Superpyth: Difference between revisions

(5 intermediate revisions by 3 users not shown)

Line 17:

| Color name = Ruti

| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]], [[5L 12s]], [[5L 17s]]

| Odd limit 1 = (2.3.7) 9 | Mistuning 1 = 9.09 | Complexity 1 = 12

| Odd limit 1 = (2.3.7) 7 | Mistuning 1 = 9.09 | Complexity 1 = 12

| Odd limit 2 = 9 | Mistuning 2 = 13.63 | Complexity 2 = 27

}}

Line 140:

The fifth of superpyth is supposed to be tuned sharp of just for the accuracy of the overall temperament. Roughly speaking, it ranges from as flat as [[Pythagorean tuning|Pythagorean]] (where 3 is tuned just) to 1/2-comma (where 7 is tuned just, between [[52edo|52b-edo]] and [[57edo|57b-edo]]), with 22edo and 27edo being typical endpoints of superpyth's optimal range.

Despite superpyth being seen as the "counterpart" of meantone for sharp fifths and septimal thirds, it is actually of considerably higher damage than meantone. The tempered comma, 64/63, is not only larger than 81/80, but it must be split over only three intervals (one minor seventh and two perfect fifths), rather than five as in meantone (one major third and four fifths). This can be shown by the fact that 1/5-comma meantone, the meantone tuning with the minimum damage to harmonics 3 and 5 ~~(meantone's 5-[[integer limit|integer-limit]] [[minimax tuning]])~~ has a tuning error on 3 and 5 of 4.3{{c}}, while 1/3-comma superpyth, the superpyth tuning with the minimum damage to harmonics 3 and 7 (the minimax tuning for the no-5 [[7-odd-limit]] [[tonality diamond]]) has a tuning error on 3 and 7 of 9.09{{c}}, over twice as much ~~as meantone~~. Therefore, tuning superpyth can be a somewhat contentious matter, as some intervals have to be essentially sacrificed for the sake of optimizing others. An additional consideration is the use of tertian triads in conventional diatonic harmony, whereby the 9/7 supermajor third may be more important than it looks from the bare math.

Despite superpyth being seen as the "counterpart" of meantone for sharp fifths and septimal thirds, it is actually of considerably higher damage than meantone. The tempered comma, 64/63, is not only larger than 81/80, but it must be split over only three intervals (one minor seventh and two perfect fifths), rather than five as in meantone (one major third and four fifths). This can be shown by the fact that 1/5-comma meantone, the meantone tuning with the minimum damage to harmonics 3 and 5, has a tuning error on 3 and 5 of 4.3{{c}}, while 1/3-comma superpyth, the superpyth tuning with the minimum damage to harmonics 3 and 7 (the minimax tuning for the no-5 [[7-odd-limit]] [[tonality diamond]]) has a tuning error on 3 and 7 of 9.09{{c}}, over twice as much. Therefore, tuning superpyth can be a somewhat contentious matter, as some intervals have to be essentially sacrificed for the sake of optimizing others. An additional consideration is the use of tertian triads in conventional diatonic harmony, whereby the 9/7 supermajor third may be more important than it looks from the bare math.

If we focus purely on the 2.3.7 subgroup for now, and as a starting point adopt an approach based on the example of [[quarter-comma meantone]], treating archy's harmonic 7 as analogous to 5 in meantone, 1/3-comma and 1/4-comma turn out to be logical solutions. In 1/3-comma superpyth, the whole tone leans towards 8/7 so that 3 and 7 are equally sharp and the minor third is tuned to exactly 7/6; 27edo is extremely close to a closed system of 1/3-comma. In 1/4-comma tuning, which is the minimax tuning for the no-5 [[9-odd-limit]], the whole tone is midway between 8/7 and 9/8 so that the 7 is twice as sharp as 3 and that the major third is exactly 9/7; 22edo is very close to a closed circle of 1/4-comma.

