Ploidacot/Diploid monocot: Difference between revisions

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== Intervals and notation ==

Diploid monocot notation is complicated as it conventionally requires either the introduction of new "[[hemipythagorean]]" ~~ordinals or~~ the use of scales other than the standard diatonic scale. As such, there is no universally accepted convention. Note and interval names are provided where diploid monocot intervals align with standard monocot intervals.

Diploid monocot notation is complicated as it conventionally requires either the introduction of new "[[hemipythagorean]]" nominals, the use of scales other than the standard diatonic scale, or the use of halved pythagorean commas. As such, there is no universally accepted convention. Note and interval names are provided where diploid monocot intervals align with standard monocot intervals.

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== Temperament interpretations ==

By definition, diploid monocot temperaments equate ~~some interval~~ to ~~its octave complement~~.

By definition, diploid monocot temperaments equate a pair of intervals, mapping them both to the semioctave.

=== ~~Diaschismic~~ ===

=== Medium accuracy ===

Diaschismic sets [[3/2]] equal to [[16/15]] plus a semioctave, setting the semioctave equal to {{nowrap|[[45/32]]~[[64/45]]}}, and mapping [[5/4]] to 2 generators down. Diaschismic naturally extends to prime 17 by setting the flat 16/15 generator also equal to [[17/16]]. Diaschismic is tuned best with fifths slightly sharp of just.

~~To extend to the 7-limit~~, ~~a simple mapping (~~[[~~pajara~~]]~~) sets 7/5~~ equal to ~~the semioctave, meaning 7~~/~~4 is~~ a semioctave ~~above~~ 5/4. ~~This suggests tuning~~ the generator ~~to about 709{{c}} (or equivalently 109{{c}})~~. ~~With a~~ slightly ~~less~~ sharp ~~fifth, septimal diaschismic maps 7/4 to 8 generators down~~.

==== Diaschismic ====

Diaschismic equates [[45/32]] with [[64/45]], setting [[3/2]] equal to [[16/15]] plus a semioctave and mapping [[5/4]] to 2 generators down - a minor seventh minus a semioctave, or equivalently half of a minor sixth. Diaschismic naturally extends to prime 17 by equating the flat 16/15 generator with [[17/16]]. Diaschismic is tuned best with fifths slightly sharp of just.

~~=== Injera ===~~

To extend to the 7-limit, [[pajara]] equates 7/5 with 10/7 (as per [[jubilismic]]), meaning 7/4 simply maps to a semioctave above 5/4. This suggests tuning the generator to about 709{{c}} (or equivalently 109{{c}}).

~~Injera extends~~ [[~~meantone~~]] ~~by setting~~ 7/5 ~~equal to~~ 10/7~~, so that both~~ [[~~5/4~~]] ~~and [[~~7/4~~]] are found at~~ 4 ~~generators up, offset by a 600~~{{c}} ~~tritone representing both 7/5 and 10/7~~.

=== ~~Kalismic~~ ===

[[Septimal diaschismic]] instead maps 7/4 to a semioctave 8 fifths down with more accuracy, and the [[pythagorean comma]] is tuned sharp enough so that its half can be used as a [[septimal comma]] or [[syntonic comma]] accidental. [[Srutal]] is the most accurate of diaschismic extensions, though the most complex, at +15 fifths.

~~Kalismic sets the~~ semioctave to [[99/70]]~[[140/99]], so that [[10/9]] and [[11/7]] are a semioctave apart, as well as are [[11/10]] and [[14/9]]. The tone is split into two [[35/33]] semitones, and the [[Pythagorean comma]] into two [[2835/2816|fwiwismas]].

==== Injera ====

Injera extends [[meantone]] by equating 7/5 with 10/7 (as per jubilismic), so that both maps of [[5/4]] and [[7/4]] are found at 4 generators up, offset by a semioctave. Either sharper fifths (as in 12edo) or flatter fifths (as in 26edo) can work.

=== High accuracy ===

==== Gariwizmic ====

[[Gariwizmic]] equates [[99/70]] with [[140/99]] (as per [[kalismic]]), so that maps of [[10/9]] and [[11/7]] are a semioctave apart, as well as are [[11/10]] and [[14/9]]; The tone is split into two [[35/33]] semitones, and the Pythagorean comma into two [[2835/2816|fwiwismas]]. Gariwizmic combines this with combining the haploid chain of fifths of [[gary]] for primes 7 and 11, with new diploid mappings for primes 5, 13, 17 and 19.

It is incredibly accurate, qualifying as a [[microtemperament]]. Despite its great accuracy, it finds other primes deep in the diploid chain, with 5/4 at +39 fifths minus a semioctave, 13/8 at -27 fifths plus a semioctave, and 17/16 as +48 fifths (4 gary commas), and 19/16 as +44 fifths minus a semioctave.

