7L 1s

From Xenharmonic Wiki
Jump to navigation Jump to search
←6L 1s7L 1s 8L 1s→
↙6L 2s ↓7L 2s 8L 2s↘
Scale structure
Step pattern LLLLLLLs
Equave 2/1 (1200.0¢)
Period 2/1 (1200.0¢)
Step size ranges
Large 1\8 to 1\7 (150.0¢ to 171.4¢)
Small 0\7 to 1\8 (0.0¢ to 150.0¢)
Generator ranges
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. pine
Related scales
Parent 1L 6s
Sister 1L 7s
Daughters 8L 7s, 7L 8s
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, in reference to porcupine temperament.

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 S10/S11 so that (4/3)/(11/10)3 is tempered results in an unusually high accuracy & efficient rank 2 temperament in the 2.3.11/10 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 100/99 = S10 and 121/120 = S11.)

Modes

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.

Mode UDP Mode name Descriptive mode name
LLLLLLLs 7|0 octopus Bright quartal
LLLLLLsL 6|1 mantis Dark quartal
LLLLLsLL 5|2 dolphin Bright major
LLLLsLLL 4|3 crab Middle major
LLLsLLLL 3|4 tuna Dark major
LLsLLLLL 2|5 salmon Bright minor
LsLLLLLL 1|6 starfish Middle minor
sLLLLLLL 0|7 whale Dark major

Scale tree

Scale tree and tuning spectrum of 7L 1s
Generator (in steps of edo) Cents Step ratio Comments
(Every tuning produces a proper scale)
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
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
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