TOP tuning - Revision history

FloraC: Terminology: absolute error is ambiguous here. Use damage instead. Eliminate cents in favor of arbitrary interval size units. Misc. cleanup

2025-08-18T12:39:46Z

Terminology: *absolute error* is ambiguous here. Use *damage* instead. Eliminate cents in favor of arbitrary interval size units. Misc. cleanup

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VectorGraphics at 04:21, 12 June 2025

2025-06-12T04:21:51Z

← Older revision		Revision as of 04:21, 12 June 2025
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	== TOP commas and TOP extensions ==		== TOP commas and TOP extensions ==

	Suppose ''T'' is a TOP tuned temperament with ''i'' intrinsic primes, ''e'' extrinsic primes, and a sharp semigroup of rank ''k'' + 1. Then the dimensionality of ''T'' is ''n'' = ''e'' + ''i''; the corank (rank of the comma group) is ''i'' - ''k'' and so the rank of the temperament is ''n'' - (''i'' - ''k'') = ''e'' + ''k''. If we move a prime from intrinsic to extrinsic, the rank is therefore increased by 1 and the corank decreased by 1, leaving the dimensionality the same. If ''ν''<sub>''p''</sub> is the valuation val from prime ''p'', meaning all coefficients but the one for ''p'' are zero and the ''p'' coefficient is 1, then this "moving" can be accomplished by adding ''ν''<sub>''p''</sub>, for some prime ''p'' which is intrinsic but not a prime or inverse prime of the sharp semigroup, as the bottom row of the val list (mapping matrix) ~~for T, or equivalently wedging it with the wedgie~~ for T. This process can continue until all intrinsic primes except those for the sharp semigroup are moved to extrinsic primes. In this case, ''i'' = ''k'' + 1 so the corank is ''i'' - ''k'' = (''k'' + 1) - ''k'' = 1, and there is only one comma, defined as usual as a rational number number greater than one which is not a square, cube or other power, generating the kernel. Since either this comma or its inverse is a product in the sharp semigroup, its absolute proportional error is equal to APE (''T''). The result is that for any regular temperament, there is a unique comma of the temperament such that the absolute proportional error in any TOP tuning is equal to the maximal absolute proportional error for the temperament. This comma we may call the ''TOP comma''. The TOP comma in a sense encapsulates the error of the temperament. Any TOP tuning of the temperament, including TIPTOP, is also a TOP tuning of the codimension one temperament defined by the TOP comma.		Suppose ''T'' is a TOP tuned temperament with ''i'' intrinsic primes, ''e'' extrinsic primes, and a sharp semigroup of rank ''k'' + 1. Then the dimensionality of ''T'' is ''n'' = ''e'' + ''i''; the corank (rank of the comma group) is ''i'' - ''k'' and so the rank of the temperament is ''n'' - (''i'' - ''k'') = ''e'' + ''k''. If we move a prime from intrinsic to extrinsic, the rank is therefore increased by 1 and the corank decreased by 1, leaving the dimensionality the same. If ''ν''<sub>''p''</sub> is the valuation val from prime ''p'', meaning all coefficients but the one for ''p'' are zero and the ''p'' coefficient is 1, then this "moving" can be accomplished by adding ''ν''<sub>''p''</sub>, for some prime ''p'' which is intrinsic but not a prime or inverse prime of the sharp semigroup, as the bottom row of the val list (mapping matrix) for T. This process can continue until all intrinsic primes except those for the sharp semigroup are moved to extrinsic primes. In this case, ''i'' = ''k'' + 1 so the corank is ''i'' - ''k'' = (''k'' + 1) - ''k'' = 1, and there is only one comma, defined as usual as a rational number number greater than one which is not a square, cube or other power, generating the kernel. Since either this comma or its inverse is a product in the sharp semigroup, its absolute proportional error is equal to APE (''T''). The result is that for any regular temperament, there is a unique comma of the temperament such that the absolute proportional error in any TOP tuning is equal to the maximal absolute proportional error for the temperament. This comma we may call the ''TOP comma''. The TOP comma in a sense encapsulates the error of the temperament. Any TOP tuning of the temperament, including TIPTOP, is also a TOP tuning of the codimension one temperament defined by the TOP comma.