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 == Intervals and notation ==
-Diploid monocot notation is complicated as it conventionally requires either the introduction of new "[[hemipythagorean]]" ordinals or the use of scales other than the standard diatonic scale. As such, there is no universally accepted convention. Note and interval names are provided where diploid monocot intervals align with standard monocot intervals.
+Diploid monocot notation is complicated as it conventionally requires either the introduction of new "[[hemipythagorean]]" nominals, the use of scales other than the standard diatonic scale, or the use of halved pythagorean commas. As such, there is no universally accepted convention. Note and interval names are provided where diploid monocot intervals align with standard monocot intervals.
 {| class="wikitable"
@@ Line 125: / Line 125: @@
 == Temperament interpretations ==
-By definition, diploid monocot temperaments equate some interval to its octave complement.
+By definition, diploid monocot temperaments equate a pair of intervals, mapping them both to the semioctave.
-=== Diaschismic ===
+=== Medium accuracy ===
-Diaschismic sets [[3/2]] equal to [[16/15]] plus a semioctave, setting the semioctave equal to {{nowrap|[[45/32]]~[[64/45]]}}, and mapping [[5/4]] to 2 generators down. Diaschismic naturally extends to prime 17 by setting the flat 16/15 generator also equal to [[17/16]]. Diaschismic is tuned best with fifths slightly sharp of just.
-To extend to the 7-limit, a simple mapping ([[pajara]]) sets 7/5 equal to the semioctave, meaning 7/4 is a semioctave above 5/4. This suggests tuning the generator to about 709{{c}} (or equivalently 109{{c}}). With a slightly less sharp fifth, septimal diaschismic maps 7/4 to 8 generators down.
+==== Diaschismic ====
+Diaschismic equates [[45/32]] with [[64/45]], setting [[3/2]] equal to [[16/15]] plus a semioctave and mapping [[5/4]] to 2 generators down - a minor seventh minus a semioctave, or equivalently half of a minor sixth. Diaschismic naturally extends to prime 17 by equating the flat 16/15 generator with [[17/16]]. Diaschismic is tuned best with fifths slightly sharp of just.
-=== Injera ===
+To extend to the 7-limit, [[pajara]] equates 7/5 with 10/7 (as per [[jubilismic]]), meaning 7/4 simply maps to a semioctave above 5/4. This suggests tuning the generator to about 709{{c}} (or equivalently 109{{c}}).
-Injera extends [[meantone]] by setting 7/5 equal to 10/7, so that both [[5/4]] and [[7/4]] are found at 4 generators up, offset by a 600{{c}} tritone representing both 7/5 and 10/7.
-=== Kalismic ===
+[[Septimal diaschismic]] instead maps 7/4 to a semioctave 8 fifths down with more accuracy, and the [[pythagorean comma]] is tuned sharp enough so that its half can be used as a [[septimal comma]] or [[syntonic comma]] accidental. [[Srutal]] is the most accurate of diaschismic extensions, though the most complex, at +15 fifths.
-Kalismic sets the semioctave to [[99/70]]~[[140/99]], so that [[10/9]] and [[11/7]] are a semioctave apart, as well as are [[11/10]] and [[14/9]]. The tone is split into two [[35/33]] semitones, and the [[Pythagorean comma]] into two [[2835/2816|fwiwismas]].
+==== Injera ====
+Injera extends [[meantone]] by equating 7/5 with 10/7 (as per jubilismic), so that both maps of [[5/4]] and [[7/4]] are found at 4 generators up, offset by a semioctave. Either sharper fifths (as in 12edo) or flatter fifths (as in 26edo) can work.
+=== High accuracy ===
+==== Gariwizmic ====
+[[Gariwizmic]] equates [[99/70]] with [[140/99]] (as per [[kalismic]]), so that maps of [[10/9]] and [[11/7]] are a semioctave apart, as well as are [[11/10]] and [[14/9]]; The tone is split into two [[35/33]] semitones, and the Pythagorean comma into two [[2835/2816|fwiwismas]]. Gariwizmic combines this with combining the haploid chain of fifths of [[gary]] for primes 7 and 11, with new diploid mappings for primes 5, 13, 17 and 19.
+It is incredibly accurate, qualifying as a [[microtemperament]]. Despite its great accuracy, it finds other primes deep in the diploid chain, with 5/4 at +39 fifths minus a semioctave, 13/8 at -27 fifths plus a semioctave, and 17/16 as +48 fifths (4 gary commas), and 19/16 as +44 fifths minus a semioctave.