BPS: Difference between revisions

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

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]].

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]], which produces [[5L 2s (3/1-equivalent)|5L 2s]] and [[5L 7s (3/1-equivalent)|5L 7s]] MOS scales (functioning as a macro-[[superpyth]]). Simple tunings include [[17edt]] and [[39edt]].

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. This produces [[8L 1s (3/1-equivalent)|8L 1s]] enneatonic and [[9L 8s (3/1-equivalent)|9L 8s]] chromatic MOS scales. Simple tunings include [[17edt]] and [[26edt]].

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.
-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]].
+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]], which produces [[5L 2s (3/1-equivalent)|5L 2s]] and [[5L 7s (3/1-equivalent)|5L 7s]] MOS scales (functioning as a macro-[[superpyth]]). Simple tunings include [[17edt]] and [[39edt]].
+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. This produces [[8L 1s (3/1-equivalent)|8L 1s]] enneatonic and [[9L 8s (3/1-equivalent)|9L 8s]] chromatic MOS scales. Simple tunings include [[17edt]] and [[26edt]].
 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.