Superpyth: Difference between revisions
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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. | 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}}) | 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. | 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. | ||
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|- | |- | ||
! rowspan="2" | | ! rowspan="2" | | ||
! colspan=" | ! colspan="3" | Euclidean | ||
|- | |- | ||
! | ! Constrained | ||
! | ! Constrained & skewed | ||
! Destretched | |||
|- | |- | ||
! Equilateral | ! Equilateral | ||
| CEE: ~3/2 = 712. | | CEE: ~3/2 = 712.8606{{c}}<br>(2/5-comma) | ||
| CSEE: ~3/2 = 711. | | CSEE: ~3/2 = 711.9997{{c}}<br>(7/19-comma) | ||
| POEE: ~3/2 = 709.6343{{c}} | |||
|- | |- | ||
! Tenney | ! Tenney | ||
| CTE: ~3/2 = 709. | | CTE: ~3/2 = 709.5948{{c}} | ||
| CWE: ~3/2 = 709. | | CWE: ~3/2 = 709.3901{{c}} | ||
| POTE: ~3/2 = 709.3213{{c}} | |||
|- | |- | ||
! Benedetti, <br>Wilson | ! Benedetti, <br>Wilson | ||
| CBE: ~3/2 = 707. | | CBE: ~3/2 = 707.7286{{c}}<br>(18/85-comma) | ||
| CSBE: ~3/2 = 707. | | CSBE: ~3/2 = 707.9869{{c}}<br>(25/113-comma) | ||
| POBE: ~3/2 = 708.6428{{c}} | |||
|} | |} | ||
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|- | |- | ||
! rowspan="2" | | ! rowspan="2" | | ||
! colspan=" | ! colspan="3" | Euclidean | ||
|- | |- | ||
! | ! Constrained | ||
! | ! Constrained & skewed | ||
! Destretched | |||
|- | |- | ||
! Equilateral | ! Equilateral | ||
| CEE: ~3/2 = 709. | | CEE: ~3/2 = 709.7805{{c}} | ||
| CSEE: ~3/2 = 710. | | CSEE: ~3/2 = 710.2428{{c}} | ||
| POEE: ~3/2 = 710.4936{{c}} | |||
|- | |- | ||
! Tenney | ! Tenney | ||
| CTE: ~3/2 = 709. | | CTE: ~3/2 = 709.5907{{c}} | ||
| CWE: ~3/2 = 710. | | CWE: ~3/2 = 710.1193{{c}} | ||
| POTE: ~3/2 = 710.2910{{c}} | |||
|- | |- | ||
! Benedetti, <br>Wilson | ! Benedetti, <br>Wilson | ||
| CBE: ~3/2 = 709. | | CBE: ~3/2 = 709.4859{{c}} | ||
| CSBE: ~3/2 = 710. | | CSBE: ~3/2 = 710.0321{{c}} | ||
| POBE: ~3/2 = 710.2421{{c}} | |||
|} | |} | ||