m + s<br/>3L + 2s<br/>3L + m + s
 | 27/20<br/>64/45<br/>45/32<br/>40/27
 | 66/49<br/>11/8<br/>16/11<br/>49/33
 |-
-!|[[TAMNAMS|6-steps]]
+! [[TAMNAMS|6-steps]]
 | 3L + m + 2s<br/>3L + 2m + s
 | 3/2<br/>128/81
 | 3/2<br/>20/13
 |-
-!|[[TAMNAMS|7-steps]]
+! [[TAMNAMS|7-steps]]
 | 3L + m + 3s<br/>3L + 2m + 2s<br/>4L + m + 2s<br/>4L + 2m + s
 | 243/160<br/>8/5<br/>5/3<br/>1280/729
 | 20/13<br/>11/7<br/>22/13<br/>12/7
 |-
-!|[[TAMNAMS|8-steps]]
+! [[TAMNAMS|8-steps]]
 | 4L + m + 3s<br/>4L + 2m + 2s
 | 27/16<br/>16/9
 | 12/7<br/>7/4, 16/9
 |-
-!|[[TAMNAMS|9-steps]]
+! [[TAMNAMS|9-steps]]
 | 5L + 2m + s<br/>5L + m + 2s<br/>4L + 2m + 2s
 | 9/5<br/>15/8<br/>160/81
@@ Line 75: / Line 77: @@
 |}
-== Code ==
+== See also ==
-Haskell function for edo tunings of aberrismic scales:
+* [[Pseudo-MOS scale]]: A related but different concept of [[CompactStar]]
-<syntaxhighlight lang="haskell">
-import Data.List
--- `k`\`edo` in cents.
-stepsOfEdoInCents :: Int -> Int -> Double
-stepsOfEdoInCents k edo = 1200*fromIntegral k/fromIntegral edo
-{- Return a list of (edo, step ratio) tuples for the `(countL)L(countM)M(countS)s` aberrismic scale where `edo <= edoBound`,
-   where the tuning's s step satisfies the bound `aberLower` <= s <= `aberUpper`.
-   Non-coprime step ratios are reduced. -}
-boundedEdosWithTernaryAberrismicScale :: Int -> Double -> Double -> Int -> Int -> Int -> [(Int, (Int, Int, Int))]
-boundedEdosWithTernaryAberrismicScale edoBound aberLower aberUpper countL countM countS =
-  let
-    sizesOfS = [1..edoBound] -- smallest s possible in n-edo is 1\n
-    sizesOfM = [2..edoBound] -- smallest m possible in n-edo is 2\n
-    sizesOfL = [3..edoBound] -- smallest L possible in n-edo is 3\n
-  in sortBy (\x y -> compare (fst x) (fst y)) -- sort the list, which is finite, by the edo
-      $ filter (\x -> (fst x) <= edoBound) -- filter edos that exceed `edoBound`
-      [ (edo, (x, y, z)) -- divide step sizes by gcd
-      | x <- sizesOfL, y <- sizesOfM, z <- sizesOfS,
-        let edo = countL*x + countM*y + countS*z,
-        let aberSize = stepsOfEdoInCents z edo, -- compute aberrisma size in given tuning
-        x > y && y > z && aberLower <= aberSize && aberSize <= aberUpper ]
-{-
-`boundedEdosWithTernaryAberrismicScale 53 20.0 60.0 5 2 3` returns:
-`[(22,(3,2,1)),(27,(4,2,1)),(29,(4,3,1)),(32,(5,2,1)),(34,(5,3,1)),(36,(5,4,1)),(37,(6,2,1)),(39,(6,3,1)),(41,(6,4,1)),(42,(6,3,2)),(42,(7,2,1)),(43,(6,5,1)),(44,(6,4,2)),(44,(7,3,1)),(46,(6,5,2)),(46,(7,4,1)),(47,(7,3,2)),(47,(8,2,1)),(48,(7,5,1)),(49,(7,4,2)),(49,(8,3,1)),(50,(7,6,1)),(51,(7,5,2)),(51,(8,4,1)),(52,(8,3,2)),(52,(9,2,1)),(53,(7,6,2)),(53,(8,5,1))]`
--}
-</syntaxhighlight>
 [[Category:Terms]]
-[[Category:Aberrismic theory|*]]<!--Main article-->
+[[Category:Aberrismic theory| ]]<!--Main article-->