Superpyth: Difference between revisions

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; 5-note mos ([[2L 3s]], proper)

In contrast to the meantone pentic scale, the superpyth pentic is much softer and mellow in quality, which is related to the fact that the intervals of the 2.3.7 subgroup cluster around [[5edo]]. As such, this system may be preferred over diatonic for interval classification, with 7/6 becoming a major interval and 8/7~9/8 becoming a minor one, and 49/48~[[28/27]] becoming a chroma. See [[Pentatonic Functional Just System]] for further ~~reading~~.

In contrast to the meantone pentic scale, the superpyth pentic is much softer and mellow in quality, which is related to the fact that the intervals of the 2.3.7 subgroup cluster around [[5edo]]. As such, this system may be preferred over diatonic for interval classification, with 7/6 becoming a major interval and 8/7~9/8 becoming a minor one, and 49/48~[[28/27]] becoming a chroma. See [[Pentatonic Functional Just System]] for a further explanation of what such a system would look like.

; 7-note mos ([[5L 2s]], improper)

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A case can also be made for tuning archy even sharper than 27edo, which involves the notion of splitting the error of 4/3 into that of 8/7 and 7/6. This is a similar logic to Zarlino's preference for [[2/7-comma meantone]], treating [[~]][[6:7:8]] as the fundamental chord of the 2.3.7 subgroup, and in this case would imply 2/5-comma archy, where [[49/48]] is tuned justly, and 8/7 and 7/6 are both 1/5 a septimal comma off, and which is closely approximated by [[32edo]]. Unlike in the case of meantone, [[CEE]] optimization agrees with the notion of such a sharp tuning, where 3 is twice as sharp as 7. In this range, the best extension to prime 5 is ultrapyth.

Finally, it may be noted that the {{w|plastic ~~number~~}} has a value of ~486.822 cents, which, taken as a generator (~4/3) and assuming a pure-octave period, constitutes an extremely sharp variety of archy. ~~This can be explained since archy equates [[21/16]] and [[4/3]]~~, ~~making~~ the ~9:12:~~16:21 chord evenly spaced by ~4/3, and when keeping {{nowrap| ~9~~ + ~~~12 {{=}} ~21 }} the generator becomes the plastic number~~.

Finally, it may be noted that the {{w|plastic ratio}} has a value of ~486.822 cents, which, taken as a generator (~4/3) and assuming a pure-octave period, constitutes an extremely sharp variety of archy. In fact, it is the tuning that makes ~6:7:8 become +1+1 [[delta-rational]].

=== Norm-based tunings ===

@@ Line 139: / Line 139: @@
 ; 5-note mos ([[2L&nbsp;3s]], proper)
-In contrast to the meantone pentic scale, the superpyth pentic is much softer and mellow in quality, which is related to the fact that the intervals of the 2.3.7 subgroup cluster around [[5edo]]. As such, this system may be preferred over diatonic for interval classification, with 7/6 becoming a major interval and 8/7~9/8 becoming a minor one, and 49/48~[[28/27]] becoming a chroma. See [[Pentatonic Functional Just System]] for further reading.
+In contrast to the meantone pentic scale, the superpyth pentic is much softer and mellow in quality, which is related to the fact that the intervals of the 2.3.7 subgroup cluster around [[5edo]]. As such, this system may be preferred over diatonic for interval classification, with 7/6 becoming a major interval and 8/7~9/8 becoming a minor one, and 49/48~[[28/27]] becoming a chroma. See [[Pentatonic Functional Just System]] for a further explanation of what such a system would look like.
 ; 7-note mos ([[5L&nbsp;2s]], improper)
@@ Line 171: / Line 171: @@
 A case can also be made for tuning archy even sharper than 27edo, which involves the notion of splitting the error of 4/3 into that of 8/7 and 7/6. This is a similar logic to Zarlino's preference for [[2/7-comma meantone]], treating [[~]][[6:7:8]] as the fundamental chord of the 2.3.7 subgroup, and in this case would imply 2/5-comma archy, where [[49/48]] is tuned justly, and 8/7 and 7/6 are both 1/5 a septimal comma off, and which is closely approximated by [[32edo]]. Unlike in the case of meantone, [[CEE]] optimization agrees with the notion of such a sharp tuning, where 3 is twice as sharp as 7. In this range, the best extension to prime 5 is ultrapyth.
-Finally, it may be noted that the {{w|plastic number}} has a value of ~486.822 cents, which, taken as a generator (~4/3) and assuming a pure-octave period, constitutes an extremely sharp variety of archy. This can be explained since archy equates [[21/16]] and [[4/3]], making the ~9:12:16:21 chord evenly spaced by ~4/3, and when keeping {{nowrap| ~9 + ~12 {{=}} ~21 }} the generator becomes the plastic number.
+Finally, it may be noted that the {{w|plastic ratio}} has a value of ~486.822 cents, which, taken as a generator (~4/3) and assuming a pure-octave period, constitutes an extremely sharp variety of archy. In fact, it is the tuning that makes ~6:7:8 become +1+1 [[delta-rational]].
 === Norm-based tunings ===