Temperament merging: Difference between revisions
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{{Beginner|Meet and join}} | {{Beginner|Meet and join}} | ||
'''Temperament merging''' is a way to find new [[regular temperaments]] by merging others. There are two ways to merge temperaments: '''map-merge | '''Temperament merging''' is a way to find new [[regular temperaments]] by merging others. There are two ways to merge temperaments: '''joining''' (or map-merge), which works by merging the temperaments' [[mapping]]s, and '''comma-merge''', which works by merging the temperaments' [[comma basis|comma bases]]. | ||
== | These are multiple ways in which a temperament can be defined in terms of the properties of another temperament. | ||
'''Joining''' two temperaments ''a'' and ''b'' (notated a & b) results in a higher-rank temperament which tempers out only the commas that both ''a'' and ''b'' temper out. Usually, this is done with two [[Equal temperament|ETs]] ([[vals]], usually written in wart notation) to receive a rank-2 temperament (sometimes called cross-breeding), and indeed, all possible rank-2 temperaments can be written as a combination of two ETs. The resulting rank-2 essentially captures the similarities between the two ETs: [[15edo|15]] & [[22edo|22]] is [[porcupine]], because both ETs have an [[11/10]] that doubles to [[6/5]] and triples to [[4/3]]. Similarly, [[19edo|19]] & [[26edo|26]] is [[flattone]], because in the diatonic scale of both edos, the [[Major third (interval region)|major third]] is 5/4 and the [[Major sixth#As a diatonic interval category|diminished seventh]] is 7/4. Higher-rank temperaments can also be joined; [[garibaldi]] & [[rodan]] is [[hemifamity]], because both garibaldi and rodan conflate [[81/80]] and [[64/63]] into a single comma-sized interval. | |||
'''Comma-merging''' two temperaments ''a'' and ''b'' (notated a | b) results in a lower-rank temperament which tempers out all of the commas that either ''a'' or ''b'' temper out. This can be done with two rank-2 temperaments to find the equal temperament which [[Support|supports]] them both. For example, [[meantone]] | [[Augmented (temperament)|augmented]] is [[12edo|12-ET]], since 12-ET both has 5/4 as its diatonic major third and has that 5/4 equal to [[3edo|1\3]] of the [[Octave|octave.]] | |||
More than two temperaments may be merged at once. For example, joining three ETs results in a [[rank-3 temperament]] (e.g. 22 & 34d & 37 is [[ares]]). | |||
Note that while a given temperament merging expression unambiguously refers to a single temperament, a given temperament can be expressed by many possible different temperament merging expressions. | |||
== With mappings == | |||
To perform the join with mappings, we vertically concatenate the matrices. In this form, the mapping does represent the temperament (and is the form used in [[Diatonic, chromatic, enharmonic, and subchromatic steps|diatonic, chromatic, enharmonic, and subchromatic]] theory), but to get a more conventional mapping, we can then [[Temperament merging#Canonicalization|canonicalize]] it. | |||
Similarly, to perform the join with comma bases, we horizontally concatenate them, and then canonicalize the result. | |||
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</math> | </math> | ||
== | == With multivals == | ||
Joining is equivalent to the [[wedge product]], and can be calculated in that manner. Wedging two vals results in the same temperament (in [[wedgie]] form) as joining them does. | |||
== Canonicalization == | == Canonicalization == | ||
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We haven't ''completely'' canonicalized yet; we didn't remove the all-zero column (highlighted in red) that was created by the [[Hermite normal form]] step. The existence of any all-zero columns like this tells us that our matrix was column-rank-deficient, or in layperson's terms, that it contained redundant commas. In other words, these two temperaments make some of the same commas vanish, and so when we merged | We haven't ''completely'' canonicalized yet; we didn't remove the all-zero column (highlighted in red) that was created by the [[Hermite normal form]] step. The existence of any all-zero columns like this tells us that our matrix was column-rank-deficient, or in layperson's terms, that it contained redundant commas. In other words, these two temperaments make some of the same commas vanish, and so when we merged them—even though the input temperaments required 2 vectors each to represent—their merged result doesn't require all 4 vectors; it can be completely represented using only 3 vectors. So once we fully [[canonical form|canonicalize]], any all-zero column(s) are removed, and we end up with: | ||
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=== Non-canonicalizing definition === | === Non-canonicalizing definition === | ||
By some definitions of the & operator, the [[defactoring]] part of canonicalization is not | By some definitions of the & operator, the [[defactoring]] part of canonicalization is not include—for example on [http://x31eq.com/temper/ Graham Breed's temperament finding tool]. This allows for things like {{nowrap|5 & 19}} to represent 2-enfactored meantone, rather than meantone itself. Instead of a full canonicalization, then, this definition merely puts the result into Hermite normal form and removes any all-zero rows or columns resulting from rank-deficiencies. | ||
== Parallel intersections == | == Parallel intersections == | ||
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[[File:Temperament merging 7-limit example.png|1000px|frameless|center]] | [[File:Temperament merging 7-limit example.png|1000px|frameless|center]] | ||
== Cross-domain temperament merging == | == Cross-domain temperament merging == | ||