In general, we would want to consider 3 somewhat more important than 7, and 7 somewhat more important than 9; in meantone, similar principles imply that an optimum is to be found sharp of 1/4-comma, though flat of [[1/5-comma meantone|1/5-comma]]. In archy, these place it sharper than 1/4-comma but flatter than 1/3-comma, which is supported by the standard [[CTE]] and [[CWE]] metrics. In fact, 22edo is slightly sharp of 1/4-comma (though still flat of the CTE optimum) and therefore pushes in the more accurate direction given the above discussion. 2/7-comma superpyth is particularly notable since it tunes the 7/6 and 9/7 equally sharp and 3/2 twice as sharp as the thirds; [[71edo]] (709.859{{c}}) is very close to a closed ~~system~~ of 2/7-comma.

In general, we would want to consider 3 somewhat more important than 7, and 7 somewhat more important than 9; in meantone, similar principles imply that an optimum is to be found sharp of 1/4-comma, though flat of [[1/5-comma meantone|1/5-comma]]. In archy, these place it sharper than 1/4-comma but flatter than 1/3-comma, which is supported by the standard [[CTE]] and [[CWE]] metrics. In fact, 22edo is slightly sharp of 1/4-comma (though still flat of the CTE optimum) and therefore pushes in the more accurate direction given the above discussion. 2/7-comma superpyth is particularly notable since it tunes the 7/6 and 9/7 equally sharp and 3/2 twice as sharp as the thirds; [[71edo]] (709.859{{c}}) and [[93edo]] with its sharp fifth of 709.677{{c}} come very close to forming closed systems of 2/7-comma.

27edo is also the point where superpyth tunes 5/4 to the familiar 400{{c}} major third of [[12edo]], and in sharper tunings different mappings of 5/4 arise with more accuracy (see [[quasiultra]] and [[ultrapyth]]), somewhat analogous to [[19edo]] (which represents [[1/3-comma meantone]] and is on the edge between septimal meantone and [[flattone]]). The same goes for flatter tunings than 22edo (see [[quasisuper]] and [[dominant (temperament)|dominant]]). Furthermore, the 11-limit extension works strictly within 22edo and 27e-edo, with 22edo conflating 11/10 with 12/11, and 27e-edo conflating 11/8 with 7/5.

27edo is also the point where superpyth tunes 5/4 to the familiar 400{{c}} major third of [[12edo]], and in sharper tunings different mappings of 5/4 arise with more accuracy (see [[quasiultra]] and [[ultrapyth]]), somewhat analogous to [[19edo]] (which represents [[1/3-comma meantone]] and is on the edge between septimal meantone and [[flattone]]). The same goes for flatter tunings than 22edo (see [[quasisuper]] and [[dominant (temperament)|dominant]]). Furthermore, the [[11-limit]] canonical extension works strictly within 22edo and 27e-edo, with 22edo conflating 11/10 with 12/11, and 27e-edo conflating 11/8 with 7/5.

Tunings flatter than 1/4-comma archy, such as 1/5-comma (close to [[39edo]]), 1/6-comma, … are analogous to the historical "modified meantones" ([[1/6-comma meantone|1/6-comma]], [[1/7-comma meantone|1/7-comma]], …), as they prioritize the tuning of 3/2 more than the accuracy of septimal harmony. The alternative 11-limit extension, suprapyth, and an alternative extension to 5, quasisuper, work best for tunings in the range of 17edo to 22edo.

Line 159:

|-

! rowspan="2" |

! colspan="2" | Euclidean

! colspan="3" | Euclidean

|-

! ~~Unskewed~~

! Constrained

! ~~Skewed~~

! Constrained & skewed

! Destretched

|-

! Equilateral

| CEE: ~3/2 = 712.~~8606¢~~ (2/5-comma)

| CEE: ~3/2 = 712.8606{{c}} (2/5-comma)

| CSEE: ~3/2 = 711.~~9997¢~~ (7/19-comma)

| CSEE: ~3/2 = 711.9997{{c}} (7/19-comma)

