Talk:17L 2s: Difference between revisions

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==== Proposed text to add to introduction section (revised) ====

From a regular temperament theory perspective, this scale is notable for corresponding to the mega chromatic scale of the [[Alphatricot family]] temperaments. Unfortunately, its generator does not have a convenient rational representation — the simple ratios [[23/16]] and even [[36/25]] are off-scale flat, while the simple ratio [[13/9]] is off-scale sharp. The Alphatricot family uses ~[[59049/40960]] as a generator. ~~Note~~ that ~~although~~ the comparitively simple 36/25 is just barely off-scale flat (being near just in the equalized endpoint [[19edo]])~~, using it effectively~~ depends upon direct approximation of ~~the 25th~~ harmonic~~, while~~ one ~~might~~ also ~~need~~ to use ~~the 5th~~ harmonic as ~~opposed to its square~~, requiring the use of a 2.3.5♯.5♭ (or 2.3.5.25) ~~subgroup temperament that includes~~ a rule on ~~when to use each flavor of 5th harmonic (or~~ when to use the ~~5th harmonic and when to use~~ direct approximation of the ~~25th harmonic). In some cases, the analogous treatment might be needed for the 9th harmonic as well (such as when using ~13/9 with [[112edo]])~~.

From a regular temperament theory perspective, this scale is notable for corresponding to the mega chromatic scale of the [[Alphatricot family]] temperaments. Unfortunately, its generator does not have a convenient rational representation — the simple ratios [[23/16]] and even [[36/25]] are off-scale flat, while the simple ratio [[13/9]] is off-scale sharp. The Alphatricot family uses ~[[59049/40960]] as a generator. 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 range close to step ratio 10:3, while the comparitively simple 36/25 that is just barely off-scale flat (being near just in the equalized endpoint [[19edo]]) fails to map correctly even for several EDOs close to the soft end of the scale's tuning spectrum; the even simpler 13/9 (off-scale sharp) is likewise affected. Using such generators outside of a narrow set of EDos depends upon direct approximation of a harmonic and/or subharmonic. 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 or 2.3.5♯.5♭ (or 2.3.9.5 or 2.3.5.25) including a rule on when to use the direct approximation or the patent val.

Added: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 15:14, 2 May 2025 (UTC)<br>

Last modified: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 15:31, 2 May 2025 (UTC)

Last modified: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 18:57, 2 May 2025 (UTC)

@@ Line 28: / Line 28: @@
 ==== Proposed text to add to introduction section (revised) ====
-From a regular temperament theory perspective, this scale is notable for corresponding to the mega chromatic scale of the [[Alphatricot family]] temperaments. Unfortunately, its generator does not have a convenient rational representation &mdash; the simple ratios [[23/16]] and even [[36/25]] are off-scale flat, while the simple ratio [[13/9]] is off-scale sharp. The Alphatricot family uses ~[[59049/40960]] as a generator. Note that although the comparitively simple 36/25 is just barely off-scale flat (being near just in the equalized endpoint [[19edo]]), using it effectively depends upon direct approximation of the 25th harmonic, while one might also need to use the 5th harmonic as opposed to its square, requiring the use of a 2.3.5♯.5♭ (or 2.3.5.25) subgroup temperament that includes a rule on when to use each flavor of 5th harmonic (or when to use the 5th harmonic and when to use direct approximation of the 25th harmonic).  In some cases, the analogous treatment might be needed for the 9th harmonic as well (such as when using ~13/9 with [[112edo]]).
+From a regular temperament theory perspective, this scale is notable for corresponding to the mega chromatic scale of the [[Alphatricot family]] temperaments. Unfortunately, its generator does not have a convenient rational representation &mdash; the simple ratios [[23/16]] and even [[36/25]] are off-scale flat, while the simple ratio [[13/9]] is off-scale sharp. The Alphatricot family uses ~[[59049/40960]] as a generator. 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 range close to step ratio 10:3, while the comparitively simple 36/25 that is just barely off-scale flat (being near just in the equalized endpoint [[19edo]]) fails to map correctly even for several EDOs close to the soft end of the scale's tuning spectrum; the even simpler 13/9 (off-scale sharp) is likewise affected. Using such generators outside of a narrow set of EDos depends upon direct approximation of a harmonic and/or subharmonic. 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 or 2.3.5♯.5♭ (or 2.3.9.5 or 2.3.5.25) including a rule on when to use the direct approximation or the patent val.
 Added:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 15:14, 2 May 2025 (UTC)<br>
-Last modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 15:31, 2 May 2025 (UTC)
+Last modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 18:57, 2 May 2025 (UTC)