Diaschismic: Difference between revisions

Line 18:

A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. The mappings of primes [[11/1|11]] and [[13/1|13]] can also be characterized by [[parapyth]], where the major third at +4 fifths represents [[14/11]], and the minor third at -3 fifths represents [[13/11]], which makes sense as the fifth is tuned slightly sharp.

Since the whole tone has been split in two halves of ~16/15, it makes good sense to equate that interval to [[17/16]] and [[18/17]] as well by tempering out [[256/255]] ({{S|16}}), [[289/288]] ({{S|17}}), and their product [[136/135]]. Therefore, diaschismic is most naturally viewed as a full [[17-limit]] temperament, tempering out 126/125, 136/135, [[176/175]], [[196/195]], and 256/255. This extension works best between [[46edo]] and [[58edo]]. The [[restriction]] to the [[2.3.5.17 subgroup|2.3.5.17-~~]][[~~subgroup]], called '''srutal archagall''', is simpler and more flexible with its tunings (any tuning between [[10edo]] and [[12edo]] works, as is the case with 5-limit diaschismic), but the amount of harmonic resource is more limited.

Since the whole tone has been split in two halves of ~16/15, it makes good sense to equate that interval to [[17/16]] and [[18/17]] as well by tempering out [[256/255]] ({{S|16}}), [[289/288]] ({{S|17}}), and their product [[136/135]]. Therefore, diaschismic is most naturally viewed as a full [[17-limit]] temperament, tempering out 126/125, 136/135, [[176/175]], [[196/195]], and 256/255. This extension works best between [[46edo]] and [[58edo]]. The [[restriction]] to the [[2.3.5.17 subgroup|2.3.5.17-subgroup]], called '''srutal archagall''', is simpler and more flexible with its tunings (any tuning between [[10edo]] and [[12edo]] works, as is the case with 5-limit diaschismic), but the amount of harmonic resource is more limited.

An alternative extension to the full 17-limit is [[srutal]], which has a more complex mapping of prime 7 at +15 fifths, tempering out [[4375/4374]]. The mappings for primes 11 and 13 follow through parapyth, and 17 is mapped as in srutal archagall. This works best for a sharper tuning, between [[34edo]] (with the 34d [[val]]) and [[46edo]]. Another option for extension is [[pajara]], which equates the semioctave to [[7/5]] and [[10/7]]. This inherits the inaccuracy of [[archy]], while providing a much simpler representation of the 7-limit.

@@ Line 18: / Line 18: @@
 A stack of twelve perfect fifths octave reduced (a [[diesis (scale theory)|diesis]]), in this tuning range, is close in size to a quartertone, and that plus a period can be used to represent [[16/11]]. Three more fifths on top of 16/11 give [[16/13]]. The mappings of primes [[11/1|11]] and [[13/1|13]] can also be characterized by [[parapyth]], where the major third at +4 fifths represents [[14/11]], and the minor third at -3 fifths represents [[13/11]], which makes sense as the fifth is tuned slightly sharp.
-Since the whole tone has been split in two halves of ~16/15, it makes good sense to equate that interval to [[17/16]] and [[18/17]] as well by tempering out [[256/255]] ({{S|16}}), [[289/288]] ({{S|17}}), and their product [[136/135]]. Therefore, diaschismic is most naturally viewed as a full [[17-limit]] temperament, tempering out 126/125, 136/135, [[176/175]], [[196/195]], and 256/255. This extension works best between [[46edo]] and [[58edo]]. The [[restriction]] to the [[2.3.5.17 subgroup|2.3.5.17-]][[subgroup]], called '''srutal archagall''', is simpler and more flexible with its tunings (any tuning between [[10edo]] and [[12edo]] works, as is the case with 5-limit diaschismic), but the amount of harmonic resource is more limited.
+Since the whole tone has been split in two halves of ~16/15, it makes good sense to equate that interval to [[17/16]] and [[18/17]] as well by tempering out [[256/255]] ({{S|16}}), [[289/288]] ({{S|17}}), and their product [[136/135]]. Therefore, diaschismic is most naturally viewed as a full [[17-limit]] temperament, tempering out 126/125, 136/135, [[176/175]], [[196/195]], and 256/255. This extension works best between [[46edo]] and [[58edo]]. The [[restriction]] to the [[2.3.5.17 subgroup|2.3.5.17-subgroup]], called '''srutal archagall''', is simpler and more flexible with its tunings (any tuning between [[10edo]] and [[12edo]] works, as is the case with 5-limit diaschismic), but the amount of harmonic resource is more limited.
 An alternative extension to the full 17-limit is [[srutal]], which has a more complex mapping of prime 7 at +15 fifths, tempering out [[4375/4374]]. The mappings for primes 11 and 13 follow through parapyth, and 17 is mapped as in srutal archagall. This works best for a sharper tuning, between [[34edo]] (with the 34d [[val]]) and [[46edo]]. Another option for extension is [[pajara]], which equates the semioctave to [[7/5]] and [[10/7]]. This inherits the inaccuracy of [[archy]], while providing a much simpler representation of the 7-limit.