A brief introduction to Regular Temperament Theory - Revision history

ArrowHead294: Changed redirect target from Dave Keenan & Douglas Blumeyer's guide to RTT: introductions#Dave.27s brief introduction to RTT to Dave Keenan & Douglas Blumeyer's guide to RTT/Introductions#Dave's brief introduction to RTT

2024-11-02T01:07:50Z

Changed redirect target from Dave Keenan & Douglas Blumeyer's guide to RTT: introductions#Dave.27s brief introduction to RTT to Dave Keenan & Douglas Blumeyer's guide to RTT/Introductions#Dave's brief introduction to RTT

← Older revision		Revision as of 01:07, 2 November 2024
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	#REDIRECT [[~~Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_introductions~~#Dave~~.27s_brief_introduction_to_RTT~~]]		#REDIRECT [[Dave Keenan & Douglas Blumeyer's guide to RTT/Introductions#Dave's brief introduction to RTT]]

Cmloegcmluin: Redirected page to Dave Keenan & Douglas Blumeyer's guide to RTT: introductions#Dave.27s brief introduction to RTT

2022-12-08T19:30:28Z

Redirected page to Dave Keenan & Douglas Blumeyer's guide to RTT: introductions#Dave.27s brief introduction to RTT

← Older revision		Revision as of 19:30, 8 December 2022
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	~~''by~~ [[Dave ~~Keenan]]''~~		#REDIRECT [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_RTT:_introductions#Dave.27s_brief_introduction_to_RTT]]

	Dissonance is easy. Consonance is rare. The most common kind of consonance occurs when two notes have their frequencies approximating<ref>where "approximating" means something like "having errors less than 30 cents".</ref> a simple<ref>where "simple" means something like "involving a pair of integers whose product is less than 1000".</ref> ratio.

	We like to have different flavors of consonance, which correspond to different simple ratios. We want to find tunings — sets of notes — that will give us enough consonances in enough flavors, without being too complex, and without having errors that are too large. The complexity of a tuning is partly about the number of notes per octave, and partly about the number of different step sizes.

	[[regular temperament theory\|Regular Temperament Theory]] is a powerful tool to aid us in finding such tunings. It observes that the problem of approximating many simple ratios can be reduced to one of approximating a few small prime numbers. It then generates all of its tunings by stacking (both up and down) a small number of intervals which are called the [[generators]]. In one extreme, the generators are the small prime numbers themselves, giving [[just intonation]] (JI) as a lattice, having [https://en.xen.wiki/w/Tenney-Euclidean_temperament_measures zero errors but high complexity]. ~~In another extreme, there is a single small generator whose iterations must approximate all the desired primes, giving an [[equal temperament]] (ET), having low complexity but high errors.~~

	These two extremes were well explored prior to RTT. What RTT did was open up a vast middle ground between JI and ET, where the number of generators is greater than one but less than the number of primes being approximated. These are called regular temperaments (RT). Only a very small region of that middle ground had been explored prior to RTT, namely the "[[meantone]]" region that approximates primes 2, 3 and 5, using two generators which are an octave (prime 2) and a slightly narrow fifth (approximate 2:3).

	~~When the generation of tunings is formulated in this way, the tools of linear algebra can be applied.~~

	The defining thing about a regular temperament is the the count of each generator required to approximate each prime number. This is called the temperament's [[Temperament_Mapping_Matrices_(M-maps)\|mapping]], and can be represented as a matrix.<ref>Strictly speaking, it is the [[Canonical_form\|canonical form]] of the mapping matrix that defines the temperament, because you can replace any set of generators with linear combinations of those generators, and change the mapping accordingly, to obtain the same temperament.</ref>

	We can then institute computer searches to find optimum mappings, with our desired balance of error versus complexity. Many such searches have been done and [[Tour_of_Regular_Temperaments\|many resulting temperaments named and catalogued]].