| POEE: ~3/2 = 709.6343{{c}}

|-

! Tenney

| CTE: ~3/2 = 709.~~5948¢~~

| CTE: ~3/2 = 709.5948{{c}}

| CWE: ~3/2 = 709.~~3901¢~~

| CWE: ~3/2 = 709.3901{{c}}

| POTE: ~3/2 = 709.3213{{c}}

|-

! Benedetti, Wilson

| CBE: ~3/2 = 707.~~7286¢~~ (18/85-comma)

| CBE: ~3/2 = 707.7286{{c}} (18/85-comma)

| CSBE: ~3/2 = 707.~~9869¢~~ (25/113-comma)

| CSBE: ~3/2 = 707.9869{{c}} (25/113-comma)

| POBE: ~3/2 = 708.6428{{c}}

|}

Line 181:

Line 185:

|-

! rowspan="2" |

! colspan="2" | Euclidean

! colspan="3" | Euclidean

|-

! ~~Unskewed~~

! Constrained

! ~~Skewed~~

! Constrained & skewed

! Destretched

|-

! Equilateral

| CEE: ~3/2 = 709.~~7805¢~~

| CEE: ~3/2 = 709.7805{{c}}

| CSEE: ~3/2 = 710.~~2428¢~~

| CSEE: ~3/2 = 710.2428{{c}}

| POEE: ~3/2 = 710.4936{{c}}

|-

! Tenney

| CTE: ~3/2 = 709.~~5907¢~~

| CTE: ~3/2 = 709.5907{{c}}

| CWE: ~3/2 = 710.~~1193¢~~

| CWE: ~3/2 = 710.1193{{c}}

| POTE: ~3/2 = 710.2910{{c}}

|-

! Benedetti, Wilson

| CBE: ~3/2 = 709.~~4859¢~~

| CBE: ~3/2 = 709.4859{{c}}

| CSBE: ~3/2 = 710.~~0321¢~~

| CSBE: ~3/2 = 710.0321{{c}}

| POBE: ~3/2 = 710.2421{{c}}

|}

@@ Line 17: / Line 17: @@
 | Color name = Ruti
 | MOS scales = [[2L&nbsp;3s]], [[5L&nbsp;2s]], [[5L&nbsp;7s]], [[5L&nbsp;12s]], [[5L&nbsp;17s]]
-| Odd limit 1 = (2.3.7) 9 | Mistuning 1 = 9.09 | Complexity 1 = 12
+| Odd limit 1 = (2.3.7) 7 | Mistuning 1 = 9.09 | Complexity 1 = 12
 | Odd limit 2 = 9 | Mistuning 2 = 13.63 | Complexity 2 = 27
 }}
@@ Line 140: / Line 140: @@
 The fifth of superpyth is supposed to be tuned sharp of just for the accuracy of the overall temperament. Roughly speaking, it ranges from as flat as [[Pythagorean tuning|Pythagorean]] (where 3 is tuned just) to 1/2-comma (where 7 is tuned just, between [[52edo|52b-edo]] and [[57edo|57b-edo]]), with 22edo and 27edo being typical endpoints of superpyth's optimal range.
-Despite superpyth being seen as the "counterpart" of meantone for sharp fifths and septimal thirds, it is actually of considerably higher damage than meantone. The tempered comma, 64/63, is not only larger than 81/80, but it must be split over only three intervals (one minor seventh and two perfect fifths), rather than five as in meantone (one major third and four fifths). This can be shown by the fact that 1/5-comma meantone, the meantone tuning with the minimum damage to harmonics 3 and 5 (meantone's 5-[[integer limit|integer-limit]] [[minimax tuning]]) has a tuning error on 3 and 5 of 4.3{{c}}, while 1/3-comma superpyth, the superpyth tuning with the minimum damage to harmonics 3 and 7 (the minimax tuning for the no-5 [[7-odd-limit]] [[tonality diamond]]) has a tuning error on 3 and 7 of 9.09{{c}}, over twice as much as meantone. Therefore, tuning superpyth can be a somewhat contentious matter, as some intervals have to be essentially sacrificed for the sake of optimizing others. An additional consideration is the use of tertian triads in conventional diatonic harmony, whereby the 9/7 supermajor third may be more important than it looks from the bare math.
