POTE tuning: Difference between revisions

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'''POTE tuning''' is the short form of '''Pure-Octaves [[Tenney-Euclidean_Tuning#Pure octaves TE tuning|Tenney-Euclidean tuning]]''', a good choice for a standard tuning enforcing a just 2/1 octave.

The POTE tuning for a [[mappings|map matrix]] such as M = [{{val|1 0 2 -1}}, {{val|0 5 1 12}}] (the [[map]] for 7-limit [[Magic_family|magic]], which consists of a linearly independent list of [[val|vals]] defining magic) can be found as follows:

#1 Form a matrix V from M by multiplying by the diagonal matrix which is zero off the diagonal and 1/log2(p) on the diagonal; in other words the diagonal is [1 1/log2(3) 1/log2(5) 1/log2(7)]. Another way to say this is that each val is "weighted" by dividing through by the logarithms, so that V = [~~<~~1 0 2/log2(5) -1/log2(7)| ~~<~~5/log2(3) 1/log2(5) 12/log2(7)]

# Form a matrix V from M by multiplying by the diagonal matrix which is zero off the diagonal and 1/log2(p) on the diagonal; in other words the diagonal is [1 1/log2(3) 1/log2(5) 1/log2(7)]. Another way to say this is that each val is "weighted" by dividing through by the logarithms, so that V = [{{val|1 0 2/log2(5) -1/log2(7)}}, {{val|5/log2(3) 1/log2(5) 12/log2(7)}}]

# Find the matrix P = V*(VV*)^(-1), where V* is the transpose matrix.

#2 Find the matrix P = V*(VV*)^(-1), where V* is the transpose matrix.

# Find T = {{val|1 1 1 1}}P.

# Find POTE = T/T[1]; in other words T scalar divided by T[1], the first element of T.

#3 Find T = ~~<~~1 1 1 1|P.

#4 Find POTE = T/T[1]; in other words T scalar divided by T[1], the first element of T.

If you carry out these operations, you should find

V ~ [~~<~~1 0 0.861 -0.356|~~, <~~0 3.155 0.431 4.274|]

* V ~ [{{val|1 0 0.861 -0.356}}, {{val|0 3.155 0.431 4.274}}]

T ~ ~~<~~1.000902 0.317246|

* T ~ {{val|1.000902 0.317246}}

POTE ~ ~~<~~1 0.3169600|

* POTE ~ {{val|1 0.3169600}}

The tuning of the POTE [[~~generator|~~generator]] corresponding to the mapping M is therefore 0.31696 octaves, or 380.~~252~~ cents. Naturally, this only gives the single POTE generator in the rank two case, and only when the map M is in period-generator form, but the POTE tuning can still be found in this way for mappings defining higher rank temperaments. The method can be generalized to subgroup temperaments so long as the group contains 2 by [[Lp_tuning|POL2 tuning]].

The tuning of the POTE [[generator]] corresponding to the mapping M is therefore 0.31696 octaves, or 380.352 cents. Naturally, this only gives the single POTE generator in the rank two case, and only when the map M is in period-generator form, but the POTE tuning can still be found in this way for mappings defining higher rank temperaments. The method can be generalized to subgroup temperaments so long as the group contains 2 by [[Lp_tuning|POL2 tuning]].

[[Category:glossary]]

[[Category:todo:clarify]]

@@ Line 1: / Line 1: @@
 '''POTE tuning''' is the short form of '''Pure-Octaves [[Tenney-Euclidean_Tuning#Pure octaves TE tuning|Tenney-Euclidean tuning]]''', a good choice for a standard tuning enforcing a just 2/1 octave.
-The POTE tuning for a [[map_matrix|map matrix]] such as M = [&lt;1 0 2 -1|, &lt;0 5 1 12|] (the [[map|map]] for 7-limit [[Magic_family|magic]], which consists of a linearly independent list of [[val|vals]] defining magic) can be found as follows:
+The POTE tuning for a [[mappings|map matrix]] such as M = [{{val|1 0 2 -1}}, {{val|0 5 1 12}}] (the [[map]] for 7-limit [[Magic_family|magic]], which consists of a linearly independent list of [[val|vals]] defining magic) can be found as follows:
-#1 Form a matrix V from M by multiplying by the diagonal matrix which is zero off the diagonal and 1/log2(p) on the diagonal; in other words the diagonal is [1 1/log2(3) 1/log2(5) 1/log2(7)]. Another way to say this is that each val is "weighted" by dividing through by the logarithms, so that V = [&lt;1 0 2/log2(5) -1/log2(7)| &lt;5/log2(3) 1/log2(5) 12/log2(7)]
+# Form a matrix V from M by multiplying by the diagonal matrix which is zero off the diagonal and 1/log2(p) on the diagonal; in other words the diagonal is [1 1/log2(3) 1/log2(5) 1/log2(7)]. Another way to say this is that each val is "weighted" by dividing through by the logarithms, so that V = [{{val|1 0 2/log2(5) -1/log2(7)}}, {{val|5/log2(3) 1/log2(5) 12/log2(7)}}]
+# Find the matrix P = V*(VV*)^(-1), where V* is the transpose matrix.
-#2 Find the matrix P = V*(VV*)^(-1), where V* is the transpose matrix.
+# Find T = {{val|1 1 1 1}}P.
+# Find POTE = T/T[1]; in other words T scalar divided by T[1], the first element of T.
-#3 Find T = &lt;1 1 1 1|P.
-#4 Find POTE = T/T[1]; in other words T scalar divided by T[1], the first element of T.
 If you carry out these operations, you should find
-V ~ [&lt;1 0 0.861 -0.356|, &lt;0 3.155 0.431 4.274|]
+* V ~ [{{val|1 0 0.861 -0.356}}, {{val|0 3.155 0.431 4.274}}]
-T ~ &lt;1.000902 0.317246|
+* T ~ {{val|1.000902 0.317246}}
-POTE ~ &lt;1 0.3169600|
+* POTE ~ {{val|1 0.3169600}}
-The tuning of the POTE [[generator|generator]] corresponding to the mapping M is therefore 0.31696 octaves, or 380.252 cents. Naturally, this only gives the single POTE generator in the rank two case, and only when the map M is in period-generator form, but the POTE tuning can still be found in this way for mappings defining higher rank temperaments. The method can be generalized to subgroup temperaments so long as the group contains 2 by [[Lp_tuning|POL2 tuning]].
+The tuning of the POTE [[generator]] corresponding to the mapping M is therefore 0.31696 octaves, or 380.352 cents. Naturally, this only gives the single POTE generator in the rank two case, and only when the map M is in period-generator form, but the POTE tuning can still be found in this way for mappings defining higher rank temperaments. The method can be generalized to subgroup temperaments so long as the group contains 2 by [[Lp_tuning|POL2 tuning]].
 [[Category:glossary]]
 [[Category:todo:clarify]]