TOP tuning: Difference between revisions

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== Proportional error ==

A ''tuning'' for a regular temperament is defined by a vector ''T'' in [[Vals and tuning space #Vals and monzos|Tenney tuning space]] whose entries are the sizes of the intervals, in cents, which the ''n'' generators of the regular temperament (often the first ''n'' primes) are mapped to. ''T'' is denoted by a {{w|Bra–ket notation|bra vector}}, and if '''m''' is a monzo then {{~~vmprod~~|''T''|'''m'''}} is the size, in cents, of the interval defined by '''m''' in the tuning ''T''. If ''k'' is the rational number which '''m''' represents, then we may also write this quantity as ''T'' (''k'').

A ''tuning'' for a regular temperament is defined by a vector ''T'' in [[Vals and tuning space #Vals and monzos|Tenney tuning space]] whose entries are the sizes of the intervals, in cents, which the ''n'' generators of the regular temperament (often the first ''n'' primes) are mapped to. ''T'' is denoted by a {{w|Bra–ket notation|bra vector}}, and if '''m''' is a monzo then {{vmp|''T''|'''m'''}} is the size, in cents, of the interval defined by '''m''' in the tuning ''T''. If ''k'' is the rational number which '''m''' represents, then we may also write this quantity as ''T'' (''k'').

For example, if '''m''' is {{monzo| -4 4 -1 }} then ''k'' = 81/80 (a [[syntonic comma]]). If ''T'' is {{val| 1200 1900 2800 }} (a multiple of [[12edo]]) then {{~~vmprod~~|''T''|'''m'''}} = -4800 + 7600 - 2800 = 0. Thus, while cents (''k'') = 21.506290, ''T'' (''k'') = 0 (i.e., the tuning tempers away the syntonic comma).

For example, if '''m''' is {{monzo| -4 4 -1 }} then ''k'' = 81/80 (a [[syntonic comma]]). If ''T'' is {{val| 1200 1900 2800 }} (a multiple of [[12edo]]) then {{vmp|''T''|'''m'''}} = -4800 + 7600 - 2800 = 0. Thus, while cents (''k'') = 21.506290, ''T'' (''k'') = 0 (i.e., the tuning tempers away the syntonic comma).

Given a tuning ''T'' and any rational number ''q'' in the domain of ''T'', the ''signed error'' of ''T'' on ''q'' is defined as Err (''q'') = ''T'' (''q'') - cents (''q''). The ''absolute error'' Arr (''q'') = |Err (''q'')| is the absolute value of the signed error.

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== Finding the tuning ==

For a temperament with both intrinsic and extrinsic primes, we may find the set of TOP tunings by first computing the tuning of the intrinsic primes. Then if ''r'' is an extrinsic prime, the tuning may be anything in the range where APE (''r'') ≤ ''E''. The limit of the [[Tp tuning|L''p'' tuning]] as ''p'' tends to 1 and the centroid of the region of TOP tunings both lead to choosing the JI tuning for ''r''. This produces the canonical TOP tuning, called '''TIPTOP'''. To find the TIPTOP tuning one method is to solve for all the potential TOP tunings of the intrinsic primes, find the tuning with least error, and then tune all the extrinsic primes purely. An alternative method is to first set up a linear programming problem; if ''T'' is a val with indeterminate coefficients ''T'' = {{val| ''t''1 ''t''2 … ''t''''k'' }} then minimize ''E'' subject to nonnegativity and the linear constraints {''t''''n''/log2(''p''''n'') - 1 ≤ ''E'', 1 - ''t''''n''/log2(''p''''n'') ≤ ''E'', {{~~vmprod~~|T|''c''''k''}} = 0} where the ''p''''n'' are the primes of the temperament, and the ''c''''k'' are the commas. We then may replace the tuning of all of the extrinsic primes with the pure JI tuning to get TIPTOP.

For a temperament with both intrinsic and extrinsic primes, we may find the set of TOP tunings by first computing the tuning of the intrinsic primes. Then if ''r'' is an extrinsic prime, the tuning may be anything in the range where APE (''r'') ≤ ''E''. The limit of the [[Tp tuning|L''p'' tuning]] as ''p'' tends to 1 and the centroid of the region of TOP tunings both lead to choosing the JI tuning for ''r''. This produces the canonical TOP tuning, called '''TIPTOP'''. To find the TIPTOP tuning one method is to solve for all the potential TOP tunings of the intrinsic primes, find the tuning with least error, and then tune all the extrinsic primes purely. An alternative method is to first set up a linear programming problem; if ''T'' is a val with indeterminate coefficients ''T'' = {{val| ''t''1 ''t''2 … ''t''''k'' }} then minimize ''E'' subject to nonnegativity and the linear constraints {''t''''n''/log2(''p''''n'') - 1 ≤ ''E'', 1 - ''t''''n''/log2(''p''''n'') ≤ ''E'', {{vmp|T|''c''''k''}} = 0} where the ''p''''n'' are the primes of the temperament, and the ''c''''k'' are the commas. We then may replace the tuning of all of the extrinsic primes with the pure JI tuning to get TIPTOP.