+Despite superpyth being seen as the "counterpart" of meantone for sharp fifths and septimal thirds, it is actually of considerably higher damage than meantone. The tempered comma, 64/63, is not only larger than 81/80, but it must be split over only three intervals (one minor seventh and two perfect fifths), rather than five as in meantone (one major third and four fifths). This can be shown by the fact that 1/5-comma meantone, the meantone tuning with the minimum damage to harmonics 3 and 5, has a tuning error on 3 and 5 of 4.3{{c}}, while 1/3-comma superpyth, the superpyth tuning with the minimum damage to harmonics 3 and 7 (the minimax tuning for the no-5 [[7-odd-limit]] [[tonality diamond]]) has a tuning error on 3 and 7 of 9.09{{c}}, over twice as much. Therefore, tuning superpyth can be a somewhat contentious matter, as some intervals have to be essentially sacrificed for the sake of optimizing others. An additional consideration is the use of tertian triads in conventional diatonic harmony, whereby the 9/7 supermajor third may be more important than it looks from the bare math.
 If we focus purely on the 2.3.7 subgroup for now, and as a starting point adopt an approach based on the example of [[quarter-comma meantone]], treating archy's harmonic 7 as analogous to 5 in meantone, 1/3-comma and 1/4-comma turn out to be logical solutions. In 1/3-comma superpyth, the whole tone leans towards 8/7 so that 3 and 7 are equally sharp and the minor third is tuned to exactly 7/6; 27edo is extremely close to a closed system of 1/3-comma. In 1/4-comma tuning, which is the minimax tuning for the no-5 [[9-odd-limit]], the whole tone is midway between 8/7 and 9/8 so that the 7 is twice as sharp as 3 and that the major third is exactly 9/7; 22edo is very close to a closed circle of 1/4-comma.
-In general, we would want to consider 3 somewhat more important than 7, and 7 somewhat more important than 9; in meantone, similar principles imply that an optimum is to be found sharp of 1/4-comma, though flat of [[1/5-comma meantone|1/5-comma]]. In archy, these place it sharper than 1/4-comma but flatter than 1/3-comma, which is supported by the standard [[CTE]] and [[CWE]] metrics. In fact, 22edo is slightly sharp of 1/4-comma (though still flat of the CTE optimum) and therefore pushes in the more accurate direction given the above discussion. 2/7-comma superpyth is particularly notable since it tunes the 7/6 and 9/7 equally sharp and 3/2 twice as sharp as the thirds; [[71edo]] (709.859{{c}}) is very close to a closed system of 2/7-comma.
+In general, we would want to consider 3 somewhat more important than 7, and 7 somewhat more important than 9; in meantone, similar principles imply that an optimum is to be found sharp of 1/4-comma, though flat of [[1/5-comma meantone|1/5-comma]]. In archy, these place it sharper than 1/4-comma but flatter than 1/3-comma, which is supported by the standard [[CTE]] and [[CWE]] metrics. In fact, 22edo is slightly sharp of 1/4-comma (though still flat of the CTE optimum) and therefore pushes in the more accurate direction given the above discussion. 2/7-comma superpyth is particularly notable since it tunes the 7/6 and 9/7 equally sharp and 3/2 twice as sharp as the thirds; [[71edo]] (709.859{{c}}) and [[93edo]] with its sharp fifth of 709.677{{c}} come very close to forming closed systems of 2/7-comma.
