Hyperpyth: Difference between revisions

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=Hyperpyth=

== Hyperpyth ==

Using the fifth harmonic as an interval of equivalence, instead of the more common octave or even tritave, the first place to look for xenharmonies is isoharmonic chords, of which there are two. 1:2:3:4:5, and 5:9:13:17:21:25. The latter is more xenharmonic, though 16ed5 suits the former and can be strange. It turns out that the key to making a scale of this chord is to use as the generator ~13/5, which, in the fifth harmonic, results in a scale of the same shape as meantone and similar placement of consonances. Thus it is a [[MacrodiatonicAndMicrodiatonic|macrodiatonic]] tuning. In this case, the most suitable scales are such as would have a sharp fifth, in octaves known as "superpythagorean", so I dub this "hyperpyth".

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Good tunings for hyperpyth are:

[[~~5ed5|~~5ed5]]

* [[5ed5]]

* [[10ed5]]

[[~~10ed5|~~10ed5]]

* [[17ed5]]

* [[22ed5]]

[[~~17ed5|~~17ed5]]

* [[29ed5]]

* [[39ed5]]

[[~~22ed5|~~22ed5]]

[[~~29ed5|~~29ed5]]

[[~~39ed5|~~39ed5]]

etc.

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[http://x31eq.com/cgi-bin/rt.cgi?ets=9ccqqfff_13p&limit=5_9_13 http://x31eq.com/cgi-bin/rt.cgi?ets=9ccqqfff_13p&limit=5_9_13]

=Hyperreich?=

== Hyperreich? ==

Looking at the primes, 7 and 11 (and 19) are "conspicuously absent" which begs comparison to the Meantone/Orgone dichotomy. The search being on, in the context of simple scales, 11/5 is close enough to the square root of 5, that one might as well just use it (1393 v the real 11/5 at 1365 cents); eventually as step sizes get closer to 60 cents or so, better approximations will abound. This would make a good period for a scale. The pure 7/5 then is around 582 cents, and among the simpler temperaments 557-cent (from [[5ed5]], [[10ed5]], 15ed5) and 596-cent (from [[14ed5]], which is a slightly compressed [[6edo]]) intervals are the closest approximations. That is, until [[19ed5]] (14+5) which is a very slightly stretched [[13edt]] (Bohlen-Pierce) scale, and [[24ed5]] which is something completely different.

Looking at the primes, 7 and 11 (and 19) are "conspicuously absent" which begs comparison to the Meantone/Orgone dichotomy. The search being on, in the context of simple scales, 11/5 is close enough to the square root of 5, that one might as well just use it (1393 v the real 11/5 at 1365 cents); eventually as step sizes get closer to 60 cents or so, better approximations will abound. This would make a good period for a scale. The pure 7/5 then is around 582 cents, and among the simpler temperaments 557-cent (from [[5ed5|5ed5]], [[10ed5|10ed5]], 15ed5) and 596-cent (from [[14ed5|14ed5]], which is a slightly compressed [[6edo|6edo]]) intervals are the closest approximations. That is, until [[19ed5|19ed5]] (14+5) which is a very slightly stretched [[13edt|13edt]] (Bohlen-Pierce) scale, and [[24ed5|24ed5]] which is something completely different.

[[Category:Fifth harmonic]]

[[Category:~~fifth_harmonic~~]]

[[Category:Temperament]]

[[Category:~~temperament~~]]

@@ Line 1: / Line 1: @@
-=Hyperpyth=
+== Hyperpyth ==
 Using the fifth harmonic as an interval of equivalence, instead of the more common octave or even tritave, the first place to look for xenharmonies is isoharmonic chords, of which there are two. 1:2:3:4:5, and 5:9:13:17:21:25. The latter is more xenharmonic, though 16ed5 suits the former and can be strange. It turns out that the key to making a scale of this chord is to use as the generator ~13/5, which, in the fifth harmonic, results in a scale of the same shape as meantone and similar placement of consonances. Thus it is a [[MacrodiatonicAndMicrodiatonic|macrodiatonic]] tuning. In this case, the most suitable scales are such as would have a sharp fifth, in octaves known as "superpythagorean", so I dub this "hyperpyth".
@@ Line 7: / Line 7: @@
 Good tunings for hyperpyth are:
-[[5ed5|5ed5]]
+* [[5ed5]]
+* [[10ed5]]
-[[10ed5|10ed5]]
+* [[17ed5]]
+* [[22ed5]]
-[[17ed5|17ed5]]
+* [[29ed5]]
+* [[39ed5]]
-[[22ed5|22ed5]]
-[[29ed5|29ed5]]
-[[39ed5|39ed5]]
 etc.
@@ Line 25: / Line 20: @@
 [http://x31eq.com/cgi-bin/rt.cgi?ets=9ccqqfff_13p&limit=5_9_13 http://x31eq.com/cgi-bin/rt.cgi?ets=9ccqqfff_13p&amp;limit=5_9_13]
-=Hyperreich?=
+== Hyperreich? ==
+Looking at the primes, 7 and 11 (and 19) are "conspicuously absent" which begs comparison to the Meantone/Orgone dichotomy. The search being on, in the context of simple scales, 11/5 is close enough to the square root of 5, that one might as well just use it (1393 v the real 11/5 at 1365 cents); eventually as step sizes get closer to 60 cents or so, better approximations will abound. This would make a good period for a scale. The pure 7/5 then is around 582 cents, and among the simpler temperaments 557-cent (from [[5ed5]], [[10ed5]], 15ed5) and 596-cent (from [[14ed5]], which is a slightly compressed [[6edo]]) intervals are the closest approximations. That is, until [[19ed5]] (14+5) which is a very slightly stretched [[13edt]] (Bohlen-Pierce) scale, and [[24ed5]] which is something completely different.
-Looking at the primes, 7 and 11 (and 19) are "conspicuously absent" which begs comparison to the Meantone/Orgone dichotomy. The search being on, in the context of simple scales, 11/5 is close enough to the square root of 5, that one might as well just use it (1393 v the real 11/5 at 1365 cents); eventually as step sizes get closer to 60 cents or so, better approximations will abound. This would make a good period for a scale. The pure 7/5 then is around 582 cents, and among the simpler temperaments 557-cent (from [[5ed5|5ed5]], [[10ed5|10ed5]], 15ed5) and 596-cent (from [[14ed5|14ed5]], which is a slightly compressed [[6edo|6edo]]) intervals are the closest approximations. That is, until [[19ed5|19ed5]] (14+5) which is a very slightly stretched [[13edt|13edt]] (Bohlen-Pierce) scale, and [[24ed5|24ed5]] which is something completely different.
+[[Category:Fifth harmonic]]
-[[Category:fifth_harmonic]]
+[[Category:Temperament]]
-[[Category:temperament]]