	For example, in the 5 and 7 limits, the TOP comma for magic temperament is 3125/3072; in the 11-limit, {{monzo\| 0 -11 15 0 -5 }}; in the 13 limit, {{monzo\| 0 0 46 0 -19 -11 }}.		For example, in the 5 and 7 limits, the TOP comma for magic temperament is 3125/3072; in the 11-limit, {{monzo\| 0 -11 15 0 -5 }}; in the 13 limit, {{monzo\| 0 0 46 0 -19 -11 }}.

Sintel: -legacy

2025-03-29T18:13:03Z

-legacy

← Older revision		Revision as of 18:13, 29 March 2025
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	~~{{Legacy}}~~
	'''TOP tuning''' is a tuning technique for [[regular temperament]]s. The acronym TOP stands for both <u>T</u>enney <u>OP</u>timal and <u>T</u>empered <u>O</u>ctaves <u>P</u>lease. The latter refers to the fact that the optimal scaling is applied to all of the prime dimensions, including the octave 2/1. The TOP tuning concept was first suggested and named by [[Paul Erlich]].		'''TOP tuning''' is a tuning technique for [[regular temperament]]s. The acronym TOP stands for both <u>T</u>enney <u>OP</u>timal and <u>T</u>empered <u>O</u>ctaves <u>P</u>lease. The latter refers to the fact that the optimal scaling is applied to all of the prime dimensions, including the octave 2/1. The TOP tuning concept was first suggested and named by [[Paul Erlich]].

ArrowHead294 at 13:45, 26 February 2025

2025-02-26T13:45:36Z

← Older revision		Revision as of 13:45, 26 February 2025
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	== Proportional error ==		== 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.		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 ==		== 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.		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.

Lériendil at 16:56, 17 February 2025

2025-02-17T16:56:45Z

← Older revision		Revision as of 16:56, 17 February 2025
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	{{Legacy~~\|TOP_tuning~~}}		{{Legacy}}
	'''TOP tuning''' is a tuning technique for [[regular temperament]]s. The acronym TOP stands for both <u>T</u>enney <u>OP</u>timal and <u>T</u>empered <u>O</u>ctaves <u>P</u>lease. The latter refers to the fact that the optimal scaling is applied to all of the prime dimensions, including the octave 2/1. The TOP tuning concept was first suggested and named by [[Paul Erlich]].		'''TOP tuning''' is a tuning technique for [[regular temperament]]s. The acronym TOP stands for both <u>T</u>enney <u>OP</u>timal and <u>T</u>empered <u>O</u>ctaves <u>P</u>lease. The latter refers to the fact that the optimal scaling is applied to all of the prime dimensions, including the octave 2/1. The TOP tuning concept was first suggested and named by [[Paul Erlich]].

Lériendil: deploying new cat

2025-02-17T16:12:25Z

deploying new cat

← Older revision		Revision as of 16:12, 17 February 2025
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			{{Legacy\|TOP_tuning}}
	'''TOP tuning''' is a tuning technique for [[regular temperament]]s. The acronym TOP stands for both <u>T</u>enney <u>OP</u>timal and <u>T</u>empered <u>O</u>ctaves <u>P</u>lease. The latter refers to the fact that the optimal scaling is applied to all of the prime dimensions, including the octave 2/1. The TOP tuning concept was first suggested and named by [[Paul Erlich]].		'''TOP tuning''' is a tuning technique for [[regular temperament]]s. The acronym TOP stands for both <u>T</u>enney <u>OP</u>timal and <u>T</u>empered <u>O</u>ctaves <u>P</u>lease. The latter refers to the fact that the optimal scaling is applied to all of the prime dimensions, including the octave 2/1. The TOP tuning concept was first suggested and named by [[Paul Erlich]].

