User:UnbihexiumFan/Temperaments: Difference between revisions

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A collection of temperaments that I have found that may or may not have yet been discovered. ~~A lot of these are~~ the ~~same as already-known temperaments but with non-octave periods. I am not very good with technical details so even though they are included as info on most temperaments I will not be putting it here~~.

A collection of temperaments that I have found that may or may not have yet been discovered. If you find any inaccuracies feel free to point them out on the [[User talk:UnbihexiumFan/Temperaments|talk page]].

== Stearnsmic 7/4-period temperaments ==

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'''Bolded''' ratios are octave-reduced harmonics up to 21.

=== ~~2.3.7.11.13~~ extension ===

=== No-5's 19-limit extension ===

~~One extension to~~ the ~~2.3.7.11.13 subgroup~~ is to ~~temper~~ [[28/27]] to be equal to [[27/26]], ~~tempering out~~ [[~~729~~/~~728~~]]~~. This maps the 13th harmonic~~, [[13/8]], to the ~~sesqui~~-~~augmented second~~, D{{~~sesquisharp~~}}.

While the major third is too sharp to be seen as [[5/4]], it can be seen as [[24/19]] or [[64/51]]. Treating it equal to both tempers out [[513/512]] and [[4131/4096]], providing a comma basis of [[2057/2052]], [[513/512]], [[154/153]], and [[243/242]]. The 17th harmonic is mapped to the minor second (-10 generators, D{{flat}} on C) and the 19th harmonic is mapped to the minor third (-6 generators, E{{flat}} on C). The 13th harmonic can be added by setting [[28/27]] equal to [[27/26]], tempering out [[729/728]]. This maps the 13th harmonic, [[13/8]], to the sesqui-augmented fifth (+23 generators, G{{sesquisharp}} on C).

~~2.3.7.11.17.19 extension:~~ [[~~2057~~/~~2052~~]], [[~~513~~/~~512~~]], [[~~154~~/~~153~~]], [[~~243~~/~~242~~]]

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-A collection of temperaments that I have found that may or may not have yet been discovered. A lot of these are the same as already-known temperaments but with non-octave periods. I am not very good with technical details so even though they are included as info on most temperaments I will not be putting it here.
+A collection of temperaments that I have found that may or may not have yet been discovered. If you find any inaccuracies feel free to point them out on the [[User talk:UnbihexiumFan/Temperaments|talk page]].
 == Stearnsmic 7/4-period temperaments ==
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 '''Bolded''' ratios are octave-reduced harmonics up to 21.
-=== 2.3.7.11.13 extension ===
+=== No-5's 19-limit extension ===
-One extension to the 2.3.7.11.13 subgroup is to temper [[28/27]] to be equal to [[27/26]], tempering out [[729/728]]. This maps the 13th harmonic, [[13/8]], to the sesqui-augmented second, D{{sesquisharp}}.
+While the major third is too sharp to be seen as [[5/4]], it can be seen as [[24/19]] or [[64/51]]. Treating it equal to both tempers out [[513/512]] and [[4131/4096]], providing a comma basis of [[2057/2052]], [[513/512]], [[154/153]], and [[243/242]]. The 17th harmonic is mapped to the minor second (-10 generators, D{{flat}} on C) and the 19th harmonic is mapped to the minor third (-6 generators, E{{flat}} on C). The 13th harmonic can be added by setting [[28/27]] equal to [[27/26]], tempering out [[729/728]]. This maps the 13th harmonic, [[13/8]], to the sesqui-augmented fifth (+23 generators, G{{sesquisharp}} on C).
-.3.7.11.17.19 extension: [[2057/2052]], [[513/512]], [[154/153]], [[243/242]]