	~~<references/>~~

	~~[[Category:Regular temperament theory~~]]

Fredg999 category edits: Categories

2022-08-14T01:10:44Z

Dave Keenan at 09:10, 7 October 2021

2021-10-07T09:10:45Z

← Older revision		Revision as of 09:10, 7 October 2021
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	When the generation of tunings is formulated in this way, the tools of linear algebra can be applied.		When the generation of tunings is formulated in this way, the tools of linear algebra can be applied.

	The defining thing about a regular temperament is the the count of each generator required to approximate each prime number.<ref>Strictly speaking, it is the [[Canonical_form\|canonical form]] of the mapping matrix that defines the temperament, because you can replace any set of generators with linear combinations of those generators, and change the mapping accordingly, to obtain the same temperament.</ref> ~~This is called the temperament's [[Temperament_Mapping_Matrices_(M-maps)\|mapping]], and can be represented as a matrix.~~		The defining thing about a regular temperament is the the count of each generator required to approximate each prime number. This is called the temperament's [[Temperament_Mapping_Matrices_(M-maps)\|mapping]], and can be represented as a matrix.<ref>Strictly speaking, it is the [[Canonical_form\|canonical form]] of the mapping matrix that defines the temperament, because you can replace any set of generators with linear combinations of those generators, and change the mapping accordingly, to obtain the same temperament.</ref>

	We can then institute computer searches to find optimum mappings, with our desired balance of error versus complexity. Many such searches have been done and [[Tour_of_Regular_Temperaments\|many resulting temperaments named and catalogued]].		We can then institute computer searches to find optimum mappings, with our desired balance of error versus complexity. Many such searches have been done and [[Tour_of_Regular_Temperaments\|many resulting temperaments named and catalogued]].

	<references/>		<references/>

Dave Keenan at 09:09, 7 October 2021

2021-10-07T09:09:30Z

← Older revision		Revision as of 09:09, 7 October 2021
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	When the generation of tunings is formulated in this way, the tools of linear algebra can be applied.		When the generation of tunings is formulated in this way, the tools of linear algebra can be applied.

	The defining thing about a regular temperament is the the count of each generator required to approximate each prime number.<ref>Strictly speaking, it is the [[Canonical_form\|canonical form]] ~~of the rows~~ of the mapping matrix that defines the temperament, because you can replace any set of generators with linear combinations of those generators, and change the mapping accordingly, to obtain the same temperament.</ref> This is called the temperament's [[Temperament_Mapping_Matrices_(M-maps)\|mapping]], and can be represented as a matrix.		The defining thing about a regular temperament is the the count of each generator required to approximate each prime number.<ref>Strictly speaking, it is the [[Canonical_form\|canonical form]] of the mapping matrix that defines the temperament, because you can replace any set of generators with linear combinations of those generators, and change the mapping accordingly, to obtain the same temperament.</ref> This is called the temperament's [[Temperament_Mapping_Matrices_(M-maps)\|mapping]], and can be represented as a matrix.

	We can then institute computer searches to find optimum mappings, with our desired balance of error versus complexity. Many such searches have been done and [[Tour_of_Regular_Temperaments\|many resulting temperaments named and catalogued]].		We can then institute computer searches to find optimum mappings, with our desired balance of error versus complexity. Many such searches have been done and [[Tour_of_Regular_Temperaments\|many resulting temperaments named and catalogued]].

	<references/>		<references/>

Dave Keenan: Removed references to exterior algebra and wedgies now that canonical mappings are available.

2021-10-07T09:08:32Z

Removed references to exterior algebra and wedgies now that canonical mappings are available.

← Older revision		Revision as of 09:08, 7 October 2021
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	These two extremes were well explored prior to RTT. What RTT did was open up a vast middle ground between JI and ET, where the number of generators is greater than one but less than the number of primes being approximated. These are called regular temperaments (RT). Only a very small region of that middle ground had been explored prior to RTT, namely the "[[meantone]]" region that approximates primes 2, 3 and 5, using two generators which are an octave (prime 2) and a slightly narrow fifth (approximate 2:3).		These two extremes were well explored prior to RTT. What RTT did was open up a vast middle ground between JI and ET, where the number of generators is greater than one but less than the number of primes being approximated. These are called regular temperaments (RT). Only a very small region of that middle ground had been explored prior to RTT, namely the "[[meantone]]" region that approximates primes 2, 3 and 5, using two generators which are an octave (prime 2) and a slightly narrow fifth (approximate 2:3).

