Tuning map

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 5^1/4; this gives a generator map of ⟨1200.000 696.578].

To obtain the tuning map T from the generator map G and the temperament mapping V:

[math]\displaystyle{ 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 can use the same generator count vector we already found. All we need to do now is map that with this different tuning map. So that gives us 4×1201.397 + (-1)×1898.446 + (-1)×2788.196 = 125.931 cents. So 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. So, JI subgroups have tuning maps too; 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.