We may solve the sharp semigroup equations exactly to obtain solutions in the transcendental extension ''Q'' (log2(''q''1), log2(''q''2), …, log2(''q''''k'')) where the ''q''''n'' are the intrinsic primes other than 2. For example, take 5-limit meantone. Since 2 and 5 divide 80 and 3 divides 81. this is an intrinsic temperament. Solving for the TOP tuning either by linear programing or checking all the potential TOP tunings, we find the sharp semigroup is generated by {2, 1/3, 5}. Solving the sharp semigroup equations gives us a TOP tuning ''T'' = {{val| 3''q''3/log2(6480) (8''q''3 + 2''q''3''q''5)/log2(6480) 8''q''3''q''5/log2(6480) }}. Here ''q''3 = log2 (3), ''q''5 = log2 (5), and the denominator can also be written 4 + 4''q''3 + q5. A more complex example including an extrinsic prime is 13-limit [[parahemif]] temperament. Setting ''D'' = 22 + ''q''11 + 5''q''13, we have ''T'' = {{val| (2''q''11 + 10''q''13)/''D'' (18''q''11 + 2''q''13)/''D'' ''q''5 (102''q''11 - 62''q''13)/''D'' 44''q''11/''D'' 44''q''13/''D'' }}. Note that all the prime tunings except for that of 5 lie in the field ''Q'' (''q''11, ''q''13), where 1/2, 11 and 13 generate the sharp semigroup; 5 is of course the extrinsic prime. The tuning of the other primes is the same as the tuning for hemif temperament, which has the same commas, generated by {144/143, 243/242, 364/363}, and the same sharp semigroup, but which tempers the 2.3.7.11.13 subgroup.

