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 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 – the 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, 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 "fifths" are ~[[17/9]], which is just near [[19edt]] in the soft range, and ~[[21/11]] which is just near [[17edt]] in the hard range.
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 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.


== 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 (1119.463c); 7/2: [[Mintra]]; 4/1: [[Mintaka]] is around here; 5/1: [[No-twos subgroup temperaments#Minalzidar|Minalzidar]]}}
{{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]]}}

Revision as of 08:21, 12 January 2025

↖ 4L 1s⟨3/1⟩ ↑ 5L 1s⟨3/1⟩ 6L 1s⟨3/1⟩ ↗
← 4L 2s⟨3/1⟩ 5L 2s (3/1-equivalent) 6L 2s⟨3/1⟩ →
↙ 4L 3s⟨3/1⟩ ↓ 5L 3s⟨3/1⟩ 6L 3s⟨3/1⟩ ↘
Scale structure
Step pattern LLLsLLs
sLLsLLL
Equave 3/1 (1902.0 ¢)
Period 3/1 (1902.0 ¢)
Generator size(edt)
Bright 4\7 to 3\5 (1086.8 ¢ to 1141.2 ¢)
Dark 2\5 to 3\7 (760.8 ¢ to 815.1 ¢)
Related MOS scales
Parent 2L 3s⟨3/1⟩
Sister 2L 5s⟨3/1⟩
Daughters 7L 5s⟨3/1⟩, 5L 7s⟨3/1⟩
Neutralized 3L 4s⟨3/1⟩
2-Flought 12L 2s⟨3/1⟩, 5L 9s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1) 4\7 (1086.8 ¢)
Supersoft (L:s = 4:3) 15\26 (1097.3 ¢)
Soft (L:s = 3:2) 11\19 (1101.1 ¢)
Semisoft (L:s = 5:3) 18\31 (1104.4 ¢)
Basic (L:s = 2:1) 7\12 (1109.5 ¢)
Semihard (L:s = 5:2) 17\29 (1114.9 ¢)
Hard (L:s = 3:1) 10\17 (1118.8 ¢)
Superhard (L:s = 4:1) 13\22 (1123.9 ¢)
Collapsed (L:s = 1:0) 3\5 (1141.2 ¢)
View • Talk • Edit

5L 2s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 5 large steps and 2 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 1086.8 ¢ to 1141.2 ¢, or from 760.8 ¢ to 815.1 ¢.

Name

The name triatonic was coined by CompactStar, and is a back-formation from "diatonic" with di- being interpreted as 2 (the octave) and replaced with tri- for 3 (the tritave). It is not an official name in TAMNAMS.

Theory

As a macrodiatonic scale

It is the macrodiatonic scale with the period of a tritave. This means it is a 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.

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 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 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 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

The modes have step patterns which are the same as the modes of the diatonic scale.

Modes of 5L 2s⟨3/1⟩
UDP Cyclic
order		 Step
pattern
6|0 1 LLLsLLs
5|1 5 LLsLLLs
4|2 2 LLsLLsL
3|3 6 LsLLLsL
2|4 3 LsLLsLL
1|5 7 sLLLsLL
0|6 4 sLLsLLL

Scale degrees

Scale degrees of the modes of 5L 2s⟨3/1⟩
UDP Cyclic
order 		Step
pattern 		Scale degree (mosdegree)
0 1 2 3 4 5 6 7
6|0 1 LLLsLLs Perf. Maj. Maj. Aug. Perf. Maj. Maj. Perf.
5|1 5 LLsLLLs Perf. Maj. Maj. Perf. Perf. Maj. Maj. Perf.
4|2 2 LLsLLsL Perf. Maj. Maj. Perf. Perf. Maj. Min. Perf.
3|3 6 LsLLLsL Perf. Maj. Min. Perf. Perf. Maj. Min. Perf.
2|4 3 LsLLsLL Perf. Maj. Min. Perf. Perf. Min. Min. Perf.
1|5 7 sLLLsLL Perf. Min. Min. Perf. Perf. Min. Min. Perf.
0|6 4 sLLsLLL Perf. Min. Min. Perf. Dim. Min. Min. Perf.

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.

