Tuning map: Difference between revisions

Line 11:

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.

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>𝒈M</math>.~~ 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]] <~~math~~>~~M^{~~+}</~~math~~>, as in ~~<math>𝒈~~ = 𝒕M^{+}~~</math>.~~ For ~~more information,~~ see ~~the explanation~~ [[~~Dave_Keenan_~~%~~26_Douglas_Blumeyer~~%~~27s_guide_to_RTT~~:~~_tuning_in_nonstandard_domains~~#9.~~_Find_pseudoinverse|here~~]].

From the generator tuning map ''G'' and the mapping ''M'', we can obtain the tuning map ''T'' as

$$T = GM$$

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''+, 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]].

== With respect to JIP ==

Line 19:

Line 27:

== Error map ==

An '''error map''', also known as '''mistuning map''' or '''retuning map''', is like a tuning map, but each entry shows the signed amount of deviation from the target value (usually [[JI]]), i.e. the [[error]]. It is therefore equal to the difference between the tempered tuning map and the just tuning map.

An '''error map''', also known as '''mistuning map''' or '''retuning map''', is like a tuning map, but each entry shows the signed amount of deviation from the target value (usually JI), i.e. the [[error]]. It is therefore equal to the difference between the tempered tuning map and the just tuning map. If we have an error map ''Ɛ'', tempered tuning map ''T'', and just tuning map ''J'', it follows that

$$\mathcal{E} = T - J$$

== Example ==

Consider meantone temperament, with the mapping {{~~rket|{{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 51/4; this gives a generator tuning map of {{map| 1200.000 696.578 }}.

Consider meantone temperament, with the mapping {{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 51/4; 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>~~ = {{~~rket|{{map~~|1 1 0~~}} {{map~~|0 1 4}}}} is ~~<math>𝒕</math>~~ = {{map|1200.000 1896.578 2786.314}}.

The tuning map from ''G'' = {{map| 1200.000 696.578 }} and ''M'' = {{mapping| 1 1 0 | 0 1 4 }} is ''T'' = {{map| 1200.000 1896.578 2786.314 }}.

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, 4×1200.000 + (-1)×1896.578 + (-1)×2786.314 = 117.108 cents.

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, 4×1200.000 + (-1)×1896.578 + (-1)×2786.314 = 117.108 cents.

Another example tuning for meantone would be the [[TE tuning]], which is the default that [http://x31eq.com/temper|Graham Breed's popular RTT web tool] provides. This gives us a tuning map of {{map|1201.397 1898.446 2788.196}}. To answer the same question about 16/15 in this tuning of meantone, we use the same prime count vector, but map it with this different tuning map. So that gives us 4×1201.397 + (-1)×1898.446 + (-1)×2788.196 = 125.931 cents. And that's our answer for TE meantone.

Another example tuning for meantone would be the [[TE tuning]], which is the default that [http://x31eq.com/temper|Graham Breed's popular RTT web tool] provides. This gives us a tuning map of {{map| 1201.397 1898.446 2788.196 }}. To answer the same question about 16/15 in this tuning of meantone, we use the same prime count vector, but map it with this different tuning map. So that gives us 4×1201.397 + (-1)×1898.446 + (-1)×2788.196 = 125.931 cents. And that is our answer for TE meantone.

== Cents versus octaves ==

@@ Line 11: / Line 11: @@
 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.
-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>𝒈M</math>. 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]] <math>M^{+}</math>, as in <math>𝒈 = 𝒕M^{+}</math>. For more information, see the explanation [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_tuning_in_nonstandard_domains#9._Find_pseudoinverse|here]].
+From the generator tuning map ''G'' and the mapping ''M'', we can obtain the tuning map ''T'' as
+$$T = GM$$
+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
+$$G = TM^{+}$$
+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 19: / Line 27: @@
 == Error map ==
-An '''error map''', also known as '''mistuning map''' or '''retuning map''', is like a tuning map, but each entry shows the signed amount of deviation from the target value (usually [[JI]]), i.e. the [[error]]. It is therefore equal to the difference between the tempered tuning map and the just tuning map.
+An '''error map''', also known as '''mistuning map''' or '''retuning map''', is like a tuning map, but each entry shows the signed amount of deviation from the target value (usually JI), i.e. the [[error]]. It is therefore equal to the difference between the tempered tuning map and the just tuning map. If we have an error map ''Ɛ'', tempered tuning map ''T'', and just tuning map ''J'', it follows that
+$$\mathcal{E} = T - J$$
 == Example ==
-Consider meantone temperament, with the mapping {{rket|{{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 }}.
+Consider meantone temperament, with the mapping {{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 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> = {{rket|{{map|1 1 0}} {{map|0 1 4}}}} is <math>𝒕</math> = {{map|1200.000 1896.578 2786.314}}.
+The tuning map from ''G'' = {{map| 1200.000 696.578 }} and ''M'' = {{mapping| 1 1 0 | 0 1 4 }} is ''T'' = {{map| 1200.000 1896.578 2786.314 }}.
-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, 4×1200.000 + (-1)×1896.578 + (-1)×2786.314 = 117.108 cents.
+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, 4×1200.000 + (-1)×1896.578 + (-1)×2786.314 = 117.108 cents.
-Another example tuning for meantone would be the [[TE tuning]], which is the default that [http://x31eq.com/temper|Graham Breed's popular RTT web tool] provides. This gives us a tuning map of {{map|1201.397 1898.446 2788.196}}. To answer the same question about 16/15 in this tuning of meantone, we use the same prime count vector, but map it with this different tuning map. So that gives us 4×1201.397 + (-1)×1898.446 + (-1)×2788.196 = 125.931 cents. And that's our answer for TE meantone.
+Another example tuning for meantone would be the [[TE tuning]], which is the default that [http://x31eq.com/temper|Graham Breed's popular RTT web tool] provides. This gives us a tuning map of {{map| 1201.397 1898.446 2788.196 }}. To answer the same question about 16/15 in this tuning of meantone, we use the same prime count vector, but map it with this different tuning map. So that gives us 4×1201.397 + (-1)×1898.446 + (-1)×2788.196 = 125.931 cents. And that is our answer for TE meantone.
 == Cents versus octaves ==