Tuning map

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A tuning map represents the tuning of a regular temperament. It can take a vector representation of an interval (monzo) as input and outputs its pitch, usually measured in cents or octaves. A tuning map has one entry for each formal prime of the temperament, giving its size in cents or octaves.

Example

Consider meantone temperament, with the mapping [1 1 0] 0 1 4]. Temperaments, as represented by mappings, remain abstract; while this mapping does convey that the generators are ~2/1 and ~3/2, it does not specify exact tunings for those approximations. One example tuning would be quarter-comma meantone, where the octave is pure and the perfect fifth is 51/4; this gives a generator tuning map of 1200.000 696.578]. The generator tuning map is like a tuning map, but each entry gives the size in cents or octaves of a different generator, rather than of a formal prime.

From the generator tuning map G and the temperament mapping V, we can obtain the tuning map T:

[math] T = GV [/math]

The tuning map from G = 1200.000 696.578] and V = [1 1 0] 0 1 4] is 1200.000 1896.578 2786.314].

So, to answer the question, "how many cents is the approximation of the interval 16/15 in quarter-comma meantone?" we use the dot product to map 16/15's prime count vector [4 -1 -1 via the tuning map given above, 4×1200.000 + (-1)×1896.578 + (-1)×2786.314 = 117.108 cents.

Another example tuning for meantone would be the TE tuning, which is the default that Breed's popular RTT web tool provides. This gives us a tuning map of 1201.397 1898.446 2788.196]. To answer the same question about 16/15 in this tuning of meantone, we use the same prime count vector, but map it with this different tuning map. So that gives us 4×1201.397 + (-1)×1898.446 + (-1)×2788.196 = 125.931 cents. And that's our answer for TE meantone.

Cents versus octaves

Sometimes you will see tuning maps given in octaves instead of cents. They work the same exact way. The only difference is that these octave-based tuning maps have each entry divided by 1200. For example, the quarter-comma meantone tuning map, in octaves, would be 1200 1896.578 2786.314]/1200 = 1 1.580 2.322]. If we dot product [4 -1 -1 with that, we get 4×1 + (-1)×1.580 + (-1)×2.322 = 0.098, which tells us that 16/15 is a little less than 1/10 of an octave here.

With respect to the JIP

JI can be conceptualized as the temperament where nothing is tempered out, and as such, the untempered primes can be thought of as its generators, or of course its formal primes. So, JI subgroups have generator tuning maps and tuning maps too; the generator tuning maps and tuning maps are always the same thing as each other, and they are all subsets of the entries of the JIP.

With respect to linear algebra

A tuning map can be thought of either as a one-row matrix or as a covector. The same is true of generator tuning maps.