Tuning map: Difference between revisions

Line 5:

It may be helpful, then, to think of the units of each entry of a tuning map as c/p (read "cents per prime"), oct/p (read "octaves per prime"), or any other logarithmic pitch unit per prime.

== ~~Generators~~ tuning map ==

== Generator tuning map ==

A '''~~generators~~ tuning map''' is like a (temperament) tuning map, but each entry gives the size in cents or octaves of a different [[generator]], rather than of a formal prime.

A '''generator tuning map''' is like a (temperament) tuning map, but each entry gives the size in cents or octaves of a different [[generator]], rather than of a formal prime.

It may be helpful, then, to think of the units of each entry of a ~~generators~~ tuning map as c/g (read "cents per generator"), oct/g (read "octaves per generator"), or any other logarithmic pitch unit per generator.

It may be helpful, then, to think of the units of each entry of a generator tuning map as c/g (read "cents per generator"), oct/g (read "octaves per generator"), or any other logarithmic pitch unit per generator.

From the ~~generators~~ tuning map <math>𝒈</math> and the mapping <math>M</math>, we can obtain the tuning map <math>𝒕</math> as <math>𝒈.M</math>.

From the generator tuning map <math>𝒈</math> and the mapping <math>M</math>, we can obtain the tuning map <math>𝒕</math> as <math>𝒈.M</math>.

== Example ==

Consider meantone temperament, with the mapping {{ket|{{map| 1 1 0 }} {{map| 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<sup>1/4</sup>; this gives a ~~generators~~ tuning map of {{map| 1200.000 696.578 }}.

Consider meantone temperament, with the mapping {{ket|{{map| 1 1 0 }} {{map| 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<sup>1/4</sup>; this gives a generator tuning map of {{map| 1200.000 696.578 }}.

The tuning map from <math>𝒈</math> = {{map| 1200.000 696.578 }} and <math>M</math> = {{ket|{{map| 1 1 0 }} {{map| 0 1 4 }} }} is <math>𝒕</math> = {{map| 1200.000 1896.578 2786.314 }}.

Line 26:

== 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 basis elements. So, JI subgroups have ~~generators~~ tuning maps and tuning maps too; the ~~generators~~ tuning maps and tuning maps are always the same thing as each other, and they are all subsets of the entries of 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 basis elements. 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 ~~generators~~ tuning maps.

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.

[[Category:Tuning]]

@@ Line 5: / Line 5: @@
 It may be helpful, then, to think of the units of each entry of a tuning map as c/p (read "cents per prime"), oct/p (read "octaves per prime"), or any other logarithmic pitch unit per prime.
-== Generators tuning map ==
+== Generator tuning map ==
-A '''generators tuning map''' is like a (temperament) tuning map, but each entry gives the size in cents or octaves of a different [[generator]], rather than of a formal prime.
+A '''generator tuning map''' is like a (temperament) tuning map, but each entry gives the size in cents or octaves of a different [[generator]], rather than of a formal prime.
-It may be helpful, then, to think of the units of each entry of a generators tuning map as c/g (read "cents per generator"), oct/g (read "octaves per generator"), or any other logarithmic pitch unit per generator.
+It may be helpful, then, to think of the units of each entry of a generator tuning map as c/g (read "cents per generator"), oct/g (read "octaves per generator"), or any other logarithmic pitch unit per generator.
-From the generators tuning map <math>𝒈</math> and the mapping <math>M</math>, we can obtain the tuning map <math>𝒕</math> as <math>𝒈.M</math>.
+From the generator tuning map <math>𝒈</math> and the mapping <math>M</math>, we can obtain the tuning map <math>𝒕</math> as <math>𝒈.M</math>.
 == Example ==
-Consider meantone temperament, with the mapping {{ket|{{map| 1 1 0 }} {{map| 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<sup>1/4</sup>; this gives a generators tuning map of {{map| 1200.000 696.578 }}.
+Consider meantone temperament, with the mapping {{ket|{{map| 1 1 0 }} {{map| 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<sup>1/4</sup>; this gives a generator tuning map of {{map| 1200.000 696.578 }}.
 The tuning map from <math>𝒈</math> = {{map| 1200.000 696.578 }} and <math>M</math> = {{ket|{{map| 1 1 0 }} {{map| 0 1 4 }} }} is <math>𝒕</math> = {{map| 1200.000 1896.578 2786.314 }}.
@@ Line 26: / Line 26: @@
 == 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 basis elements. So, JI subgroups have generators tuning maps and tuning maps too; the generators tuning maps and tuning maps are always the same thing as each other, and they are all subsets of the entries of 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 basis elements. 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 generators tuning maps.
+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.
 [[Category:Tuning]]