BPS: Difference between revisions

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Several extensions of this temperament are possible to incorporate additional harmonics.

~~While~~ [[11-limit]] extensions can be proposed, tempering out [[77/75]] in the flat range and [[1375/1323]] in the sharp range, neither of these are of particular accuracy. ~~The convenient comma here is [[1331/1323]] that can be tempered out~~, ~~but this~~ splits the 9/7 generator ~~(plus a tritave)~~ in ~~three~~ and ~~therefore functions instead as a weak extension of BPS, while a strong add-5 extension~~ of [[~~Mintaka~~]].

In the [[11-limit]], [[1331/1323]] is the most convenient comma that can be tempered out, which produces ''Mintra'' temperament; this splits the 9/7 generator (plus a tritave) in three and therefore functions instead as a weak extension of BPS, and a strong add-5 extension of [[Mintaka]].

While strong 11-limit extensions can be proposed, tempering out [[77/75]] in the flat range and [[1375/1323]] in the sharp range, neither of these are of particular accuracy; more accurate extensions would be of considerably higher complexity.

Another weak extension to add prime 17, known as ''Rigil'', splits the 9/7 BPS generator in half, by tempering out [[2025/2023]] and equating two of [[17/15]] to 9/7.

Sharp tunings generally possess a more convenient 13th harmonic than 11th, by tempering out [[637/625]] and identifying (25/21)<sup>2</sup> with [[13/9]], which is optimal near the 30edt tuning. It is then very easy to insert in 19 by tempering [[247/245]], and identifying [[13/9]] with [[27/19]], therefore placing the 19th harmonic 10 generators down; this extension is optimal near the 56edt tuning.

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 Several extensions of this temperament are possible to incorporate additional harmonics.
-While [[11-limit]] extensions can be proposed, tempering out [[77/75]] in the flat range and [[1375/1323]] in the sharp range, neither of these are of particular accuracy. The convenient comma here is [[1331/1323]] that can be tempered out, but this splits the 9/7 generator (plus a tritave) in three and therefore functions instead as a weak extension of BPS, while a strong add-5 extension of [[Mintaka]].
+In the [[11-limit]], [[1331/1323]] is the most convenient comma that can be tempered out, which produces ''Mintra'' temperament; this splits the 9/7 generator (plus a tritave) in three and therefore functions instead as a weak extension of BPS, and a strong add-5 extension of [[Mintaka]].
+While strong 11-limit extensions can be proposed, tempering out [[77/75]] in the flat range and [[1375/1323]] in the sharp range, neither of these are of particular accuracy; more accurate extensions would be of considerably higher complexity.
+Another weak extension to add prime 17, known as ''Rigil'', splits the 9/7 BPS generator in half, by tempering out [[2025/2023]] and equating two of [[17/15]] to 9/7.
 Sharp tunings generally possess a more convenient 13th harmonic than 11th, by tempering out [[637/625]] and identifying (25/21)<sup>2</sup> with [[13/9]], which is optimal near the 30edt tuning. It is then very easy to insert in 19 by tempering [[247/245]], and identifying [[13/9]] with [[27/19]], therefore placing the 19th harmonic 10 generators down; this extension is optimal near the 56edt tuning.