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. | '''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. | ||
== Computing TE and POTE == | == Computing TE and POTE tuning == | ||
The TE and 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: | The TE and 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: | ||
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# Form a matrix V from M by multiplying by the diagonal matrix which is zero off the diagonal and 1/log<sub>2</sub>''p'' on the diagonal; in other words the diagonal is [1 1/log<sub>2</sub>3 1/log<sub>2</sub>5 1/log<sub>2</sub>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)}}] | # Form a matrix V from M by multiplying by the diagonal matrix which is zero off the diagonal and 1/log<sub>2</sub>''p'' on the diagonal; in other words the diagonal is [1 1/log<sub>2</sub>3 1/log<sub>2</sub>5 1/log<sub>2</sub>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 pseudoinverse of the matrix V<sup>+</sup> = V<sup>T</sup>(VV<sup>T</sup>)<sup>-1</sup>. | # Find the pseudoinverse of the matrix V<sup>+</sup> = V<sup>T</sup>(VV<sup>T</sup>)<sup>-1</sup>. | ||
# Find the TE generators | # Find the TE generators g = {{val|1 1 1 1}}V<sup>+</sup>. | ||
# Find the TE tuning map: T = | # Find the TE tuning map: T = gV. | ||
# Find the POTE generators | # Find the POTE generators g<nowiki/>' = g/T<sub>1</sub>; in other words g scalar divided by the first entry of T. | ||
If you carry out these operations, you should find | If you carry out these operations, you should find | ||
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* V ~ [{{val|1 0 0.861 -0.356}}, {{val|0 3.155 0.431 4.274}}] | * V ~ [{{val|1 0 0.861 -0.356}}, {{val|0 3.155 0.431 4.274}}] | ||
* | * g ~ {{val|1.000902 0.317246}} | ||
* | * g<nowiki/>' ~ {{val|1.000000 0.316960}} | ||
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]]. | 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]]. | ||
Revision as of 15:09, 22 July 2020
POTE tuning is the short form of Pure-Octaves Tenney-Euclidean tuning, a good choice for a standard tuning enforcing a just 2/1 octave.
Computing TE and POTE tuning
The TE and POTE tuning for a map matrix such as M = [⟨1 0 2 -1], ⟨0 5 1 12]] (the map for 7-limit magic, which consists of a linearly independent list of vals defining magic) can be found as follows:
- Form a matrix V from M by multiplying by the diagonal matrix which is zero off the diagonal and 1/log2p on the diagonal; in other words the diagonal is [1 1/log23 1/log25 1/log27]. 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)]]
- Find the pseudoinverse of the matrix V+ = VT(VVT)-1.
- Find the TE generators g = ⟨1 1 1 1]V+.
- Find the TE tuning map: T = gV.
- Find the POTE generators g' = g/T1; in other words g scalar divided by the first entry 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]]
- g ~ ⟨1.000902 0.317246]
- g' ~ ⟨1.000000 0.316960]
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 POL2 tuning.
Computer Program for TE and POTE
Below is a Python program that takes a map and gives TE and POTE generators.
Note: this program depends on Scipy.
import numpy as np
from scipy import linalg
def find_te (map, subgroup):
dimension = len (subgroup)
subgroup_octaves = np.log2 (subgroup)
weight = np.eye (dimension)
for i in range (0, dimension):
weight[i][i] = 1/np.log2 (subgroup[i])
map = map @ weight
subgroup_octaves = subgroup_octaves @ weight
te_gen = linalg.lstsq (np.transpose (map), subgroup_octaves)[0]
te_map = te_gen @ map
print (1200*te_gen)
pote_gen = te_gen/te_map[0]
print (1200*pote_gen)
Take 7-limit magic as an example, to find TE and POTE you input:
seven_limit = [2, 3, 5, 7] map_magic = [[1, 0, 2, -1], [0, 5, 1, 12]] find_te (map_magic, seven_limit)
Output:
[1201.08240941 380.695113 ] [1200. 380.35203249]