5L 2s (3/1-equivalent): Difference between revisions
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== Theory == | == Theory == | ||
=== As a macrodiatonic scale === | === As a macrodiatonic scale === | ||
It is the [[macrodiatonic and microdiatonic scales|macrodiatonic scale]] with the period of a [[3/1|tritave]]. This means it is a [[5L 2s| | It is the [[macrodiatonic and microdiatonic scales|macrodiatonic scale]] with the period of a [[3/1|tritave]]. This means it is a [[5L 2s|diatonic scale]], but its intervals are all stretched in a way that makes them unrecognizable—the diatonic fifth is now the size of a major seventh, and octaves are stretched out to tritaves. Interestingly, [[19edt]], an approximation of [[12edo]], has a tuning of this scale, meaning it contains both a diatonic scale (which approximates 12edo's diatonic scale) and a triatonic scale. | ||
diatonic scale]], but | |||
=== Temperament interpretations === | === Temperament interpretations === | ||
It is possible to construct no-twos [[rank-2 temperament]] interpretations of this scale, although most of these do not fit neatly into the 3.5.7 [[subgroup]] used for [[ | It is possible to construct no-twos [[rank-2 temperament]] interpretations of this scale, although most of these do not fit neatly into the 3.5.7 [[subgroup]] used for [[Bohlen–Pierce]]. Two intervals that can serve as macrodiatonic generators are ~[[17/9]], which is just near [[19edt]] in the soft range, and ~[[21/11]] which is just near [[17edt]] in the hard range. | ||
Very soft scales (in the range between [[26edt]] and [[45edt]], serving as a macro-[[flattone]]) can be interpreted in the 3.5.7.17 subgroup as [[no-twos subgroup temperaments#Mizar|Mizar]], in which the generator of a flattened ~17/9 stacks twice and tritave-reduces to [[25/21]], which generates [[Sirius]] temperament. Scales close to basic have an interpretation in the 3.13.17 subgroup, documented as [[no-twos subgroup temperaments#Sadalmelik|Sadalmelik]] in which the generator (the stretched counterpart of the fifth) is also ~17/9 and a stack of 4 generators tritave-reduced (equivalent to the major third) is ~[[13/9]]; see also the page for [[12edt]]. Harder scales can be interpreted in [[Mintaka]] temperament in the 3.7.11 subgroup, which tempers out [[1331/1323]] so that the dark generator (the stretched counterpart of the fourth) is ~[[11/7]], a stack of 2 generators (equivalent to the minor seventh) is ~[[27/11]], and a stack of three generators (equivalent to the minor third) is ~[[9/7]]. | Very soft scales (in the range between [[26edt]] and [[45edt]], serving as a macro-[[flattone]]) can be interpreted in the 3.5.7.17 subgroup as [[no-twos subgroup temperaments#Mizar|Mizar]], in which the generator of a flattened ~17/9 stacks twice and tritave-reduces to [[25/21]], which generates [[Sirius]] temperament. Scales close to basic have an interpretation in the 3.13.17 subgroup, documented as [[no-twos subgroup temperaments#Sadalmelik|Sadalmelik]] in which the generator (the stretched counterpart of the fifth) is also ~17/9 and a stack of 4 generators tritave-reduced (equivalent to the major third) is ~[[13/9]]; see also the page for [[12edt]]. Harder scales can be interpreted in [[Mintaka]] temperament in the 3.7.11 subgroup, which tempers out [[1331/1323]] so that the dark generator (the stretched counterpart of the fourth) is ~[[11/7]], a stack of 2 generators (equivalent to the minor seventh) is ~[[27/11]], and a stack of three generators (equivalent to the minor third) is ~[[9/7]]. | ||
==Modes== | == Modes == | ||
The modes have step patterns which are the same as the modes of the diatonic scale. | The modes have step patterns which are the same as the modes of the diatonic scale. | ||
{{MOS modes}} | {{MOS modes}} | ||
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== Notation == | == Notation == | ||
Being a macrodiatonic scale, it can notated using the traditional diatonic notation, if all intervals are reinterpreted as their stretched versions (like octaves as tritaves). However, this approach involves 1-based indexing for a non-diatonic MOS which is generally discouraged. Alternatively, a generic MOS notation may be used like [[diamond | Being a macrodiatonic scale, it can notated using the traditional diatonic notation, if all intervals are reinterpreted as their stretched versions (like octaves as tritaves). However, this approach involves 1-based indexing for a non-diatonic MOS which is generally discouraged. Alternatively, a generic MOS notation may be used like [[diamond-mos notation]], which enables 0-based indexing at the cost of obscuring the connection to the standard diatonic scale. | ||
== Intervals == | == Intervals == | ||
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== Scale tree == | == Scale tree == | ||
{{Scale tree|name=triatonic|Comments=11/8:[[No-twos subgroup temperaments#Mizar|Mizar]]; 3/2:Just [[17/9]] generator (1101.045¢); 2/1:[[CTE tuning]] for the b12 & b5 temperament (1109.689¢); 3/1: Just [[21/11]] generator (1119. | {{Scale tree|name=triatonic|Comments=11/8:[[No-twos subgroup temperaments#Mizar|Mizar]]; 3/2:Just [[17/9]] generator (1101.045¢); 2/1:[[CTE tuning]] for the b12 & b5 temperament (1109.689¢); 3/1: Just [[21/11]] generator (1119.463¢); 7/2: [[Mintra]]; 4/1: [[Mintaka]] is around here; 5/1: [[No-twos subgroup temperaments#Minalzidar|Minalzidar]]}} | ||