17L 2s: Difference between revisions

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{{MOS intro}} From a regular temperament theory perspective, this scale is notable for corresponding to the mega chromatic scale of the [[Alphatricot family]] temperaments. Three bright generators can be interpreted to stack to [[3/1]], but unfortunately, the generator of 17L 2s itself does not have a very convenient rational representation, since the simple ratio [[23/16]] is off-scale flat, as is (just barely) the compound ratio [[36/25]], while the prime-over-compound ratio [[13/9]] is off-scale sharp. Using very high prime harmonics/subharmonics, we can make the interpretations of [[~]][[62/43]] (bright generator) or ~[[43/31]] (dark generator); the aforementioned Alphatricot family uses the highly compound ~[[59049/40960]] as a generator; and probably the best rational that falls within the scale is ~[[75/52]], three of which differ from 3/1 by the ~~unnoticeable~~ comma of [[140625/140608]], the catasma.

{{MOS intro}} From a regular temperament theory perspective, this scale is notable for corresponding to the mega chromatic scale of the [[Alphatricot family]] temperaments. Three bright generators can be interpreted to stack to [[3/1]], but unfortunately, the generator of 17L 2s itself does not have a very convenient rational representation, since the simple ratio [[23/16]] is off-scale flat, as is (just barely) the compound ratio [[36/25]], while the prime-over-compound ratio [[13/9]] is off-scale sharp. Using very high prime harmonics/subharmonics, we can make the interpretations of [[~]][[62/43]] (bright generator) or ~[[43/31]] (dark generator); the aforementioned Alphatricot family uses the highly compound ~[[59049/40960]] as a generator; and probably the best rational that falls within the scale is ~[[75/52]], three of which differ from 3/1 by the 0.2-cent comma of [[140625/140608]], the catasma.

A pitfall of the use of compound harmonics and subharmonics in a generator is that they multiply the effect of shifts in mapping of their respective primes with scale hardness — for instance, ~59049/40960 only maps correctly within a narrow step ratio range close to 10:3, while ~36/25 fails to map correctly even for several EDOs close to the soft end of the scale's tuning spectrum (as does the simpler but flatter ~23/16); the even simpler ~13/9 (off-scale sharp) is likewise affected. Using such generators outside of a narrow subset of the EDOs supporting the scale depends upon direct approximation of a compound harmonic and/or subharmonic such as 9 or 25. This is awkward when one also needs to use a component harmonic as specified in the patent vals of the EDOs, thus requiring the use of nonstandard conditional subgroup temperaments such as 2.3♯.3♭.5 and 2.3.5♯.5♭ (or 2.3.9.5 and 2.3.5.25), with provision of a rule specifying when to use the direct approximation as opposed to the patent val mapping.

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 {{Infobox MOS}}
-{{MOS intro}} From a regular temperament theory perspective, this scale is notable for corresponding to the mega chromatic scale of the [[Alphatricot family]] temperaments. Three bright generators can be interpreted to stack to [[3/1]], but unfortunately, the generator of 17L&nbsp;2s itself does not have a very convenient rational representation, since the simple ratio [[23/16]] is off-scale flat, as is (just barely) the compound ratio [[36/25]], while the prime-over-compound ratio [[13/9]] is off-scale sharp. Using very high prime harmonics/subharmonics, we can make the interpretations of [[~]][[62/43]] (bright generator) or ~[[43/31]] (dark generator); the aforementioned Alphatricot family uses the highly compound ~[[59049/40960]] as a generator; and probably the best rational that falls within the scale is ~[[75/52]], three of which differ from 3/1 by the unnoticeable comma of [[140625/140608]], the catasma.
+{{MOS intro}} From a regular temperament theory perspective, this scale is notable for corresponding to the mega chromatic scale of the [[Alphatricot family]] temperaments. Three bright generators can be interpreted to stack to [[3/1]], but unfortunately, the generator of 17L&nbsp;2s itself does not have a very convenient rational representation, since the simple ratio [[23/16]] is off-scale flat, as is (just barely) the compound ratio [[36/25]], while the prime-over-compound ratio [[13/9]] is off-scale sharp. Using very high prime harmonics/subharmonics, we can make the interpretations of [[~]][[62/43]] (bright generator) or ~[[43/31]] (dark generator); the aforementioned Alphatricot family uses the highly compound ~[[59049/40960]] as a generator; and probably the best rational that falls within the scale is ~[[75/52]], three of which differ from 3/1 by the 0.2-cent comma of [[140625/140608]], the catasma.
 A pitfall of the use of compound harmonics and subharmonics in a generator is that they multiply the effect of shifts in mapping of their respective primes with scale hardness &mdash; for instance, ~59049/40960 only maps correctly within a narrow step ratio range close to 10:3, while ~36/25 fails to map correctly even for several EDOs close to the soft end of the scale's tuning spectrum (as does the simpler but flatter ~23/16); the even simpler ~13/9 (off-scale sharp) is likewise affected. Using such generators outside of a narrow subset of the EDOs supporting the scale depends upon direct approximation of a compound harmonic and/or subharmonic such as 9 or 25. This is awkward when one also needs to use a component harmonic as specified in the patent vals of the EDOs, thus requiring the use of nonstandard conditional subgroup temperaments such as 2.3♯.3♭.5 and 2.3.5♯.5♭ (or 2.3.9.5 and 2.3.5.25), with provision of a rule specifying when to use the direct approximation as opposed to the patent val mapping.