Support: Difference between revisions
The "nontechnically-speaking sense" wasn't equivalent to the "technically-speaking sense" given just below, and not much less technical anyway. Implement an actually nontechnically-speaking explanation |
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A [[regular temperament]] is '''supported''' by an [[equal temperament]] that | A [[regular temperament]] is '''supported''' by an [[equal temperament]] that makes all of its [[comma]]s [[vanish]].<ref>The original definition of support in this RTT sense was given by [[Gene Ward Smith]] in [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_13712#13967], "Suppose T is a wedgie, and v is an equal temperament val. Then v supports T if and only if T^v = 0."</ref> The equal temperament thus '''supports''' this temperament, or equivalently stated, the equal temperament ''is a temperament of'' this temperament, in the same sense as that any temperament is a temperament of [[just intonation]]. | ||
For example, [[22edo|22-ET]] supports [[pajara]], because pajara | For example, [[22edo|22-ET]] supports [[pajara]], because pajara makes [[225/224]] and [[64/63]] vanish, and so does 22-ET. The supporting temperament will make at least one additional comma vanish; in this example, 22-ET makes [[245/243]] vanish. | ||
== Generalization == | == Generalization == | ||
An equal temperament is the same thing as a rank-1 temperament, and the initial definition given here where the supporting temperament is rank-1 is | An equal temperament is the same thing as a rank-1 temperament, and the initial definition given here where the supporting temperament is rank-1 is the most common use case as of 2022. However, in general, we can say that any lower-[[nullity]] (higher-[[rank]]) temperament is supported by a higher-nullity (lower-rank) temperament if the higher-nullity temperament also makes all the commas that the lower-nullity temperament vanishes vanish.<ref>As an edge case, JI is conceptualized as a temperament where no intervals are made to vanish, so any temperament "supports" JI; in other words, any temperament is a temperament of JI.</ref><ref>Example uses of this sense can be found on the following pages: [[Subgroup temperament families, relationships, and genes#Support]], [[Meet and join #Intra-Subgroup Temperament Meet and Join]], and [[Interior product #Applications]].</ref> Technically speaking, we would say that the lower-nullity temperament's comma space is a [[subspace]] of the higher-nullity temperament's comma space. | ||
An equivalent generalized definition of "support" would be to say that the lower-''rank'' temperament | An equivalent generalized definition of "support" would be to say that the lower-''rank'' temperament maps all intervals the same way as the higher-''rank'' temperament does. In this case, the technical definition would be that the lower-rank temperament's mapping-row space is a subspace of the higher-rank temperament's mapping-row space. Another way to say this is that one can find forms of the mappings for these two temperaments where the higher-rank mapping is identical to the lower-rank mapping but with additional mapping rows. To use the 22-ET and pajara example above, we can see that pajara has a mapping form {{rket|{{map|12 19 28 34}} {{map|22 35 51 62}}}}, which contains 22-ET {{map|22 35 51 62}} as its second row. | ||
== Other informal | == Other informal usage == | ||
The word "supports" is also used in a more informal and generic sense, both in and outside of regular temperament theory: a pitch structure '''A''' supports another pitch structure '''B''' when '''A''' ''can be used for'' '''B'''. For example, a temperament or tuning may be said to support a scale or a chord, or a temperament may be said to support a tuning of another temperament, or scale may be said to support a chord or harmony within a certain [[prime limit|prime-]] or [[odd-limit]] or [[domain basis]], etc. | |||
Most typically, this sense of "support" is reserved for such cases where not only is '''B''' merely possible or ''valid'' with '''A''', but '''A''' is actually ''good'' for '''B'''. The technical RTT sense of "supports" defined above is strictly mathematical and makes no such stipulation of aesthetic goodness, however. According to it, the [[wart notation|2c]] [[map]] for [[2edo|2-ET]] supports [[meantone]] despite tuning the fifth to 600¢, well outside the [[diamond tuning ranges]] for meantone. Furthermore, since [[Single-pitch tuning|0-ET]] makes ''every'' comma vanish, it therefore supports ''every temperament'', though clearly it does not do so in any musically useful sense. Due to this key difference between the technical RTT definition and the informal general definition, occasionally usages may conflict and surprise some readers. | |||
== Cover == | |||
Historically, the term "cover" was used in a way that is related to the modern technical RTT sense of "support". It was used in the opposite direction, that is, if 12-ET supports meantone, then meantone ''covers'' 12-ET. This usage can still be found on Graham Breed's website.<ref>[http://x31eq.com/catalog.htm Catalog of Linear Temperaments]</ref> However, usage was inconsistent; Gene Ward Smith used "cover" as synonymous to "support".<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_45560#45763 Yahoo! Tuning Groups | ''Also... Werkmeister III and WTC question...'']</ref> | |||
== See also == | == See also == | ||
* [[Lists of edo-distinct temperaments|EDO-distinct temperaments]] | |||
* [[ | |||
== Footnotes == | == Footnotes == | ||
<references/> | <references/> | ||
Latest revision as of 21:50, 12 June 2025
A regular temperament is supported by an equal temperament that makes all of its commas vanish.[1] The equal temperament thus supports this temperament, or equivalently stated, the equal temperament is a temperament of this temperament, in the same sense as that any temperament is a temperament of just intonation.
