Tuning map

A tuning map represents the tuning of a regular temperament. It takes a vector representation of an interval as input and outputs its pitch, usually measured in cents or octaves. A tuning map has one entry for each generator 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 ⁴√5; this gives a tuning map of ⟨1200 696.578].

So, to answer the question, "how many cents is the approximation of the interval 16/15 in quarter-comma meantone?" first we use the dot product to map 16/15's prime count vector [4 -1 -1⟩ via the meantone mapping given above, which gives us the meantone generator count vector [3 -5⟩. So while 16/15 is up four 2/1's, down one 3/1, and down 5/1 in JI, in meantone, it's up three ~2/1's, and down five ~3/2's.

But now we need to know how many cents each ~2/1 and each ~3/2 map to, and the tuning map is what answers that question. So we take this [3 -5⟩ and dot product it in turn with the tuning map ⟨1200 696.578], which gives us 3·1200 + -5·696.578 = 117.110. So that's our answer!

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 ⟨1201.397 695.652] (when we take the octave complement of the generator so that it matches our mapping). To answer the same question about 16/15 in this tuning of meantone, we can use the same generator count vector we already found; the [3 -5⟩ part is the same in any tuning of meantone. All we need to do now is map that with this different tuning map. So that gives us 3·1201.397 + -5·695.652 = 125.931. 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 696.578]/1200 = ⟨1 0.581]. If we dot product [3 -5⟩ with that, we get 3·1 + -5·0.581 = 0.095, which tells us that 16/15 is a little less than 1/10th 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.