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5L 2s (3/1-equivalent): Difference between revisions

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{{MOS intro|Other Names=triatonic}}  
{{MOS intro|Other Names=triatonic}}  
== Name ==
== Name ==
The name '''triatonic''' was coined by [[User:CompactStar|CompactStar]], and is a back-formation from "diatonic" with di- being interpreted as 2 (the [[2/1|octave]]) and replaced with tri- for 3 (the [[3/1|tritave]]). It is not an official name in [[TAMNAMS]].
The name '''triatonic''' was coined by [[User:CompactStar|CompactStar]], as a back-formation from "diatonic" with di- being interpreted as 2 (the [[2/1|octave]]) and tri- being interpreted as 3 (the [[3/1|tritave]]), though it is not an official name in [[TAMNAMS]].
 
== Scale properties ==
{{TAMNAMS use}}
 
=== Intervals ===
{{MOS intervals}}
 
=== Generator chain ===
{{MOS genchain}}
 
=== Modes ===
The modes of {{MOS scalesig|5L 2s <3/1>}} have step patterns which are the same as the modes of the diatonic scale.
{{MOS mode degrees}}


== 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|
{{MOS scalesig|5L 2s <3/1>}} is a [[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 to the point of being 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 has [[2/1|octave]]s stretched out to the size of a [[3/1|tritave]]. Other intervals are also stretched in a way that makes the unrecognizable–them diatonic fifth is now the size of a major seventh. 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.


=== Temperament interpretations ===
=== Temperament interpretations ===
It is possible to construct no-twos [[rank-2 temperament]] interpretations of this scale, but it is difficult to interpret within commonly-studied [[no-twos subgroup temperaments|no-twos subgroup]]s like the 3.5.7 [[subgroup]] used for [[Bohlen-Pierce]]. Scales close to basic have an interpretation in the as-of-yet unnamed b12 & b5 temperament in the 3.13.17 subgroup, in which the generator (the stretched counterpart of the fifth) is ~[[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 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]].
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.


==Modes==
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]].
The modes have step patterns which are the same as the modes of the diatonic scale.
{{MOS modes}}
 
=== Scale degrees ===
{{MOS mode degrees}}


== 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 MOS notation]], which enables 0-based indexing at the cost of obscuring the connection to the standard diatonic scale.
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.


== 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 (1119.463c); 4/1: [[Mintaka]] is around here}}
{{MOS tuning spectrum
| Name=triatonic
| 11/8 = [[No-twos subgroup temperaments#Mizar|Mizar]]
| 3/2 = Just [[17/9]] generator (1101.045{{c}})
| 2/1 = [[CTE tuning]] for the {{nowrap|b12 &amp; b5}} temperament (1109.689{{c}})
| 3/1 = Just [[21/11]] generator (1119.463{{c}})
| 7/2 = [[Mintra]]
| 4/1 = [[Mintaka]] is around here
| 5/1 = [[No-twos subgroup temperaments#Minalzidar|Minalzidar]]
}}
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