Tuning map: Difference between revisions

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== Generator tuning map ==

~~{{Todo|clarify|inline=1|text=What is a "formal prime" in tempered interval space, and how is it distinct from a generator?}}~~

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]].

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 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.

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$$T = GM$$

To go the other ~~way — that~~ is, to find the generator tuning map from the (primes) tuning ~~map — we~~ can multiply the tuning map by any right-inverse of the mapping, such as the [[pseudoinverse]] ''M''<sup>+</sup>, as in

To go the other way – that is, to find the generator tuning map from the (primes) tuning map – we can multiply the tuning map by any right-inverse of the mapping, such as the [[pseudoinverse]] ''M''<sup>+</sup>, as in

$$G = TM^{+}$$

For a detailed explanation see [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Tuning in nonstandard domains#9. Find pseudoinverse]].

For a detailed explanation see [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Tuning in nonstandard domains #9. Find pseudoinverse]].

== With respect to JIP ==

@@ Line 6: / Line 6: @@
 == Generator tuning map ==
-{{Todo|clarify|inline=1|text=What is a "formal prime" in tempered interval space, and how is it distinct from a generator?}}
+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]].
-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 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 15: / Line 14: @@
 $$T = GM$$
-To go the other way — that is, to find the generator tuning map from the (primes) tuning map — we can multiply the tuning map by any right-inverse of the mapping, such as the [[pseudoinverse]] ''M''<sup>+</sup>, as in
+To go the other way – that is, to find the generator tuning map from the (primes) tuning map – we can multiply the tuning map by any right-inverse of the mapping, such as the [[pseudoinverse]] ''M''<sup>+</sup>, as in
 $$G = TM^{+}$$
-For a detailed explanation see [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Tuning in nonstandard domains#9. Find pseudoinverse]].
+For a detailed explanation see [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Tuning in nonstandard domains #9. Find pseudoinverse]].
 == With respect to JIP ==