-edo is also the point where superpyth tunes 5/4 to the familiar 400{{c}} major third of [[12edo]], and in sharper tunings different mappings of 5/4 arise with more accuracy (see [[quasiultra]] and [[ultrapyth]]), somewhat analogous to [[19edo]] (which represents [[1/3-comma meantone]] and is on the edge between septimal meantone and [[flattone]]). The same goes for flatter tunings than 22edo (see [[quasisuper]] and [[dominant (temperament)|dominant]]). Furthermore, the 11-limit extension works strictly within 22edo and 27e-edo, with 22edo conflating 11/10 with 12/11, and 27e-edo conflating 11/8 with 7/5.
+edo is also the point where superpyth tunes 5/4 to the familiar 400{{c}} major third of [[12edo]], and in sharper tunings different mappings of 5/4 arise with more accuracy (see [[quasiultra]] and [[ultrapyth]]), somewhat analogous to [[19edo]] (which represents [[1/3-comma meantone]] and is on the edge between septimal meantone and [[flattone]]). The same goes for flatter tunings than 22edo (see [[quasisuper]] and [[dominant (temperament)|dominant]]). Furthermore, the [[11-limit]] canonical extension works strictly within 22edo and 27e-edo, with 22edo conflating 11/10 with 12/11, and 27e-edo conflating 11/8 with 7/5.
 Tunings flatter than 1/4-comma archy, such as 1/5-comma (close to [[39edo]]), 1/6-comma, … are analogous to the historical "modified meantones" ([[1/6-comma meantone|1/6-comma]], [[1/7-comma meantone|1/7-comma]], …), as they prioritize the tuning of 3/2 more than the accuracy of septimal harmony. The alternative 11-limit extension, suprapyth, and an alternative extension to 5, quasisuper, work best for tunings in the range of 17edo to 22edo.
@@ Line 159: / Line 159: @@
 |-
 ! rowspan="2" |
-! colspan="2" | Euclidean
+! colspan="3" | Euclidean
 |-
-! Unskewed
+! Constrained
-! Skewed
+! Constrained & skewed
+! Destretched
 |-
 ! Equilateral
-| CEE: ~3/2 = 712.8606¢<br>(2/5-comma)
+| CEE: ~3/2 = 712.8606{{c}}<br>(2/5-comma)
-| CSEE: ~3/2 = 711.9997¢<br>(7/19-comma)
+| CSEE: ~3/2 = 711.9997{{c}}<br>(7/19-comma)
+| POEE: ~3/2 = 709.6343{{c}}
 |-
 ! Tenney
-| CTE: ~3/2 = 709.5948¢
+| CTE: ~3/2 = 709.5948{{c}}
-| CWE: ~3/2 = 709.3901¢
+| CWE: ~3/2 = 709.3901{{c}}
+| POTE: ~3/2 = 709.3213{{c}}
 |-
 ! Benedetti, <br>Wilson
-| CBE: ~3/2 = 707.7286¢<br>(18/85-comma)
+| CBE: ~3/2 = 707.7286{{c}}<br>(18/85-comma)
-| CSBE: ~3/2 = 707.9869¢<br>(25/113-comma)
+| CSBE: ~3/2 = 707.9869{{c}}<br>(25/113-comma)
+| POBE: ~3/2 = 708.6428{{c}}
 |}
@@ Line 181: / Line 185: @@
 |-
 ! rowspan="2" |
-! colspan="2" | Euclidean
+! colspan="3" | Euclidean
 |-
-! Unskewed
+! Constrained
-! Skewed
+! Constrained & skewed
+! Destretched
 |-
 ! Equilateral
-| CEE: ~3/2 = 709.7805¢
+| CEE: ~3/2 = 709.7805{{c}}
-| CSEE: ~3/2 = 710.2428¢
+| CSEE: ~3/2 = 710.2428{{c}}
+| POEE: ~3/2 = 710.4936{{c}}
 |-
 ! Tenney
-| CTE: ~3/2 = 709.5907¢
+| CTE: ~3/2 = 709.5907{{c}}
-| CWE: ~3/2 = 710.1193¢
+| CWE: ~3/2 = 710.1193{{c}}
+| POTE: ~3/2 = 710.2910{{c}}
 |-
 ! Benedetti, <br>Wilson
-| CBE: ~3/2 = 709.4859¢
+| CBE: ~3/2 = 709.4859{{c}}
-| CSBE: ~3/2 = 710.0321¢
+| CSBE: ~3/2 = 710.0321{{c}}
+| POBE: ~3/2 = 710.2421{{c}}
 |}