For example, 22-ET supports pajara, because pajara makes 225/224 and 64/63 vanish, and so does 22-ET. The supporting temperament will make at least one additional comma vanish; in this example, 22-ET makes 245/243 vanish.
Generalization
An equal temperament is the same thing as a rank-1 temperament, and the initial definition given here where the supporting temperament is rank-1 is the most common use case as of 2022. However, in general, we can say that any lower-nullity (higher-rank) temperament is supported by a higher-nullity (lower-rank) temperament if the higher-nullity temperament also makes all the commas that the lower-nullity temperament vanishes vanish.[2][3] Technically speaking, we would say that the lower-nullity temperament's comma space is a subspace of the higher-nullity temperament's comma space.
An equivalent generalized definition of "support" would be to say that the lower-rank temperament maps all intervals the same way as the higher-rank temperament does. In this case, the technical definition would be that the lower-rank temperament's mapping-row space is a subspace of the higher-rank temperament's mapping-row space. Another way to say this is that one can find forms of the mappings for these two temperaments where the higher-rank mapping is identical to the lower-rank mapping but with additional mapping rows. To use the 22-ET and pajara example above, we can see that pajara has a mapping form [⟨12 19 28 34] ⟨22 35 51 62]}, which contains 22-ET ⟨22 35 51 62] as its second row.
Other informal usage
The word "supports" is also used in a more informal and generic sense, both in and outside of regular temperament theory: a pitch structure A supports another pitch structure B when A can be used for B. For example, a temperament or tuning may be said to support a scale or a chord, or a temperament may be said to support a tuning of another temperament, or scale may be said to support a chord or harmony within a certain prime- or odd-limit or domain basis, etc.
Most typically, this sense of "support" is reserved for such cases where not only is B merely possible or valid with A, but A is actually good for B. The technical RTT sense of "supports" defined above is strictly mathematical and makes no such stipulation of aesthetic goodness, however. According to it, the 2c map for 2-ET supports meantone despite tuning the fifth to 600¢, well outside the diamond tuning ranges for meantone. Furthermore, since 0-ET makes every comma vanish, it therefore supports every temperament, though clearly it does not do so in any musically useful sense. Due to this key difference between the technical RTT definition and the informal general definition, occasionally usages may conflict and surprise some readers.
Cover
Historically, the term "cover" was used in a way that is related to the modern technical RTT sense of "support". It was used in the opposite direction, that is, if 12-ET supports meantone, then meantone covers 12-ET. This usage can still be found on Graham Breed's website.[4] However, usage was inconsistent; Gene Ward Smith used "cover" as synonymous to "support".[5]
See also
Footnotes
- ↑ The original definition of support in this RTT sense was given by Gene Ward Smith in [1], "Suppose T is a wedgie, and v is an equal temperament val. Then v supports T if and only if T^v = 0."
- ↑ As an edge case, JI is conceptualized as a temperament where no intervals are made to vanish, so any temperament "supports" JI; in other words, any temperament is a temperament of JI.
- ↑ Example uses of this sense can be found on the following pages: Subgroup temperament families, relationships, and genes#Support, Meet and join #Intra-Subgroup Temperament Meet and Join, and Interior product #Applications.
- ↑ Catalog of Linear Temperaments
- ↑ Yahoo! Tuning Groups | Also... Werkmeister III and WTC question...