5L 2s (3/1-equivalent)

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↖ 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¢)

5L 2s⟨3/1⟩, also called triatonic, 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 has octaves stretched out to the size of a 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

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 subgroups like 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.

Scale tree

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 Mizar
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⟩
Just 17/9 generator (1101.045¢)
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
CTE tuning for the b12 & b5 temperament (1109.689¢)
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⟩
Just 21/11 generator (1119.463c)
33\56 1120.795 781.160 10:3 3.333
23\39 1121.666 780.289 7:2 3.500 Mintra
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⟩
Mintaka is around here
29\49 1125.647 776.308 9:2 4.500
16\27 1127.084 774.871 5:1 5.000 Minalzidar
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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