Technical data guide for regular temperaments
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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 the frequency ratio of 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) intervals.
In a subgroup, all intervals are reachable by stacking (up and down) copies of a few "generating intervals", called 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 (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 (2/3 × 2/1 = 4/3).
The entirety of JI can be generated by the infinite set of prime numbers (2, 3, 5, 7, …). In practice, most subgroups are generated by a few primes only (hence the term subgroup, where JI is the larger group). A common kind of subgroups are prime limits, which are generated by all prime harmonics up to a certain limit. For example, the 5-limit is generated by all primes up to 5 (i.e. 2, 3 and 5).
A subgroup is generally expressed as a list of its generators separated by dots. For example, "2.3.5" denotes the aforementioned 5-limit. Primes are not required to be consecutive; 2.3.7 is an equally valid subgroup.
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. Composites or fractions treated as primes in this context are often called "formal primes" or "basis elements."
Comma list
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 34 = 24 × 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 2−4 × 34 × 5−1) to the unison, in other words tempering out 81/80.
The comma basis is a list of such intervals–commas–that are tempered out by the temperament, thereby restricting the set of possible tunings. If a tuning only tempers out one comma, then the only intervals within the subgroup that are set to the unison are that comma, and its positive and negative powers (in the case above, 80/81, 6561/6400 = (81/80)2, etc.); therefore there is only one logical choice for "which comma" you claim is tempered, that being the simplest of these powers greater than the unison. Rank-2 temperaments in subgroups including 3 primes, such as the 5-limit, only temper out one comma, for instance.
However, if a tuning tempers out multiple independent commas, the situation gets more complicated, for the set of tempered intervals in fact forms a lattice generated by more than one generator (in other words, a nontrivial subgroup of JI), and the choice of which specific intervals to consider generators (which in this context are basis commas) is not always obvious. For instance, septimal meantone tempers out the intervals 126/125 = 2 × 32 × 5−3 × 7; 225/224 = 2−5 × 32 × 52 × 7−1, and 81/80, but 81/80 = (126/125) × (225/224), and therefore these three commas are not all independent–but all of them are useful, in that all three define prominent temperament families (collections of regular temperaments that share a tempered comma in common): 81/80 defines meantone, 126/125 defines starling, and 225/224 defines marvel. Various methods exist for choosing which commas are selected to be basis commas, which are associated with the technique of matrix echelon forms; in the case of septimal meantone, the basis commas are chosen to be 81/80 and 126/125 at the price of obscuring the fact that it also tempers out 225/224.
As a last note, factorizations are generally abbreviated in the form of a (subgroup) monzo, which is simply a list of the exponents in a factorization that are attached to each (formal) prime in the subgroup, so that for instance 225/224 would be [-5 2 2 -1⟩ (in this case the subgroup is 2.3.5.7; it should be specified if there is any ambiguity, but if not it can be assumed to be the temperament's subgroup).
Mapping and sval mapping
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.
Therefore, it is useful to specify a way to "translate" between JI intervals and stacks of temperament generators—for example, to know how a given JI interval is retuned under the temperament (and a specific tuning of the generators). The mapping provides this way, by specifying how each of the prime harmonics (subgroup generators) are equated to a stack of generators of the temperament.
For an example, let us look at meanpop, an 11-limit extension of meantone. Its mapping is given by [⟨1 0 -4 -13 24], ⟨0 1 4 10 -13]], with the "mapping generators" being ~2 and ~3 (with the tilde used to indicate tunings of these intervals under the temperament), where each vector within the mapping indicates the number of each generator in the stack used to reach a prime harmonic. This particular mapping tells us that 2/1 and 3/1 are reached by one of the generators ~2 and ~3 (trivially) each; that 5/1 is reached by 4 times ~3 upward and 4 times ~2 downward; that 7/1 is reached by 10 times ~3 upward and 13 times ~2 downward; and that 11/1 is reached by 24 times ~2 upward and 13 times ~3 downward. Therefore, the 11th harmonic in this temperament is quite complex (even if we regard ~2, the octave, as "free"), and intervals of 11 will not appear until quite a long way down the circle of fifths, and only in rather large scales built out of tempered intervals.
One last note is that mappings may use a slightly different (if equivalent) set of generators from elsewhere in the temperament data: for meanpop, for instance, the "canonical" generator, for which optimal tunings are specified, is in fact ~3/2, rather than ~3. In these cases, the mapping should (but does not always) specify the set of generators used for the mapping.
Advanced properties
Gencom mapping
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Todo: complete section |
Mapping to lattice
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Wedgie
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Todo: complete section |
Associated temperament
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Complexity spectrum
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Todo: complete section |
Tuning properties
While an abstract regular temperament does not specify the values to which its intervals (meaning, its generators or valuations of primes) are tuned, it is clear that there are ways to set these values that are reasonable, and those that are absurd. For instance, meantone could be tuned with its "3/2" set to 675c, and therefore its "5/4" ~ "81/64" set to 4×675¢ − 2400¢ = 300¢. However, this is an absurd tuning for meantone since 300¢ has a far better interpretation as 6/5 than 5/4, and the temperament providing that interpretation is instead mavila.
Therefore, one can speak of temperaments as having finite "tuning ranges" for their generator, which is useful in the picture of building MOS scales as finite subsets of the intervals available in the temperament.
Optimal tuning(s)
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.
It is conventional on the wiki 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. The rational interpretation of the period is depicted as equated to this fraction. However, the other generators are tuned to inexact values expressed in cents, and appear as rational interpretations equated to these values. Multiple optimization algorithms (most commonly CTE and POTE) may appear; different algorithms have subtle differences and one or the other may be chosen for a specific use. However, optimal tunings are used more often 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).
In the future, temperaments may appear with optimal tunings of prime harmonics (and their deviation from just) in a collapsible table, though this will be merely for the sake of convenience as all tunings of intervals can be derived from the generators and the mapping.
Optimal ET sequence
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Badness
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Advanced properties
Minimax tuning(s)
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Tuning ranges
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Projection pair
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Scale properties
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Scales
- See also: Category:Pages with Scala files
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