7L 1s
← 6L 1s | 7L 1s | 8L 1s → |
↙ 6L 2s | ↓ 7L 2s | 8L 2s ↘ |
┌╥╥╥╥╥╥╥┬┐ │║║║║║║║││ ││││││││││ └┴┴┴┴┴┴┴┴┘
sLLLLLLL
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
The intervals of 7L 1s are named after the number of mossteps (L and s) they subtend. Each interval, apart from the root and octave (perfect 0-pinestep and perfect 8-pinestep), has two varieties, or sizes, each. Interval varieties are named major and minor for the large and small sizes, respectively, and augmented, perfect, and diminished for the scale's generators.
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
A chain of bright generators, each a perfect 1-pinestep, produces the following scale degrees. A chain of 8 bright generators contains the scale degrees of one of the modes of 7L 1s. Expanding the chain to 15 scale degrees produces the modes of either 8L 7s (for soft-of-basic tunings) or 7L 8s (for hard-of-basic tunings).
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
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.
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 |
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.)
Scale tree
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 | ||||||
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 |