@@ Line 6: / Line 6: @@
 == Proportional error ==
-A ''tuning'' for a regular temperament is defined by a vector ''T'' in [[Vals and tuning space #Vals and monzos|Tenney tuning space]] whose entries are the sizes of the intervals, in cents, which the ''n'' generators of the regular temperament (often the first ''n'' primes) are mapped to. ''T'' is denoted by a {{w|Bra–ket notation|bra vector}}, and if '''m''' is a monzo then {{vmprod|''T''|'''m'''}} is the size, in cents, of the interval defined by '''m''' in the tuning ''T''. If ''k'' is the rational number which '''m''' represents, then we may also write this quantity as ''T'' (''k'').
+A ''tuning'' for a regular temperament is defined by a vector ''T'' in [[Vals and tuning space #Vals and monzos|Tenney tuning space]] whose entries are the sizes of the intervals, in cents, which the ''n'' generators of the regular temperament (often the first ''n'' primes) are mapped to. ''T'' is denoted by a {{w|Bra–ket notation|bra vector}}, and if '''m''' is a monzo then {{vmp|''T''|'''m'''}} is the size, in cents, of the interval defined by '''m''' in the tuning ''T''. If ''k'' is the rational number which '''m''' represents, then we may also write this quantity as ''T'' (''k'').
-For example, if '''m''' is {{monzo| -4 4 -1 }} then ''k'' = 81/80 (a [[syntonic comma]]). If ''T'' is {{val| 1200 1900 2800 }} (a multiple of [[12edo]]) then {{vmprod|''T''|'''m'''}} = -4800 + 7600 - 2800 = 0. Thus, while cents (''k'') = 21.506290, ''T'' (''k'') = 0 (i.e., the tuning tempers away the syntonic comma).
+For example, if '''m''' is {{monzo| -4 4 -1 }} then ''k'' = 81/80 (a [[syntonic comma]]). If ''T'' is {{val| 1200 1900 2800 }} (a multiple of [[12edo]]) then {{vmp|''T''|'''m'''}} = -4800 + 7600 - 2800 = 0. Thus, while cents (''k'') = 21.506290, ''T'' (''k'') = 0 (i.e., the tuning tempers away the syntonic comma).
 Given a tuning ''T'' and any rational number ''q'' in the domain of ''T'', the ''signed error'' of ''T'' on ''q'' is defined as Err (''q'') = ''T'' (''q'') - cents (''q''). The ''absolute error'' Arr (''q'') = |Err (''q'')| is the absolute value of the signed error.
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 == Finding the tuning ==
-For a temperament with both intrinsic and extrinsic primes, we may find the set of TOP tunings by first computing the tuning of the intrinsic primes. Then if ''r'' is an extrinsic prime, the tuning may be anything in the range where APE (''r'') ≤ ''E''. The limit of the [[Tp tuning|L<sup>''p''</sup> tuning]] as ''p'' tends to 1 and the centroid of the region of TOP tunings both lead to choosing the JI tuning for ''r''. This produces the canonical TOP tuning, called '''TIPTOP'''. To find the TIPTOP tuning one method is to solve for all the potential TOP tunings of the intrinsic primes, find the tuning with least error, and then tune all the extrinsic primes purely. An alternative method is to first set up a linear programming problem; if ''T'' is a val with indeterminate coefficients ''T'' = {{val| ''t''<sub>1</sub> ''t''<sub>2</sub> … ''t''<sub>''k''</sub> }} then minimize ''E'' subject to nonnegativity and the linear constraints {''t''<sub>''n''</sub>/log<sub>2</sub>(''p''<sub>''n''</sub>) - 1 ≤ ''E'', 1 - ''t''<sub>''n''</sub>/log<sub>2</sub>(''p''<sub>''n''</sub>) ≤ ''E'', {{vmprod|T|''c''<sub>''k''</sub>}} = 0} where the ''p''<sub>''n''</sub> are the primes of the temperament, and the ''c''<sub>''k''</sub> are the commas. We then may replace the tuning of all of the extrinsic primes with the pure JI tuning to get TIPTOP.
+For a temperament with both intrinsic and extrinsic primes, we may find the set of TOP tunings by first computing the tuning of the intrinsic primes. Then if ''r'' is an extrinsic prime, the tuning may be anything in the range where APE (''r'') ≤ ''E''. The limit of the [[Tp tuning|L<sup>''p''</sup> tuning]] as ''p'' tends to 1 and the centroid of the region of TOP tunings both lead to choosing the JI tuning for ''r''. This produces the canonical TOP tuning, called '''TIPTOP'''. To find the TIPTOP tuning one method is to solve for all the potential TOP tunings of the intrinsic primes, find the tuning with least error, and then tune all the extrinsic primes purely. An alternative method is to first set up a linear programming problem; if ''T'' is a val with indeterminate coefficients ''T'' = {{val| ''t''<sub>1</sub> ''t''<sub>2</sub> … ''t''<sub>''k''</sub> }} then minimize ''E'' subject to nonnegativity and the linear constraints {''t''<sub>''n''</sub>/log<sub>2</sub>(''p''<sub>''n''</sub>) - 1 ≤ ''E'', 1 - ''t''<sub>''n''</sub>/log<sub>2</sub>(''p''<sub>''n''</sub>) ≤ ''E'', {{vmp|T|''c''<sub>''k''</sub>}} = 0} where the ''p''<sub>''n''</sub> are the primes of the temperament, and the ''c''<sub>''k''</sub> are the commas. We then may replace the tuning of all of the extrinsic primes with the pure JI tuning to get TIPTOP.
 We may solve the sharp semigroup equations exactly to obtain solutions in the transcendental extension ''Q'' (log<sub>2</sub>(''q''<sub>1</sub>), log<sub>2</sub>(''q''<sub>2</sub>), …, log<sub>2</sub>(''q''<sub>''k''</sub>)) where the ''q''<sub>''n''</sub> are the intrinsic primes other than 2. For example, take 5-limit meantone. Since 2 and 5 divide 80 and 3 divides 81. this is an intrinsic temperament. Solving for the TOP tuning either by linear programing or checking all the potential TOP tunings, we find the sharp semigroup is generated by {2, 1/3, 5}. Solving the sharp semigroup equations gives us a TOP tuning ''T'' = {{val| 3''q''<sub>3</sub>/log<sub>2</sub>(6480) (8''q''<sub>3</sub> + 2''q''<sub>3</sub>''q''<sub>5</sub>)/log<sub>2</sub>(6480) 8''q''<sub>3</sub>''q''<sub>5</sub>/log<sub>2</sub>(6480) }}. Here ''q''<sub>3</sub> = log<sub>2</sub> (3), ''q''<sub>5</sub> = log<sub>2</sub> (5), and the denominator can also be written 4 + 4''q''<sub>3</sub> + q<sub>5</sub>. A more complex example including an extrinsic prime is 13-limit [[parahemif]] temperament. Setting ''D'' = 22 + ''q''<sub>11</sub> + 5''q''<sub>13</sub>, we have ''T'' = {{val| (2''q''<sub>11</sub> + 10''q''<sub>13</sub>)/''D'' (18''q''<sub>11</sub> + 2''q''<sub>13</sub>)/''D'' ''q''<sub>5</sub> (102''q''<sub>11</sub> - 62''q''<sub>13</sub>)/''D'' 44''q''<sub>11</sub>/''D'' 44''q''<sub>13</sub>/''D'' }}. Note that all the prime tunings except for that of 5 lie in the field ''Q'' (''q''<sub>11</sub>, ''q''<sub>13</sub>), where 1/2, 11 and 13 generate the sharp semigroup; 5 is of course the extrinsic prime. The tuning of the other primes is the same as the tuning for hemif temperament, which has the same commas, generated by {144/143, 243/242, 364/363}, and the same sharp semigroup, but which tempers the 2.3.7.11.13 subgroup.