Technical data guide for regular temperaments: Difference between revisions

(2 intermediate revisions by 2 users not shown)

Line 28:

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 {{monzo|-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 ===

=== Mapping and subgroup-val 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.

Line 37:

For an example, let us look at meanpop, an [[11-limit]] extension of meantone. Its mapping is given by {{mapping| 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"), especially because it is reached the ''opposite'' way that 3, 5, and 7 are and so ratios of 11 with these other primes are even more complex. Thus intervals of 11 will not appear until quite a long way down the [[chain of fifths]], and only in rather large scales built out of tempered intervals.

In subgroups other than full prime-limits, mappings are sometimes called ~~"sval~~ mappings"; the only distinction here is that the columns of the mapping do not indicate all consecutive primes but only the basis elements of the subgroup. These are distinct from "gencom mappings" with zero entries for primes not included in the subgroup.

In subgroups other than full prime-limits, mappings are sometimes called ''subgroup-val mappings''; the only distinction here is that the columns of the mapping do not indicate all consecutive primes but only the basis elements of the subgroup. These are distinct from ''gencom mappings'' with zero entries for primes not included in the subgroup.

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''.

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 conventional 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''.

=== Extensions and restrictions ===

Line 53:

~~{{todo|complete section|inline=1}}~~

~~==== Wedgie ====~~

~~{{Main|Plücker coordinates}}~~

Line 68:

Line 64:

== 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 {{nowrap|"[[5/4]]" ~ "81/64"}} set to {{nowrap|~~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]].

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 675{{c}}, and therefore its {{nowrap| "[[5/4]]" ~ "81/64" }} set to {{nowrap| 4×675{{c}} − 2400{{c}} {{=}} 300{{c}} }}. However, this is an absurd tuning for meantone since 300{{c}} 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~~ scale]]s as finite subsets of the intervals available in the temperament.

Therefore, one can speak of temperaments as having finite tuning ranges for their generator, which is useful in the picture of building [[mos scale]]s 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).

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. Multiple optimization algorithms are offered, for both tempered-octave (WE, in progress) and pure-octave tuning (most commonly CTE and POTE, but we're unifying at CWE as of late); 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.

@@ Line 28: / Line 28: @@
 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 {{monzo|-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 ===
+=== Mapping and subgroup-val mapping ===
-{{Main|Mapping}}
+{{Main| Mapping }}
-{{See also|Smonzos and svals}}
+{{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.
@@ Line 37: / Line 37: @@
 For an example, let us look at meanpop, an [[11-limit]] extension of meantone. Its mapping is given by {{mapping| 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"), especially because it is reached the ''opposite'' way that 3, 5, and 7 are and so ratios of 11 with these other primes are even more complex. Thus intervals of 11 will not appear until quite a long way down the [[chain of fifths]], and only in rather large scales built out of tempered intervals.
-In subgroups other than full prime-limits, mappings are sometimes called "sval mappings"; the only distinction here is that the columns of the mapping do not indicate all consecutive primes but only the basis elements of the subgroup. These are distinct from "gencom mappings" with zero entries for primes not included in the subgroup.
+In subgroups other than full prime-limits, mappings are sometimes called ''subgroup-val mappings''; the only distinction here is that the columns of the mapping do not indicate all consecutive primes but only the basis elements of the subgroup. These are distinct from ''gencom mappings'' with zero entries for primes not included in the subgroup.
-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''.
+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 conventional 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''.
 === Extensions and restrictions ===
@@ Line 53: / Line 53: @@
 {{Main|Mapping to lattice}}
 {{See also|Harmonic lattice diagram}}
-{{todo|complete section|inline=1}}
-==== Wedgie ====
-{{Main|Plücker coordinates}}
 {{todo|complete section|inline=1}}
@@ Line 68: / Line 64: @@
 == 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 {{nowrap|"[[5/4]]" ~ "81/64"}} set to {{nowrap|4×675¢ &minus; 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]].
+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 675{{c}}, and therefore its {{nowrap| "[[5/4]]" ~ "81/64" }} set to {{nowrap| 4×675{{c}} − 2400{{c}} {{=}} 300{{c}} }}. However, this is an absurd tuning for meantone since 300{{c}} 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 scale]]s as finite subsets of the intervals available in the temperament.
+Therefore, one can speak of temperaments as having finite tuning ranges for their generator, which is useful in the picture of building [[mos scale]]s as finite subsets of the intervals available in the temperament.
 === Optimal tuning(s) ===
-{{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.
-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).
+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. Multiple optimization algorithms are offered, for both tempered-octave (WE, in progress) and pure-octave tuning (most commonly CTE and POTE, but we're unifying at CWE as of late); 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.