Technical data guide for regular temperaments: Difference between revisions

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~~{{todo|complete section|inline=1}}~~

~~==== Wedgie ====~~

~~{{Main|Plücker coordinates}}~~

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== Tuning properties ==

While an abstract regular temperament does not specify the values to which its intervals (meaning, its generators or valuations of primes) are tuned, it is clear that there are ways to set these values that are reasonable, and those that are absurd. For instance, meantone could be tuned with its "[[3/2]]" set to ~~675c~~, and therefore its {{nowrap|"[[5/4]]" ~ "81/64"}} set to {{nowrap|~~4×675¢ − 2400¢~~ {{=}} ~~300¢~~}}. However, this is an absurd tuning for meantone since ~~300¢~~ has a far better interpretation as [[6/5]] than 5/4, and the temperament providing that interpretation is instead [[mavila]].

While an abstract regular temperament does not specify the values to which its intervals (meaning, its generators or valuations of primes) are tuned, it is clear that there are ways to set these values that are reasonable, and those that are absurd. For instance, meantone could be tuned with its "[[3/2]]" set to 675{{c}}, and therefore its {{nowrap| "[[5/4]]" ~ "81/64" }} set to {{nowrap| 4×675{{c}} − 2400{{c}} {{=}} 300{{c}} }}. However, this is an absurd tuning for meantone since 300{{c}} has a far better interpretation as [[6/5]] than 5/4, and the temperament providing that interpretation is instead [[mavila]].

Therefore, one can speak of temperaments as having finite "tuning ranges" for their generator, which is useful in the picture of building [[~~MOS~~ scale]]s as finite subsets of the intervals available in the temperament.

Therefore, one can speak of temperaments as having finite tuning ranges for their generator, which is useful in the picture of building [[mos scale]]s as finite subsets of the intervals available in the temperament.

=== Optimal tuning(s) ===

Tuning ''optimization'' is, essentially, the task of finding a tuning for a given regular temperament that has the lowest error in some way. While many approaches exist to going about this, the most widely used are algorithms based on the ''TE metric'', which weight all intervals in the infinite set available to the temperament by a measure of their complexity, and tune in order to minimize deviation from just across all of them.

It is ~~conventional on the wiki~~ to optimize under the constraint that the octave (or equave) is tuned pure, and therefore that the generator known as the ''period'' is either an exact equave or a fraction thereof~~. The rational interpretation of the period is depicted as equated to this fraction. However~~, the other ''generators'' are tuned to inexact values ~~expressed~~ in cents~~, and appear as rational interpretations equated to these values~~. Multiple optimization algorithms (most commonly CTE and POTE) ~~may appear~~; different algorithms have subtle differences and one or the other may be chosen for a specific use. However, optimal tunings are used more often as guidelines for where "good tunings" of a temperament are than as exact ways to tune, and for that purpose the algorithms usually agree sufficiently (aside from extreme exotemperaments and other special cases).

It is common to optimize under the constraint that the octave (or equave) is tuned pure, and therefore that the generator known as the ''period'' is either an exact equave or a fraction thereof, whereas the other ''generators'' are tuned to inexact values. The simplest rational interpretation of these intervals are given in cents. Multiple optimization algorithms are offered, for both tempered-octave (WE, in progress) and pure-octave tuning (most commonly CTE and POTE, but we're unifying at CWE as of late); different algorithms have subtle differences and one or the other may be chosen for a specific use. However, optimal tunings are used more often as guidelines for where good tunings of a temperament are than as exact ways to tune, and for that purpose the algorithms usually agree sufficiently (aside from extreme exotemperaments and other special cases).

In the future, temperaments may appear with optimal tunings of ''prime harmonics'' (and their deviation from just) in a collapsible table, though this will be merely for the sake of convenience as all tunings of intervals can be derived from the generators and the mapping.

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 {{Main|Mapping to lattice}}
 {{See also|Harmonic lattice diagram}}
-{{todo|complete section|inline=1}}
-==== Wedgie ====
-{{Main|Plücker coordinates}}
 {{todo|complete section|inline=1}}
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 == Tuning properties ==
-While an abstract regular temperament does not specify the values to which its intervals (meaning, its generators or valuations of primes) are tuned, it is clear that there are ways to set these values that are reasonable, and those that are absurd. For instance, meantone could be tuned with its "[[3/2]]" set to 675c, and therefore its {{nowrap|"[[5/4]]" ~ "81/64"}} set to {{nowrap|4×675¢ &minus; 2400¢ {{=}} 300¢}}. However, this is an absurd tuning for meantone since 300¢ has a far better interpretation as [[6/5]] than 5/4, and the temperament providing that interpretation is instead [[mavila]].
+While an abstract regular temperament does not specify the values to which its intervals (meaning, its generators or valuations of primes) are tuned, it is clear that there are ways to set these values that are reasonable, and those that are absurd. For instance, meantone could be tuned with its "[[3/2]]" set to 675{{c}}, and therefore its {{nowrap| "[[5/4]]" ~ "81/64" }} set to {{nowrap| 4×675{{c}} − 2400{{c}} {{=}} 300{{c}} }}. However, this is an absurd tuning for meantone since 300{{c}} has a far better interpretation as [[6/5]] than 5/4, and the temperament providing that interpretation is instead [[mavila]].
-Therefore, one can speak of temperaments as having finite "tuning ranges" for their generator, which is useful in the picture of building [[MOS scale]]s as finite subsets of the intervals available in the temperament.
+Therefore, one can speak of temperaments as having finite tuning ranges for their generator, which is useful in the picture of building [[mos scale]]s as finite subsets of the intervals available in the temperament.
 === Optimal tuning(s) ===
-{{Main|Optimization}}
+{{Main| Optimization }}
 Tuning ''optimization'' is, essentially, the task of finding a tuning for a given regular temperament that has the lowest error in some way. While many approaches exist to going about this, the most widely used are algorithms based on the ''TE metric'', which weight all intervals in the infinite set available to the temperament by a measure of their complexity, and tune in order to minimize deviation from just across all of them.
-It is conventional on the wiki to optimize under the constraint that the octave (or equave) is tuned pure, and therefore that the generator known as the ''period'' is either an exact equave or a fraction thereof. The rational interpretation of the period is depicted as equated to this fraction. However, the other ''generators'' are tuned to inexact values expressed in cents, and appear as rational interpretations equated to these values. Multiple optimization algorithms (most commonly CTE and POTE) may appear; different algorithms have subtle differences and one or the other may be chosen for a specific use. However, optimal tunings are used more often as guidelines for where "good tunings" of a temperament are than as exact ways to tune, and for that purpose the algorithms usually agree sufficiently (aside from extreme exotemperaments and other special cases).
+It is common to optimize under the constraint that the octave (or equave) is tuned pure, and therefore that the generator known as the ''period'' is either an exact equave or a fraction thereof, whereas the other ''generators'' are tuned to inexact values. The simplest rational interpretation of these intervals are given in cents. Multiple optimization algorithms are offered, for both tempered-octave (WE, in progress) and pure-octave tuning (most commonly CTE and POTE, but we're unifying at CWE as of late); different algorithms have subtle differences and one or the other may be chosen for a specific use. However, optimal tunings are used more often as guidelines for where good tunings of a temperament are than as exact ways to tune, and for that purpose the algorithms usually agree sufficiently (aside from extreme exotemperaments and other special cases).
 In the future, temperaments may appear with optimal tunings of ''prime harmonics'' (and their deviation from just) in a collapsible table, though this will be merely for the sake of convenience as all tunings of intervals can be derived from the generators and the mapping.