7L 1s

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Scale structure
Step pattern LLLLLLLs
sLLLLLLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 1\8 to 1\7 (150.0 ¢ to 171.4 ¢)
Dark 6\7 to 7\8 (1028.6 ¢ to 1050.0 ¢)
TAMNAMS information
Name pine
Prefix pine-
Abbrev. p
Related MOS scales
Parent 1L 6s
Sister 1L 7s
Daughters 8L 7s, 7L 8s
Neutralized 6L 2s
2-Flought 15L 1s, 7L 9s
Equal tunings
Equalized (L:s = 1:1) 1\8 (150.0 ¢)
Supersoft (L:s = 4:3) 4\31 (154.8 ¢)
Soft (L:s = 3:2) 3\23 (156.5 ¢)
Semisoft (L:s = 5:3) 5\38 (157.9 ¢)
Basic (L:s = 2:1) 2\15 (160.0 ¢)
Semihard (L:s = 5:2) 5\37 (162.2 ¢)
Hard (L:s = 3:1) 3\22 (163.6 ¢)
Superhard (L:s = 4:1) 4\29 (165.5 ¢)
Collapsed (L:s = 1:0) 1\7 (171.4 ¢)

7L 1s, named pine in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 7 large steps and 1 small step, repeating every octave. Generators that produce this scale range from 150 ¢ to 171.4 ¢, or from 1028.6 ¢ to 1050 ¢. Scales of this form are always proper because there is only one small step.

Name

TAMNAMS suggests the temperament-agnostic name pine as the name of 7L 1s. The name is an abstraction of porcupine temperament.

Scale properties

Intervals

Intervals of 7L 1s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-pinestep Perfect 0-pinestep P0ps 0 0.0 ¢
1-pinestep Diminished 1-pinestep d1ps s 0.0 ¢ to 150.0 ¢
Perfect 1-pinestep P1ps L 150.0 ¢ to 171.4 ¢
2-pinestep Minor 2-pinestep m2ps L + s 171.4 ¢ to 300.0 ¢
Major 2-pinestep M2ps 2L 300.0 ¢ to 342.9 ¢
3-pinestep Minor 3-pinestep m3ps 2L + s 342.9 ¢ to 450.0 ¢
Major 3-pinestep M3ps 3L 450.0 ¢ to 514.3 ¢
4-pinestep Minor 4-pinestep m4ps 3L + s 514.3 ¢ to 600.0 ¢
Major 4-pinestep M4ps 4L 600.0 ¢ to 685.7 ¢
5-pinestep Minor 5-pinestep m5ps 4L + s 685.7 ¢ to 750.0 ¢
Major 5-pinestep M5ps 5L 750.0 ¢ to 857.1 ¢
6-pinestep Minor 6-pinestep m6ps 5L + s 857.1 ¢ to 900.0 ¢
Major 6-pinestep M6ps 6L 900.0 ¢ to 1028.6 ¢
7-pinestep Perfect 7-pinestep P7ps 6L + s 1028.6 ¢ to 1050.0 ¢
Augmented 7-pinestep A7ps 7L 1050.0 ¢ to 1200.0 ¢
8-pinestep Perfect 8-pinestep P8ps 7L + s 1200.0 ¢

Generator chain

Generator chain of 7L 1s
Bright gens Scale degree Abbrev.
14 Augmented 6-pinedegree A6pd
13 Augmented 5-pinedegree A5pd
12 Augmented 4-pinedegree A4pd
11 Augmented 3-pinedegree A3pd
10 Augmented 2-pinedegree A2pd
9 Augmented 1-pinedegree A1pd
8 Augmented 0-pinedegree A0pd
7 Augmented 7-pinedegree A7pd
6 Major 6-pinedegree M6pd
5 Major 5-pinedegree M5pd
4 Major 4-pinedegree M4pd
3 Major 3-pinedegree M3pd
2 Major 2-pinedegree M2pd
1 Perfect 1-pinedegree P1pd
0 Perfect 0-pinedegree
Perfect 8-pinedegree		 P0pd
P8pd
−1 Perfect 7-pinedegree P7pd
−2 Minor 6-pinedegree m6pd
−3 Minor 5-pinedegree m5pd
−4 Minor 4-pinedegree m4pd
−5 Minor 3-pinedegree m3pd
−6 Minor 2-pinedegree m2pd
−7 Diminished 1-pinedegree d1pd
−8 Diminished 8-pinedegree d8pd
−9 Diminished 7-pinedegree d7pd
−10 Diminished 6-pinedegree d6pd
−11 Diminished 5-pinedegree d5pd
−12 Diminished 4-pinedegree d4pd
−13 Diminished 3-pinedegree d3pd
−14 Diminished 2-pinedegree d2pd

