Technical data guide for regular temperaments: Difference between revisions
Created page with "{{Beginner}} Regular temperaments are often described with several mathematical properties. This information can be condensed in the form of ''temperament data'' tables, w..." |
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{{Main|Just intonation subgroup}} | {{Main|Just intonation subgroup}} | ||
{{See also|Domain basis}} | {{See also|Domain basis}} | ||
The ''subgroup'' (or ''domain basis'') 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 [[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]] intervals. | The ''subgroup'' (or ''domain basis'') 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 [[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), aka ''rational'', intervals. | ||
In a subgroup, all intervals must be reachable by stacking (up and down) copies of a few "generating intervals", or ''[[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. 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 (i.e. 2/3 × 2/1 = 4/3). | In a subgroup, all intervals must be reachable by stacking (up and down) copies of a few "generating intervals", or ''[[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. 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 (i.e. 2/3 × 2/1 = 4/3). | ||
A subgroup is generally expressed as a list of its generators separated by dots: e.g. 2.3.5 is the subgroup of all intervals consisting of combinations of [[2/1]], [[3/1]] and [[5/1]]. The 2.3.5 subgroup is equivalent to the [[5-limit]], | The generators of the entirety of JI are the infinite set of prime numbers: 2, 3, 5, 7, etc.; therefore the most common type of subgroup of JI uses a subset of primes (or, if 2 is in the subset, equivalently octave-reduced prime harmonics such as 3/2, 5/4, 7/4, etc.) as its generators. A subgroup is generally expressed as a list of its generators separated by dots: e.g. 2.3.5 is the subgroup of all intervals consisting of combinations of [[2/1]], [[3/1]] and [[5/1]]. The 2.3.5 subgroup is equivalent to the [[5-limit]], the subgroup defined by the [[prime harmonics]] up to 5, though for maximum clarity the temperament tables currently prefer spelling out the primes explicitly. | ||
However, it may be reasonable in some cases to include composite numbers in a subgroup: the subgroup 2.7.9.11.15 includes ''some'' intervals that contain 3 and 5 in their factorization (such as 9/7, 15/8, or 5/3 - the last being interpreted as 15/9), but not others (it would not contain an interval like 3/2 or 5/4, since these can't be reached from multiplying and dividing 9 and 15 with primes); or even fractions, like the subgroup 2.3.11.13/5.17 (note that this is interpreted as 2.3.11.(13/5).17), which includes intervals of 13 and intervals of 5, but only when a power of 13 is matched by an equal power of 5 on the other side of the fraction. | |||
=== Comma list === | === Comma list === | ||
Revision as of 22:38, 15 August 2024
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This is a beginner page. It is written to allow new readers to learn about the basics of the topic easily. |
Regular temperaments are often described with several mathematical properties. This information can be condensed in the form of temperament data tables, which are typically found on wiki pages for temperament families and clans (e.g. Meantone family) or in the output of temperament finding scripts (e.g. Graham Breed's or Sintel's).
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.
Structure properties
Subgroup (domain basis)
The subgroup (or domain basis) of a regular temperament is the set of all intervals which are considered to be approximated by the temperament. For example, it is common to consider that 3/2 is approximated by 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), aka rational, intervals.
In a subgroup, all intervals must be reachable by stacking (up and down) copies of a few "generating intervals", or generators. Continuing the previous example, if 3/2 is taken as a generator of the subgroup, then 9/4 is also included in the subgroup. 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 (i.e. 2/3 × 2/1 = 4/3).
The generators of the entirety of JI are the infinite set of prime numbers: 2, 3, 5, 7, etc.; therefore the most common type of subgroup of JI uses a subset of primes (or, if 2 is in the subset, equivalently octave-reduced prime harmonics such as 3/2, 5/4, 7/4, etc.) as its generators. A subgroup is generally expressed as a list of its generators separated by dots: e.g. 2.3.5 is the subgroup of all intervals consisting of combinations of 2/1, 3/1 and 5/1. The 2.3.5 subgroup is equivalent to the 5-limit, the subgroup defined by the prime harmonics up to 5, though for maximum clarity the temperament tables currently prefer spelling out the primes explicitly.
However, it may be reasonable in some cases to include composite numbers in a subgroup: the subgroup 2.7.9.11.15 includes some intervals that contain 3 and 5 in their factorization (such as 9/7, 15/8, or 5/3 - the last being interpreted as 15/9), but not others (it would not contain an interval like 3/2 or 5/4, since these can't be reached from multiplying and dividing 9 and 15 with primes); or even fractions, like the subgroup 2.3.11.13/5.17 (note that this is interpreted as 2.3.11.(13/5).17), which includes intervals of 13 and intervals of 5, but only when a power of 13 is matched by an equal power of 5 on the other side of the fraction.
Comma list
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Mapping
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Sval mapping
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Gencom mapping
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Mapping to lattice
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Wedgie
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Associated temperament
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Complexity spectrum
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Tuning properties
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Optimal tuning(s)
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Minimax tuning(s)
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Tuning ranges
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Optimal ET sequence
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Projection pair
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Badness
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Scale properties
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Scales
- See also: Category:Pages with Scala files
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