Lériendil: Changed protection level for "TOP tuning" ([Edit=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)))

2025-02-17T16:01:43Z

Changed protection level for "TOP tuning" ([Edit=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)))

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FloraC: Style and fix typos

2024-11-26T09:02:16Z

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FloraC: Intro; style out letterwise bold; categories

2023-03-12T15:44:42Z

Intro; style out letterwise bold; categories

← Older revision		Revision as of 15:44, 12 March 2023
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	In '''TOP tuning'''~~, the~~ acronym TOP stands for both ~~'''~~T~~'''~~enney ~~'''~~OP~~'''~~timal and ~~'''~~T~~'''~~empered ~~'''~~O~~'''~~ctaves ~~'''~~P~~'''~~lease. The latter refers to the fact that the optimal scaling is applied to all of the prime dimensions, including the octave 2/1. The TOP tuning concept was first suggested and named by [[Paul Erlich]].		'''TOP tuning''' is a tuning technique for [[regular temperament]]s. The acronym TOP stands for both <u>T</u>enney <u>OP</u>timal and <u>T</u>empered <u>O</u>ctaves <u>P</u>lease. The latter refers to the fact that the optimal scaling is applied to all of the prime dimensions, including the octave 2/1. The TOP tuning concept was first suggested and named by [[Paul Erlich]].

	For any tuning T, we may define the absolute proportional error APE (T) of T as the [http://mathworld.wolfram.com/Supremum.html supremum] (maximum) of the absolute proportional errors of all ''q'' belonging to the domain of T. A TOP tuning for a regular temperament is a tuning supporting the temperament (i.e. one which sends commas of the temperament to unison) with minimal APE. This minimal proportional error is called the TOP error.		For any tuning T, we may define the absolute proportional error APE (T) of T as the [http://mathworld.wolfram.com/Supremum.html supremum] (maximum) of the absolute proportional errors of all ''q'' belonging to the domain of T. A TOP tuning for a regular temperament is a tuning supporting the temperament (i.e. one which sends commas of the temperament to unison) with minimal APE. This minimal proportional error is called the TOP error.
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	Most importantly, the above result holds without any hitches for prime-limit subgroups as the limit tends to infinity, and in particular for infinite-limit generalized patent vals, where the TOP tuning minimizes the error on all rationals regardless of whether those rationals are mapped "consistently" given the mapping on the primes, or "inconsistently" given their direct rounding to the nearest edostep.		Most importantly, the above result holds without any hitches for prime-limit subgroups as the limit tends to infinity, and in particular for infinite-limit generalized patent vals, where the TOP tuning minimizes the error on all rationals regardless of whether those rationals are mapped "consistently" given the mapping on the primes, or "inconsistently" given their direct rounding to the nearest edostep.

	~~[[Category:Regular temperament theory]]~~
	~~[[Category:Terms]]~~
	[[Category:Acronyms]]		[[Category:Acronyms]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:~~Tuning]]~~		[[Category:Regular temperament tuning]]
	~~[[Category:Tuning technique~~]]

FloraC: Improve readability (2/2)

2022-03-13T15:34:05Z

Improve readability (2/2)

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TOP tuning - Revision history

FloraC: Terminology: *absolute error* is ambiguous here. Use *damage* instead. Eliminate cents in favor of arbitrary interval size units. Misc. cleanup

VectorGraphics at 04:21, 12 June 2025

Sintel: -legacy

ArrowHead294 at 13:45, 26 February 2025

Lériendil at 16:56, 17 February 2025

Lériendil: deploying new cat

Lériendil: Changed protection level for "TOP tuning" ([Edit=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:01, 17 February 2025 (UTC)))

FloraC: Style and fix typos

FloraC: Intro; style out letterwise bold; categories

FloraC: Improve readability (2/2)

FloraC: Terminology: absolute error is ambiguous here. Use damage instead. Eliminate cents in favor of arbitrary interval size units. Misc. cleanup