Tuning map: Difference between revisions

(5 intermediate revisions by 3 users not shown)

Line 1:

A '''tuning map''' represents the tuning of a [[regular temperament]]. It specifies the tuning of a temperament in terms of logarithmic [[interval size unit]]s (such as [[cent]]s or [[octave]]s) rather than scale steps. It can take a vector representation of an interval ([[monzo]]) as input and outputs its tempered size.

A tuning map has one entry for each [[basis element]] of the temperament's JI subgroup (e.g. 2, 9, 5 in a 2.9.5-subgroup temperament), giving its size in cents, octaves, or any other logarithmic interval size unit.

A tempered-prime tuning map has one entry for each [[basis element]] of the temperament's JI subgroup (e.g. 2, 9, 5 in a 2.9.5-subgroup temperament), giving its size in cents, octaves, or any other logarithmic interval size unit.

It may be helpful, then, to think of the units of each entry of a tuning map as <math>{\large\mathsf{¢}}\small /𝗽</math> (read "cents per prime"), <math>\small \mathsf{oct}/𝗽</math> (read "octaves per prime"), or any other logarithmic pitch unit per prime (for more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]]).

== Generator tuning map ==

A '''generator tuning map''' is similar to a tuning map, but it specifies a tuning for a temperament by giving the sizes of its generators. Each entry of the generator tuning map gives the size of a different [[generator]]. It takes a vector in tempered interval space (a "[[tmonzos and tvals|tmonzo]]").

A '''generator tuning map''' is similar to a (tempered-prime) tuning map, but it specifies a tuning for a temperament by giving the sizes of its generators. Each entry of the generator tuning map gives the size of a different [[generator]]. It takes a vector in tempered interval space (a "[[tmonzos and tvals|tmonzo]]").

It may be helpful, then, to think of the units of each entry of a generator tuning map as <math>{\large\mathsf{¢}}\small /𝗴</math> (read "cents per generator"), <math>\small \mathsf{oct}/𝗴</math> (read "octaves per generator"), or any other logarithmic pitch unit per generator.

Line 43:

== 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 error maps and generator tuning maps.

A tuning map is a real-valued linear form (or covector). If we identify interval space with the integer lattice, it is a linear map <math>\varphi: \mathbb{Z}^n \to \mathbb{R}</math>, which projects abstract intervals onto the real line which measures pitch. It can also be thought of as a matrix with a single row. The same is true of error maps and generator tuning maps.

[[Category:Regular temperament tuning| ]]

@@ Line 1: / Line 1: @@
 A '''tuning map''' represents the tuning of a [[regular temperament]]. It specifies the tuning of a temperament in terms of logarithmic [[interval size unit]]s (such as [[cent]]s or [[octave]]s) rather than scale steps. It can take a vector representation of an interval ([[monzo]]) as input and outputs its tempered size.
-A tuning map has one entry for each [[basis element]] of the temperament's JI subgroup (e.g. 2, 9, 5 in a 2.9.5-subgroup temperament), giving its size in cents, octaves, or any other logarithmic interval size unit.
+A tempered-prime tuning map has one entry for each [[basis element]] of the temperament's JI subgroup (e.g. 2, 9, 5 in a 2.9.5-subgroup temperament), giving its size in cents, octaves, or any other logarithmic interval size unit.
 It may be helpful, then, to think of the units of each entry of a tuning map as <math>{\large\mathsf{¢}}\small /𝗽</math> (read "cents per prime"), <math>\small \mathsf{oct}/𝗽</math> (read "octaves per prime"), or any other logarithmic pitch unit per prime (for more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]]).
 == Generator tuning map ==
-A '''generator tuning map''' is similar to a tuning map, but it specifies a tuning for a temperament by giving the sizes of its generators. Each entry of the generator tuning map gives the size of a different [[generator]]. It takes a vector in tempered interval space (a "[[tmonzos and tvals|tmonzo]]").
+A '''generator tuning map''' is similar to a (tempered-prime) tuning map, but it specifies a tuning for a temperament by giving the sizes of its generators. Each entry of the generator tuning map gives the size of a different [[generator]]. It takes a vector in tempered interval space (a "[[tmonzos and tvals|tmonzo]]").
 It may be helpful, then, to think of the units of each entry of a generator tuning map as <math>{\large\mathsf{¢}}\small /𝗴</math> (read "cents per generator"), <math>\small \mathsf{oct}/𝗴</math> (read "octaves per generator"), or any other logarithmic pitch unit per generator.
@@ Line 43: / Line 43: @@
 == 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 error maps and generator tuning maps.
+A tuning map is a real-valued linear form (or covector). If we identify interval space with the integer lattice, it is a linear map <math>\varphi: \mathbb{Z}^n \to \mathbb{R}</math>, which projects abstract intervals onto the real line which measures pitch. It can also be thought of as a matrix with a single row. The same is true of error maps and generator tuning maps.
 [[Category:Regular temperament tuning| ]] <!-- Main article -->