Temperament merging: Difference between revisions

'''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]]s ([[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 equal temperaments. The resulting rank-2 essentially captures the similarities between the two equal temperaments: [[15edo|15]] & [[22edo|22]] is [[porcupine]], because both equal temperaments 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]] 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 [[aberschismic]], because both garibaldi and rodan conflate [[81/80]] and [[64/63]] into a single comma-sized interval. 

== 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.  

We have not ''completely'' canonicalized yet; we did not 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 does not 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:

The greatest factor of this matrix is 2, because we can produce the row {{val| 24 38 56 }} as a coprime linear combination of its rows (that's {{val| 5 8 12 }} + {{val| 19 30 44 }}), and the entries of this row have a GCD of 2, so in other words this matrix is 2-enfactored. If we merely put it into Hermite normal form, we receive:

By some definitions of the & operator, the [[defactoring]] part of canonicalization is not included—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.