Tuning map: Difference between revisions

(6 intermediate revisions by 3 users not shown)

Line 1:

A '''tuning map''' represents the tuning of a [[regular temperament]]. It ~~is similar to a standard [[val]], but 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~~. Some people consider it a type of val~~. It can take a vector representation of an interval ([[monzo]]) as input and outputs its tempered size.

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 ~~monzo~~ 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 35:

The tuning map from {{nowrap|''G'' {{=}} {{map| 1200.000 696.578 }}}} and {{nowrap|''M'' {{=}} {{mapping| 1 1 0 | 0 1 4 }}}} is {{nowrap|''T'' {{=}} {{map| 1200.000 1896.578 2786.314 }}}}. For the error map we use {{nowrap|''J'' {{=}} {{val| 1200.000 1901.955 2786.314 }}}} and find {{nowrap|''Ɛ'' {{=}} {{val| 0.000 -5.377 0.000 }}}}, showing us prime 3 is tempered flat by 5.377 cents while primes 2 and 5 are pure.

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 [[~~monzo~~]] {{vector| 4 -1 -1 }} via the tuning map given above, {{nowrap|4×1200.000 + (−1) × 1896.578 + (−1) × 2786.314 {{=}} 117.108{{cent}}}}. Similarly, to answer "how many cents is the approximation different from JI?" we go through the same process via the error map: {{nowrap| 4 × 0.000 + (−1) × (-5.377) + (−1) × 0.000 {{=}} +5.377{{c}} }}.

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]] {{vector| 4 -1 -1 }} via the tuning map given above, {{nowrap|4×1200.000 + (−1) × 1896.578 + (−1) × 2786.314 {{=}} 117.108{{cent}}}}. Similarly, to answer "how many cents is the approximation different from JI?" we go through the same process via the error map: {{nowrap| 4 × 0.000 + (−1) × (-5.377) + (−1) × 0.000 {{=}} +5.377{{c}} }}.

Another example tuning for meantone would be the [[TE tuning]], which is the default that [http://x31eq.com/temper|Breed's popular RTT web tool] provides. This gives us a tuning map of {{map| 1201.397 1898.446 2788.196 }} and error map of {{val| +1.397 -3.509 +1.882 }}. To answer the same questions about 16/15 in this tuning of meantone, we use the same prime count vector, but map it with these different tuning and error maps. So that gives us {{nowrap| 4 × 1201.397 + (−1) × 1898.446 + (−1) × 2788.196 {{=}} 118.946{{c}} }} and {{nowrap| 4 × 1.397 + (−1) × (−3.509) + (−1) × 1.882 {{=}} +7.215{{c}} }}, respectively. And that is our answer for TE meantone.

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 is similar to a standard [[val]], but 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. Some people consider it a type of val. It can take a vector representation of an interval ([[monzo]]) as input and outputs its tempered size.
+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 monzo 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 35: / Line 35: @@
 The tuning map from {{nowrap|''G'' {{=}} {{map| 1200.000 696.578 }}}} and {{nowrap|''M'' {{=}} {{mapping| 1 1 0 | 0 1 4 }}}} is {{nowrap|''T'' {{=}} {{map| 1200.000 1896.578 2786.314 }}}}. For the error map we use {{nowrap|''J'' {{=}} {{val| 1200.000 1901.955 2786.314 }}}} and find {{nowrap|''Ɛ'' {{=}} {{val| 0.000 -5.377 0.000 }}}}, showing us prime 3 is tempered flat by 5.377 cents while primes 2 and 5 are pure.
-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 [[monzo]] {{vector| 4 -1 -1 }} via the tuning map given above, {{nowrap|4×1200.000 + (−1) × 1896.578 + (−1) × 2786.314 {{=}} 117.108{{cent}}}}. Similarly, to answer "how many cents is the approximation different from JI?" we go through the same process via the error map: {{nowrap| 4 × 0.000 + (−1) × (-5.377) + (−1) × 0.000 {{=}} +5.377{{c}} }}.
+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]] {{vector| 4 -1 -1 }} via the tuning map given above, {{nowrap|4×1200.000 + (−1) × 1896.578 + (−1) × 2786.314 {{=}} 117.108{{cent}}}}. Similarly, to answer "how many cents is the approximation different from JI?" we go through the same process via the error map: {{nowrap| 4 × 0.000 + (−1) × (-5.377) + (−1) × 0.000 {{=}} +5.377{{c}} }}.
 Another example tuning for meantone would be the [[TE tuning]], which is the default that [http://x31eq.com/temper|Breed's popular RTT web tool] provides. This gives us a tuning map of {{map| 1201.397 1898.446 2788.196 }} and error map of {{val| +1.397 -3.509 +1.882 }}. To answer the same questions about 16/15 in this tuning of meantone, we use the same prime count vector, but map it with these different tuning and error maps. So that gives us {{nowrap| 4 × 1201.397 + (−1) × 1898.446 + (−1) × 2788.196 {{=}} 118.946{{c}} }} and {{nowrap| 4 × 1.397 + (−1) × (−3.509) + (−1) × 1.882 {{=}} +7.215{{c}} }}, respectively. And that is our answer for TE meantone.
@@ 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 -->