Ultrapyth: Difference between revisions

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'''Ultrapyth''' is an alternative extension of the [[archy]] chain of fifths to [[superpyth]]. Like superpyth, it is a temperament generated by a perfect fifth, where stacking two of them reaches the interval of [[8/7]] (tempering out [[64/63]]). The difference is that instead of extending to 2.3.5.7 by mapping 5 to +9 generators, it extends to the 2.3.7.13/5 subgroup (known as '''oceanfront''') by mapping the ultramajor third [[13/10]] to +4 generators (~~a supermajor third~~ which is also the diatonic major third), tempering out [[91/90]]. This makes sense because the tunings of 2.3.7 archy that optimize for the simplest 2.3.7 intervals (8/7 and [[7/6]]) are sharp of the optimal tuning for 9/7, making that third more ultramajor than supermajor. If intervals of 5 and 13 independently are desired (i.e. [[5/4]], [[13/8]]), then oceanfront may be extended to ultrapyth by mapping 5 to +14 fifths (a ~~doubly~~ augmented unison) and 13 to +18 fifths (a ~~doubly~~-augmented third). The best tunings for ultrapyth are between 712 and 714 cents.

'''Ultrapyth''' is an alternative [[extension]] of the [[archy]] [[chain of fifths]] to [[superpyth]]. Like superpyth, it is a [[regular temperament|temperament]] generated by a perfect fifth, where stacking two of them reaches the interval of [[8/7]][[~]][[9/8]], tempering out [[64/63]]. The difference is that instead of extending to 2.3.5.7 by mapping 5 to +9 generators, it extends to the 2.3.7.13/5 subgroup (known as '''oceanfront''') by mapping the ultramajor third [[13/10]] to +4 generators (which is also the diatonic major third), tempering out [[91/90]]. This makes sense because the tunings of 2.3.7 archy that optimize for the simplest 2.3.7 intervals (8/7 and [[7/6]]) are sharp of the optimal tuning for 9/7, making that third more ultramajor than supermajor. If intervals of 5 and 13 independently are desired (i.e. [[5/4]], [[13/8]]), then oceanfront may be extended to ultrapyth by mapping 5 to +14 fifths (a double-augmented unison) and 13 to +18 fifths (a double-augmented third). The best tunings for ultrapyth are between 712 and 714 cents.

If intervals of 11 are desired, [[14/11]] may be mapped to +9 generators, implying [[16/11]] is (fittingly) mapped to +11 generators and [[11/9]] is tempered together with 6/5 (a feature common to many systems with sharp fifths).

The oceanfront ~~MOS scales~~ take the form of 5L (2~~+5n~~)s, for n up to 7. Most of these scales are extremely close to [[5edo]]. [[37edo]] makes a good tuning of oceanfront or ultrapyth.

The oceanfront [[mos scale]]s take the form of 5L ({{nowrap| 5''n'' + 2 }})s, for ''n'' up to 7. Most of these scales are extremely close to [[5edo]]. [[37edo]] makes a good tuning of oceanfront or ultrapyth.

~~View~~ technical ~~temperament~~ data [[The Biosphere#Oceanfront~~|here~~]] ~~for oceanfront~~ and [[Archytas clan#Ultrapyth~~|here~~]] ~~for ultrapyth~~.

For technical data, see [[The Biosphere #Oceanfront]] and [[Archytas clan #Ultrapyth]].

== Generator chain ==

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-'''Ultrapyth''' is an alternative extension of the [[archy]] chain of fifths to [[superpyth]]. Like superpyth, it is a temperament generated by a perfect fifth, where stacking two of them reaches the interval of [[8/7]] (tempering out [[64/63]]). The difference is that instead of extending to 2.3.5.7 by mapping 5 to +9 generators, it extends to the 2.3.7.13/5 subgroup (known as '''oceanfront''') by mapping the ultramajor third [[13/10]] to +4 generators (a supermajor third which is also the diatonic major third), tempering out [[91/90]]. This makes sense because the tunings of 2.3.7 archy that optimize for the simplest 2.3.7 intervals (8/7 and [[7/6]]) are sharp of the optimal tuning for 9/7, making that third more ultramajor than supermajor. If intervals of 5 and 13 independently are desired (i.e. [[5/4]], [[13/8]]), then oceanfront may be extended to ultrapyth by mapping 5 to +14 fifths (a doubly augmented unison) and 13 to +18 fifths (a doubly-augmented third). The best tunings for ultrapyth are between 712 and 714 cents.
+'''Ultrapyth''' is an alternative [[extension]] of the [[archy]] [[chain of fifths]] to [[superpyth]]. Like superpyth, it is a [[regular temperament|temperament]] generated by a perfect fifth, where stacking two of them reaches the interval of [[8/7]][[~]][[9/8]], tempering out [[64/63]]. The difference is that instead of extending to 2.3.5.7 by mapping 5 to +9 generators, it extends to the 2.3.7.13/5 subgroup (known as '''oceanfront''') by mapping the ultramajor third [[13/10]] to +4 generators (which is also the diatonic major third), tempering out [[91/90]]. This makes sense because the tunings of 2.3.7 archy that optimize for the simplest 2.3.7 intervals (8/7 and [[7/6]]) are sharp of the optimal tuning for 9/7, making that third more ultramajor than supermajor. If intervals of 5 and 13 independently are desired (i.e. [[5/4]], [[13/8]]), then oceanfront may be extended to ultrapyth by mapping 5 to +14 fifths (a double-augmented unison) and 13 to +18 fifths (a double-augmented third). The best tunings for ultrapyth are between 712 and 714 cents.
 If intervals of 11 are desired, [[14/11]] may be mapped to +9 generators, implying [[16/11]] is (fittingly) mapped to +11 generators and [[11/9]] is tempered together with 6/5 (a feature common to many systems with sharp fifths).
-The oceanfront MOS scales take the form of 5L (2+5n)s, for n up to 7. Most of these scales are extremely close to [[5edo]]. [[37edo]] makes a good tuning of oceanfront or ultrapyth.
+The oceanfront [[mos scale]]s take the form of 5L ({{nowrap| 5''n'' + 2 }})s, for ''n'' up to 7. Most of these scales are extremely close to [[5edo]]. [[37edo]] makes a good tuning of oceanfront or ultrapyth.
-View technical temperament data [[The Biosphere#Oceanfront|here]] for oceanfront and [[Archytas clan#Ultrapyth|here]] for ultrapyth.
+For technical data, see [[The Biosphere #Oceanfront]] and [[Archytas clan #Ultrapyth]].
 == Generator chain ==