	When the generation of tunings is formulated in this way, the tools of linear algebra can be applied.<ref>or our homegrown variety of multilinear algebra, which includes two copies of exterior algebra and uses an extended bra-ket notation, for which, unfortunately, tools are not readily available.</ref>		When the generation of tunings is formulated in this way, the tools of linear algebra can be applied.

	The defining thing about a regular temperament is the the count of each generator required to approximate each prime number.<ref>Strictly speaking, it is the [[~~Wedgies_and_Multivals~~\|~~wedge product~~]] of the rows of the mapping matrix that defines the temperament, because you can replace any set of generators with linear combinations of those generators, and change the mapping accordingly, to obtain the same temperament.		The defining thing about a regular temperament is the the count of each generator required to approximate each prime number.<ref>Strictly speaking, it is the [[Canonical_form\|canonical form]] of the rows of the mapping matrix that defines the temperament, because you can replace any set of generators with linear combinations of those generators, and change the mapping accordingly, to obtain the same temperament.</ref> This is called the temperament's [[Temperament_Mapping_Matrices_(M-maps)\|mapping]], and can be represented as a matrix.

	Alternatively, you can avoid wedge products, by agreeing on rules that determine a canonical set of generators, and therefore a canonical form for the mapping matrix. For example, the first generator, called the period, can be given as a unit fraction of the lowest prime, usually the octave (2). The second generator can be given as a unit fraction of the simplest ratio that allows it to be smaller than the period, and so on.</ref> This is called the temperament's [[Temperament_Mapping_Matrices_(M-maps)\|mapping]], and can be represented as a matrix.

	We can then institute computer searches to find optimum mappings, with our desired balance of error versus complexity. Many such searches have been done and [[Tour_of_Regular_Temperaments\|many resulting temperaments named and catalogued]].		We can then institute computer searches to find optimum mappings, with our desired balance of error versus complexity. Many such searches have been done and [[Tour_of_Regular_Temperaments\|many resulting temperaments named and catalogued]].

	<references/>		<references/>

Dave Keenan: Removed ref to geometric algebra. Included ref to multilinear algebra.

2021-06-27T07:30:20Z

Removed ref to geometric algebra. Included ref to multilinear algebra.

← Older revision		Revision as of 07:30, 27 June 2021
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	These two extremes were well explored prior to RTT. What RTT did was open up a vast middle ground between JI and ET, where the number of generators is greater than one but less than the number of primes being approximated. These are called regular temperaments (RT). Only a very small region of that middle ground had been explored prior to RTT, namely the "[[meantone]]" region that approximates primes 2, 3 and 5, using two generators which are an octave (prime 2) and a slightly narrow fifth (approximate 2:3).		These two extremes were well explored prior to RTT. What RTT did was open up a vast middle ground between JI and ET, where the number of generators is greater than one but less than the number of primes being approximated. These are called regular temperaments (RT). Only a very small region of that middle ground had been explored prior to RTT, namely the "[[meantone]]" region that approximates primes 2, 3 and 5, using two generators which are an octave (prime 2) and a slightly narrow fifth (approximate 2:3).

	When the generation of tunings is formulated in this way, the tools of linear algebra can be applied.<ref>or ~~the slightly more powerful but less well-known and less available tools~~ of ~~a related~~ algebra ~~called~~ exterior algebra, ~~Grassman algebra or geometric algebra~~.</ref>		When the generation of tunings is formulated in this way, the tools of linear algebra can be applied.<ref>or our homegrown variety of multilinear algebra, which includes two copies of exterior algebra and uses an extended bra-ket notation, for which, unfortunately, tools are not readily available.</ref>

	The defining thing about a regular temperament is the the count of each generator required to approximate each prime number.<ref>Strictly speaking, it is the [[Wedgies_and_Multivals\|wedge product]] of the rows of the mapping matrix that defines the temperament, because you can replace any set of generators with linear combinations of those generators, and change the mapping accordingly, to obtain the same temperament.		The defining thing about a regular temperament is the the count of each generator required to approximate each prime number.<ref>Strictly speaking, it is the [[Wedgies_and_Multivals\|wedge product]] of the rows of the mapping matrix that defines the temperament, because you can replace any set of generators with linear combinations of those generators, and change the mapping accordingly, to obtain the same temperament.