Intervals

Intervals of 5L 2s⟨3/1⟩
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-mosstep Perfect 0-mosstep P0ms 0 0.0 ¢
1-mosstep Minor 1-mosstep m1ms s 0.0 ¢ to 271.7 ¢
Major 1-mosstep M1ms L 271.7 ¢ to 380.4 ¢
2-mosstep Minor 2-mosstep m2ms L + s 380.4 ¢ to 543.4 ¢
Major 2-mosstep M2ms 2L 543.4 ¢ to 760.8 ¢
3-mosstep Perfect 3-mosstep P3ms 2L + s 760.8 ¢ to 815.1 ¢
Augmented 3-mosstep A3ms 3L 815.1 ¢ to 1141.2 ¢
4-mosstep Diminished 4-mosstep d4ms 2L + 2s 760.8 ¢ to 1086.8 ¢
Perfect 4-mosstep P4ms 3L + s 1086.8 ¢ to 1141.2 ¢
5-mosstep Minor 5-mosstep m5ms 3L + 2s 1141.2 ¢ to 1358.5 ¢
Major 5-mosstep M5ms 4L + s 1358.5 ¢ to 1521.6 ¢
6-mosstep Minor 6-mosstep m6ms 4L + 2s 1521.6 ¢ to 1630.2 ¢
Major 6-mosstep M6ms 5L + s 1630.2 ¢ to 1902.0 ¢
7-mosstep Perfect 7-mosstep P7ms 5L + 2s 1902.0 ¢

Scale tree

Template: Scale tree is deprecated. Please use Template: MOS tuning spectrum instead. Details:
Use of a single Comments parameter has become unmaintainable. Existing scale trees should be migrated to the new template, where comments are entered using a step ratio p/q as a parameter: 
{{MOS tuning spectrum
| 3/2 = Example comment
| 4/3 = Another example comment
}}


The parameters tuning and depth have been replaced with Scale Signature and Depth, respectively.


Scale tree and tuning spectrum of 5L 2s⟨3/1⟩
Generator(edt) Cents Step ratio Comments
Bright Dark L:s Hardness
4\7 1086.831 815.124 1:1 1.000 Equalized 5L 2s⟨3/1⟩
23\40 1093.624 808.331 6:5 1.200
19\33 1095.065 806.890 5:4 1.250
34\59 1096.042 805.913 9:7 1.286
15\26 1097.282 804.673 4:3 1.333 Supersoft 5L 2s⟨3/1⟩
41\71 1098.312 803.643 11:8 1.375
26\45 1098.907 803.048 7:5 1.400
37\64 1099.568 802.387 10:7 1.429
11\19 1101.132 800.823 3:2 1.500 Soft 5L 2s⟨3/1⟩
40\69 1102.583 799.372 11:7 1.571
29\50 1103.134 798.821 8:5 1.600
47\81 1103.604 798.351 13:8 1.625
18\31 1104.361 797.594 5:3 1.667 Semisoft 5L 2s⟨3/1⟩
43\74 1105.190 796.765 12:7 1.714
25\43 1105.788 796.167 7:4 1.750
32\55 1106.592 795.363 9:5 1.800
7\12 1109.474 792.481 2:1 2.000 Basic 5L 2s⟨3/1⟩
Scales with tunings softer than this are proper
31\53 1112.464 789.491 9:4 2.250
24\41 1113.340 788.615 7:3 2.333
41\70 1114.002 787.953 12:5 2.400
17\29 1114.939 787.016 5:2 2.500 Semihard 5L 2s⟨3/1⟩
44\75 1115.814 786.141 13:5 2.600
27\46 1116.365 785.590 8:3 2.667
37\63 1117.021 784.934 11:4 2.750
10\17 1118.797 783.158 3:1 3.000 Hard 5L 2s⟨3/1⟩
33\56 1120.795 781.160 10:3 3.333
23\39 1121.666 780.289 7:2 3.500
36\61 1122.465 779.490 11:3 3.667
13\22 1123.883 778.073 4:1 4.000 Superhard 5L 2s⟨3/1⟩
29\49 1125.647 776.308 9:2 4.500
16\27 1127.084 774.871 5:1 5.000
19\32 1129.286 772.669 6:1 6.000
3\5 1141.173 760.782 1:0 → ∞ Collapsed 5L 2s⟨3/1⟩
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