Tp tuning: Difference between revisions

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For an example, consider [[Chromatic pairs #Indium|indium temperament]], with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the [[Tenney-Euclidean tuning|usual methods]], in particular the pseudoinverse, we find that the T2 (TE) tuning map is {{val| 1199.552 1901.846 2783.579 3371.401 4153.996 }}. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.000/1199.552 gives a POT2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting {{val| 1200 1200 1200 1200 1200 }} gives {{val| -0.4475 -0.0685 -1.1778 0.9172 0.7741 }}. The ordinary Euclidean norm of this, i.e. the square root of the dot product, is 1.7414, and dividing by sqrt (5) gives the RMS error, 0.77879 cents.

This is called '''subgroup TE''' in Graham Breed's temperament finder. Subgroup TE does not depend on the basis, because it is always a restriction of the corresponding full prime-limit TE temperament.

This is called '''subgroup TE''' in Graham Breed's temperament finder. Subgroup TE does not depend on the basis, because it is always a restriction of the TE tuning for the corresponding full prime-limit temperament.

[[Category:Math]]

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 For an example, consider [[Chromatic pairs #Indium|indium temperament]], with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the [[Tenney-Euclidean tuning|usual methods]], in particular the pseudoinverse, we find that the T2 (TE) tuning map is {{val| 1199.552 1901.846 2783.579 3371.401 4153.996 }}. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.000/1199.552 gives a POT2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting {{val| 1200 1200 1200 1200 1200 }} gives {{val| -0.4475 -0.0685 -1.1778 0.9172 0.7741 }}. The ordinary Euclidean norm of this, i.e. the square root of the dot product, is 1.7414, and dividing by sqrt (5) gives the RMS error, 0.77879 cents.
-This is called '''subgroup TE''' in Graham Breed's temperament finder. Subgroup TE does not depend on the basis, because it is always a restriction of the corresponding full prime-limit TE temperament.
+This is called '''subgroup TE''' in Graham Breed's temperament finder. Subgroup TE does not depend on the basis, because it is always a restriction of the TE tuning for the corresponding full prime-limit temperament.
 [[Category:Math]]