Cmloegcmluin at 17:51, 23 June 2021

2021-06-23T17:51:15Z

← Older revision		Revision as of 17:51, 23 June 2021
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			''by [[Dave Keenan]]''

	Dissonance is easy. Consonance is rare. The most common kind of consonance occurs when two notes have their frequencies approximating<ref>where "approximating" means something like "having errors less than 30 cents".</ref> a simple<ref>where "simple" means something like "involving a pair of integers whose product is less than 1000".</ref> ratio.		Dissonance is easy. Consonance is rare. The most common kind of consonance occurs when two notes have their frequencies approximating<ref>where "approximating" means something like "having errors less than 30 cents".</ref> a simple<ref>where "simple" means something like "involving a pair of integers whose product is less than 1000".</ref> ratio.

Cmloegcmluin: Cmloegcmluin moved page Dave Keenan's Introduction to RTT to A brief introduction to Regular Temperament Theory: author's request to remove name from title

2021-04-21T15:22:26Z

Cmloegcmluin moved page Dave Keenan's Introduction to RTT to A brief introduction to Regular Temperament Theory: author's request to remove name from title

← Older revision	Revision as of 15:22, 21 April 2021
(No difference)

Dave Keenan: Extracted mention of exterior algebra to a footnote. Simplified the canonical mapping description.

2021-04-20T22:34:46Z

Extracted mention of exterior algebra to a footnote. Simplified the canonical mapping description.

← Older revision		Revision as of 22:34, 20 April 2021
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	Dissonance is easy. Consonance is rare. The most common kind of consonance occurs when two notes have their frequencies approximating<ref>~~and~~ where "approximating" means something like "having errors less than 30 cents"</ref> a simple<ref>~~and~~ "simple" means something like involving a pair of integers whose product is less than ~~105~~</ref> ratio.		Dissonance is easy. Consonance is rare. The most common kind of consonance occurs when two notes have their frequencies approximating<ref>where "approximating" means something like "having errors less than 30 cents".</ref> a simple<ref>where "simple" means something like "involving a pair of integers whose product is less than 1000".</ref> ratio.

	We like to have different ~~flavours~~ of consonance, which correspond to different simple ratios. We want to find tunings — sets of notes — that will give us enough consonances in enough ~~flavours~~, without being too complex, and without having errors that are too large. The complexity of a tuning is partly about the number of notes per octave, and partly about the number of different step sizes.		We like to have different flavors of consonance, which correspond to different simple ratios. We want to find tunings — sets of notes — that will give us enough consonances in enough flavors, without being too complex, and without having errors that are too large. The complexity of a tuning is partly about the number of notes per octave, and partly about the number of different step sizes.

