Technical data guide for regular temperaments: Difference between revisions
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Not all temperament tables provide the same information, nor do they all provide it in exactly the same way, but the following properties should cover most needs. | Not all temperament tables provide the same information, nor do they all provide it in exactly the same way, but the following properties should cover most needs. | ||
{{Tip| See [[XW:TEMPCAT]] if you wish to document a temperament. }} | |||
== Structure properties == | == Structure properties == | ||
=== Subgroup | === Subgroup === | ||
{{Main|Just intonation subgroup}} | {{Main|Just intonation subgroup}} | ||
{{See also|Domain basis}} | {{See also|Domain basis}} | ||
The ''subgroup'' (or ''domain | |||
The ''subgroup'' (or ''domain'') of a regular temperament is the set of all [[interval]]s which are considered to be approximated by the temperament. For example, it is common to consider that the [[frequency ratio]] of [[3/2]] is approximated by [[12edo|12-tone equal temperament]], therefore 3/2 would be included in this set, but other intervals like [[11/8]] could be excluded. Most of the time, a subgroup exclusively contains [[just intonation]] (JI) intervals. | |||
In a subgroup, all intervals are reachable by stacking (up and down) copies of a few "generating intervals", called ''[[Periods and generators|generator]]s''. Continuing the previous example, if [[3/2]] is taken as a generator of the subgroup, then [[9/4]] is also included in the subgroup {{nowrap|(3/2 × 3/2 {{=}} 9/4)}}, and so on. If [[2/1]] is added to the list of subgroup generators, then intervals like [[4/3]] can be reached by combining a 3/2 down with a 2/1 up {{nowrap|(2/3 × 2/1 {{=}} 4/3)}}. | In a subgroup, all intervals are reachable by stacking (up and down) copies of a few "generating intervals", called ''[[Periods and generators|generator]]s''. Continuing the previous example, if [[3/2]] is taken as a generator of the subgroup, then [[9/4]] is also included in the subgroup {{nowrap|(3/2 × 3/2 {{=}} 9/4)}}, and so on. If [[2/1]] is added to the list of subgroup generators, then intervals like [[4/3]] can be reached by combining a 3/2 down with a 2/1 up {{nowrap|(2/3 × 2/1 {{=}} 4/3)}}. | ||
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=== Comma list === | === Comma list === | ||
{{Main|Comma basis}} | {{Main|Comma basis}} | ||
An abstract regular temperament can be thought of as a family of ''valuations'' (''tunings'' of the temperament) of the primes in its subgroup that satisfy certain equations; if we bold the numbers to make it clear that we are speaking of them as abstract variables, an example of such an equation would be {{nowrap|'''3'''<sup>4</sup> {{=}} '''2'''<sup>4</sup> × '''5'''}}. Each of these equations corresponds to setting a JI interval to be equal to the unison ([[1/1]]); the equation specified here sets [[81/80]] (with factorization {{nowrap|2<sup>−4</sup> × 3<sup>4</sup> × 5<sup>−1</sup>}}) to the unison, in other words ''tempering out'' 81/80. | An abstract regular temperament can be thought of as a family of ''valuations'' (''tunings'' of the temperament) of the primes in its subgroup that satisfy certain equations; if we bold the numbers to make it clear that we are speaking of them as abstract variables, an example of such an equation would be {{nowrap|'''3'''<sup>4</sup> {{=}} '''2'''<sup>4</sup> × '''5'''}}. Each of these equations corresponds to setting a JI interval to be equal to the unison ([[1/1]]); the equation specified here sets [[81/80]] (with factorization {{nowrap|2<sup>−4</sup> × 3<sup>4</sup> × 5<sup>−1</sup>}}) to the unison, in other words ''tempering out'' 81/80. | ||
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{{Main| Mapping }} | {{Main| Mapping }} | ||
{{See also| Subgroup monzos and vals }} | {{See also| Subgroup monzos and vals }} | ||
A regular temperament has a structure defined by a set of ''generators'', whose number is equivalent to the ''rank'' of the temperament. Like JI itself, the set of all distinct intervals available to the regular temperament can be created by stacking these generators. Unlike JI, the determination of which intervals are generators is often highly nontrivial given the comma basis or other information. | A regular temperament has a structure defined by a set of ''generators'', whose number is equivalent to the ''rank'' of the temperament. Like JI itself, the set of all distinct intervals available to the regular temperament can be created by stacking these generators. Unlike JI, the determination of which intervals are generators is often highly nontrivial given the comma basis or other information. | ||
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=== Advanced properties === | === Advanced properties === | ||
==== Gencom mapping ==== | ==== Gencom mapping ==== | ||
{{Main|Gencom}} | {{Main|Gencom #Gencom mapping}} | ||
The gencom mapping is similar to the subgroup-val mapping for subgroup temperaments, but can be fractional and can involve zero columns for elements not included in the subgroup. Whilst the subgroup-val mapping takes subgroup monzos and outputs its steps in the temperament, the gencom mapping takes ordinary monzos and outputs the same steps. | |||
==== Mapping to lattice ==== | ==== Mapping to lattice ==== | ||
{{Main|Mapping to lattice}} | {{Main|Mapping to lattice}} | ||
{{See also| | {{See also|Lattice}} | ||
{{todo|complete section|inline=1}} | {{todo|complete section|inline=1}} | ||
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{{Main| Optimization }} | {{Main| Optimization }} | ||
Tuning ''optimization'' is, essentially, the task of finding a tuning for a given regular temperament that has the lowest error in some way. While many approaches exist to going about this, the most widely used are algorithms based on the ''TE metric'', which weight all intervals in the infinite set available to the temperament by a measure of their complexity, and tune in order to minimize deviation from just across all of them. | Tuning ''optimization'' is, essentially, the task of finding a tuning for a given regular temperament that has the lowest error in some way. While many approaches exist to going about this, the most widely used are algorithms based on the ''Tenney–Euclidean'' (''TE'') ''metric'', which weight all intervals in the infinite set available to the temperament by a measure of their complexity, and tune in order to minimize deviation from just across all of them. | ||
It is common to optimize under the constraint that the octave (or equave) is tuned pure, and therefore that the generator known as the ''period'' is either an exact equave or a fraction thereof, whereas the other ''generators'' are tuned to inexact values. The simplest rational interpretation of these intervals are given in cents. | It is common to optimize under the constraint that the octave (or equave) is tuned pure, and therefore that the generator known as the ''period'' is either an exact equave or a fraction thereof, whereas the other ''generators'' are tuned to inexact values. The simplest rational interpretation of these intervals are given in cents. Optima are offered for both tempered-octave (WE, in progress) and pure-octave tuning (CWE, in progress); other algorithms have subtle differences and each may be chosen for a specific use. Besides, optimal tunings are often used as guidelines for where good tunings of a temperament are than as exact ways to tune, and for that purpose the algorithms usually agree sufficiently (aside from extreme exotemperaments and other special cases). | ||
Optimal tunings come with [[error map]]s (in progress). These represent the prime harmonics' deviations from just, though it is merely for the sake of convenience as all tunings of intervals can be derived from the generators and the mapping. | |||
=== Optimal ET sequence === | === Optimal ET sequence === | ||