Aberrismic theory: Difference between revisions

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At times, a scale pattern has varying temperaments according to the tuning. For example, 5L2m3s may be given the temperament structure of either untempered 2.3.5 or Flutterpyth temperament.

~~<!--~~=== 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 [[Flutterpyth]] 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.

* Flutterpyth, 2.3.~~5.7~~.11~~.13~~[32 & 37], is a rank-2 diatonic temperament that ~~tempers out 64/63 like~~ Superpyth ~~but~~ tunes the 3/2 even sharper, as a result tempering C-D# to a 14/11 and C-Fb to a 13/11. [[37edo]] provides a nearly optimal tuning.

* Flutterpyth, 2.3.13/11.14/11[32 & 37], is a rank-2 diatonic temperament that is a neogothic analogue to 5-limit Superpyth and tunes the 3/2 even sharper, as a result tempering C-D# to a 14/11 and C-Fb to a 13/11. For sake of presentation, this table uses an extension, 2.3.5.7.11.13[32 & 37], incorporating 7/4 and 13/10; [[37edo]] provides a nearly optimal tuning for the extension.

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

|+ Blackdye intervals in two temperaments

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! Flutterpyth

|-

!|[[TAMNAMS|1-~~step~~]]

!|[[TAMNAMS|1-steps]]

| s m L

| 81/80 16/15 10/9

| 143/140 22/21 160/143

|-

!|[[TAMNAMS|2-~~step~~]]

!|[[TAMNAMS|2-steps]]

| L + s L + m

| 9/8 32/27

| 8/7, 9/8 7/6

|-

!|[[TAMNAMS|3-~~step~~]]

!|[[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-~~step~~]]

!|[[TAMNAMS|4-steps]]

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

| 81/64 4/3

| 13/10 4/3

|-

!|[[TAMNAMS|5-~~step~~]]

!|[[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-~~step~~]]

!|[[TAMNAMS|6-steps]]

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

| 3/2 128/81

| 3/2 20/13

|-

!|[[TAMNAMS|7-~~step~~]]

!|[[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-~~step~~]]

!|[[TAMNAMS|8-steps]]

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

| 27/16 16/9

| 12/7 7/4

| 12/7 7/4, 16/9

|-

!|[[TAMNAMS|9-~~step~~]]

!|[[TAMNAMS|9-steps]]

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

| 9/5 15/8 160/81

| 143/80 21/11 280/143

|}

~~-->~~

[[Category:Terms]]

[[Category:Aberrismic theory|*]]

@@ Line 9: / Line 9: @@
 At times, a scale pattern has varying temperaments according to the tuning. For example, 5L2m3s may be given the temperament structure of either untempered 2.3.5 or Flutterpyth temperament.
-<!--=== 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 [[Flutterpyth]] 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.
-* Flutterpyth, 2.3.5.7.11.13[32 & 37], is a rank-2 diatonic temperament that tempers out 64/63 like Superpyth but tunes the 3/2 even sharper, as a result tempering C-D# to a 14/11 and C-Fb to a 13/11. [[37edo]] provides a nearly optimal tuning.
+* Flutterpyth, 2.3.13/11.14/11[32 & 37], is a rank-2 diatonic temperament that is a neogothic analogue to 5-limit Superpyth and tunes the 3/2 even sharper, as a result tempering C-D# to a 14/11 and C-Fb to a 13/11. For sake of presentation, this table uses an extension, 2.3.5.7.11.13[32 & 37], incorporating 7/4 and 13/10; [[37edo]] provides a nearly optimal tuning for the extension.
 {| class="wikitable right-2 right-3 right-4 right-5"
 |+ Blackdye intervals in two temperaments
@@ Line 21: / Line 21: @@
 ! Flutterpyth
 |-
-!|[[TAMNAMS|1-step]]
+!|[[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-step]]
+!|[[TAMNAMS|2-steps]]
 | L + s<br/>L + m
 | 9/8<br/>32/27
 | 8/7, 9/8<br/>7/6
 |-
-!|[[TAMNAMS|3-step]]
+!|[[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-step]]
+!|[[TAMNAMS|4-steps]]
 | 2L + 2s<br/>2L + m + s
 | 81/64<br/>4/3
 | 13/10<br/>4/3
 |-
-!|[[TAMNAMS|5-step]]
+!|[[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-step]]
+!|[[TAMNAMS|6-steps]]
 | 3L + m + 2s<br/>3L + 2m + s
 | 3/2<br/>128/81
 | 3/2<br/>20/13
 |-
-!|[[TAMNAMS|7-step]]
+!|[[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-step]]
+!|[[TAMNAMS|8-steps]]
 | 4L + m + 3s<br/>4L + 2m + 2s
 | 27/16<br/>16/9
-| 12/7<br/>7/4
+| 12/7<br/>7/4, 16/9
 |-
-!|[[TAMNAMS|9-step]]
+!|[[TAMNAMS|9-steps]]
 | 5L + 2m + s<br/>5L + m + 2s<br/>4L + 2m + 2s
 | 9/5<br/>15/8<br/>160/81
 | 143/80<br/>21/11<br/>280/143
 |}
--->
 [[Category:Terms]]
 [[Category:Aberrismic theory|*]]