	[[regular temperament theory\|Regular Temperament Theory]] is a powerful tool to aid us in finding such tunings. It observes that the problem of approximating many simple ratios can be reduced to one of approximating a few small prime numbers. It then generates all of its tunings by stacking (both up and down) a small number of intervals which are called the [[generators]]. In one extreme, the generators are the small prime numbers themselves, giving [[just intonation]] (JI) as a lattice, having [https://en.xen.wiki/w/Tenney-Euclidean_temperament_measures zero errors but high complexity]. In another extreme, there is a single small generator whose iterations must approximate all the desired primes, giving an [[equal temperament]] (ET), having low complexity but high errors.		[[regular temperament theory\|Regular Temperament Theory]] is a powerful tool to aid us in finding such tunings. It observes that the problem of approximating many simple ratios can be reduced to one of approximating a few small prime numbers. It then generates all of its tunings by stacking (both up and down) a small number of intervals which are called the [[generators]]. In one extreme, the generators are the small prime numbers themselves, giving [[just intonation]] (JI) as a lattice, having [https://en.xen.wiki/w/Tenney-Euclidean_temperament_measures zero errors but high complexity]. In another extreme, there is a single small generator whose iterations must approximate all the desired primes, giving an [[equal temperament]] (ET), having low complexity but high errors.
Line 7:		Line 7:
	These two extremes were well explored prior to RTT. What RTT did was open up a vast middle ground between JI and ET, where the number of generators is greater than one but less than the number of primes being approximated. These are called regular temperaments (RT). Only a very small region of that middle ground had been explored prior to RTT, namely the "[[meantone]]" region that approximates primes 2, 3 and 5, using two generators which are an octave (prime 2) and a slightly narrow fifth (approximate 2:3).		These two extremes were well explored prior to RTT. What RTT did was open up a vast middle ground between JI and ET, where the number of generators is greater than one but less than the number of primes being approximated. These are called regular temperaments (RT). Only a very small region of that middle ground had been explored prior to RTT, namely the "[[meantone]]" region that approximates primes 2, 3 and 5, using two generators which are an octave (prime 2) and a slightly narrow fifth (approximate 2:3).

	When the generation of tunings is formulated in this way, the tools of linear algebra can be applied (or the slightly more powerful but less well-known and less available tools of a related algebra called exterior algebra, Grassman algebra or geometric algebra).		When the generation of tunings is formulated in this way, the tools of linear algebra can be applied.<ref>or the slightly more powerful but less well-known and less available tools of a related algebra called exterior algebra, Grassman algebra or geometric algebra.</ref>

	The defining thing about a regular temperament is the the count of each generator required to approximate each prime number.<ref>Strictly speaking, it is the [[Wedgies_and_Multivals\|wedge product]] of the rows of the mapping matrix that defines the temperament, because you can replace any set of generators with linear combinations of those generators, and change the mapping accordingly, to obtain the same temperament.		The defining thing about a regular temperament is the the count of each generator required to approximate each prime number.<ref>Strictly speaking, it is the [[Wedgies_and_Multivals\|wedge product]] of the rows of the mapping matrix that defines the temperament, because you can replace any set of generators with linear combinations of those generators, and change the mapping accordingly, to obtain the same temperament.

	Alternatively, you can avoid wedge products, by agreeing on rules that determine a canonical set of generators. For example, the first generator ~~alone should generate~~ the lowest prime ~~using a positive count~~. ~~Then the first and~~ second ~~generators alone should generate the next prime using~~ a ~~positive count~~ of the ~~second generator~~, and so on~~. This is equivalent to requiring the mapping matrix to be in a certain form. I don't know if this form has a name. I note that an RT mapping matrix is never a square matrix (otherwise it would be JI)~~.</ref> This is called the temperament's [[Temperament_Mapping_Matrices_(M-maps)\|mapping]], and can be represented as a matrix.		Alternatively, you can avoid wedge products, by agreeing on rules that determine a canonical set of generators, and therefore a canonical form for the mapping matrix. For example, the first generator, called the period, can be given as a unit fraction of the lowest prime, usually the octave (2). The second generator can be given as a unit fraction of the simplest ratio that allows it to be smaller than the period, and so on.</ref> This is called the temperament's [[Temperament_Mapping_Matrices_(M-maps)\|mapping]], and can be represented as a matrix.

	We can then institute computer searches to find optimum mappings, with our desired balance of error versus complexity. Many such searches have been done and [[Tour_of_Regular_Temperaments\|many resulting temperaments named and catalogued]].		We can then institute computer searches to find optimum mappings, with our desired balance of error versus complexity. Many such searches have been done and [[Tour_of_Regular_Temperaments\|many resulting temperaments named and catalogued]].

	<references/>		<references/>

← Older revision		Revision as of 01:10, 14 August 2022
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	<references/>		<references/>

			[[Category:Regular temperament theory]]