Modes

Scale degrees of the modes of 7L 1s
UDP Cyclic
order		 Step
pattern		 Scale degree (pinedegree)
0 1 2 3 4 5 6 7 8
7|0 1 LLLLLLLs Perf. Perf. Maj. Maj. Maj. Maj. Maj. Aug. Perf.
6|1 2 LLLLLLsL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Perf. Perf.
5|2 3 LLLLLsLL Perf. Perf. Maj. Maj. Maj. Maj. Min. Perf. Perf.
4|3 4 LLLLsLLL Perf. Perf. Maj. Maj. Maj. Min. Min. Perf. Perf.
3|4 5 LLLsLLLL Perf. Perf. Maj. Maj. Min. Min. Min. Perf. Perf.
2|5 6 LLsLLLLL Perf. Perf. Maj. Min. Min. Min. Min. Perf. Perf.
1|6 7 LsLLLLLL Perf. Perf. Min. Min. Min. Min. Min. Perf. Perf.
0|7 8 sLLLLLLL Perf. Dim. Min. Min. Min. Min. Min. Perf. Perf.

Proposed names

Mode names are from Porcupine temperament modal harmony. Descriptive mode names are based on using 1–4–7, i.e. 3+3 triads as a basis for harmony.

Modes of 7L 1s
UDP Cyclic
order		 Step
pattern		 Name Origin
7|0 1 LLLLLLLs Bright quartal
6|1 2 LLLLLLsL Dark quartal
5|2 3 LLLLLsLL Bright major
4|3 4 LLLLsLLL Middle major
3|4 5 LLLsLLLL Dark major
2|5 6 LLsLLLLL Bright minor
1|6 7 LsLLLLLL Middle minor
0|7 8 sLLLLLLL Dark minor

Theory

Low harmonic entropy scales

There are three notable harmonic entropy minima with this mos pattern.

  • The lowest accuracy one is porcupine, in which two generators make a 6/5 and three make a 4/3. The range of porcupine tunings is about 2\15 to 3\22.
  • Less well-known and more accurate is greeley, in which two generators are still 6/5 but three fall quite short of a 4/3, but the scale happens to closely approximate a lot of higher-complexity intervals like 10/7, 11/7, etc.
  • Thirdly and finally, tempering out S10/S11 so that (4/3)/(11/10)3 is tempered out results in an unusually high accuracy and efficient rank-2 temperament in the 2.3.11/5 subgroup for which interpretation as a rank-3 temperament in 2.3.5.11 (the no-7's 11-limit) is natural, making 10/9 and 12/11 equidistant from 11/10 and offering many fruitful tempering opportunities. Note therefore that porkypine can be seen as a trivial tuning of pine tempering out 100/99 = S10 and 121/120 = S11.

Scale tree

Scale tree and tuning spectrum of 7L 1s
Generator(edo) Cents Step ratio Comments(always proper)
Bright Dark L:s Hardness
1\8 150.000 1050.000 1:1 1.000 Equalized 7L 1s
6\47 153.191 1046.809 6:5 1.200
5\39 153.846 1046.154 5:4 1.250
9\70 154.286 1045.714 9:7 1.286
4\31 154.839 1045.161 4:3 1.333 Supersoft 7L 1s
11\85 155.294 1044.706 11:8 1.375
7\54 155.556 1044.444 7:5 1.400
10\77 155.844 1044.156 10:7 1.429 General range of greeley
3\23 156.522 1043.478 3:2 1.500 Soft 7L 1s
11\84 157.143 1042.857 11:7 1.571
8\61 157.377 1042.623 8:5 1.600
13\99 157.576 1042.424 13:8 1.625 Golden porcupine/hemikleismic
5\38 157.895 1042.105 5:3 1.667 Semisoft 7L 1s
12\91 158.242 1041.758 12:7 1.714
7\53 158.491 1041.509 7:4 1.750
9\68 158.824 1041.176 9:5 1.800
2\15 160.000 1040.000 2:1 2.000 Basic 7L 1s
Optimum rank range for porcupine
9\67 161.194 1038.806 9:4 2.250
7\52 161.538 1038.462 7:3 2.333
12\89 161.798 1038.202 12:5 2.400
5\37 162.162 1037.838 5:2 2.500 Semihard 7L 1s
General range of porcupine
13\96 162.500 1037.500 13:5 2.600
8\59 162.712 1037.288 8:3 2.667
11\81 162.963 1037.037 11:4 2.750
3\22 163.636 1036.364 3:1 3.000 Hard 7L 1s
10\73 164.384 1035.616 10:3 3.333
7\51 164.706 1035.294 7:2 3.500
11\80 165.000 1035.000 11:3 3.667
4\29 165.517 1034.483 4:1 4.000 Superhard 7L 1s
9\65 166.154 1033.846 9:2 4.500
5\36 166.667 1033.333 5:1 5.000
6\43 167.442 1032.558 6:1 6.000
1\7 171.429 1028.571 1:0 → ∞ Collapsed 7L 1s
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