4L 7s
| ↖3L 6s | ↑4L 6s | 5L 6s↗ |
| ←3L 7s | 4L 7s | 5L 7s→ |
| ↙3L 8s | ↓4L 8s | 5L 8s↘ |
┌╥┬╥┬┬╥┬┬╥┬┬┐ │║│║││║││║│││ │││││││││││││ └┴┴┴┴┴┴┴┴┴┴┴┘
ssLssLssLsL
4L 11s
4L 7s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 4 large steps and 7 small steps, repeating every octave. 4L 7s is a child scale of 4L 3s, expanding it by 4 tones. Generators that produce this scale range from 872.7¢ to 900¢, or from 300¢ to 327.3¢.
One of the harmonic entropy minimums in this range is Kleismic/Hanson.
The TAMNAMS name for this scale used to be kleistonic, but is now simply called p-chro smitonic in the latest extension (the euphonic name being smipechromic). The prefix for mossteps is klei-.
Notation
The notation used in this article is LssLsLssLss = АВГДЕЅЗИѲІѦА, based on old Cyrillic numerals 1-10, and the addition of the small yus (Ѧ) for 11 (old "ya" symbolically representing І҃А҃=11). A titlo can be optionally used as a numeric sign (А҃), depending on font rendering, clarity, and style. Chromas are represented by regular sharps and flats.
Thus the 15edo gamut is as follows: А А#/Вb В Г Д Д#/Еb Е Ѕ Ѕ#/Зb З И Ѳ Ѳ#/Іb І Ѧ А
Letter names
The letters can be named in English as such: Az, Vede, Glagol, Dobro, Yest, Dzelo, Zemlya, Izhe, Thita, I (Ee), Yas. They can also be named as numbers 1-11.
Intervals
| Generators | Notation (1/1 = А҃) | Interval category name | Generators | Notation of 2/1 inverse | Interval category name |
|---|---|---|---|---|---|
| The 11-note MOS has the following intervals (from some root): | |||||
| 0 | А | perfect unison | 0 | А | dodecave (same as octave) |
| 1 | Д | perfect kleifourth (minor third) | -1 | Ѳ | perfect kleininth (major sixth) |
| 2 | Зb | minor kleiseventh | -2 | Ѕ | major kleisixth |
| 3 | Іb | minor kleitenth | -3 | Г | major kleithird |
| 4 | Вb | minor kleisecond | -4 | Ѧ | major kleieleventh |
| 5 | Еb | minor kleififth | -5 | И | major kleieighth |
| 6 | Иb | minor kleieighth | -6 | Е | major kleififth |
| 7 | Ѧb | minor kleieleventh | -7 | В | major kleisecond |
| 8 | Гb | minor kleithird | -8 | І | major kleitenth |
| 9 | Ѕb | minor kleisixth | -9 | З | major kleiseventh |
| 10 | Ѳb | diminished kleininth | -10 | Д# | augmented kleithird |
| The chromatic 15-note MOS (either 4L 11s, 11L 4s, or 15edo) also has the following intervals (from some root): | |||||
| 11 | Аb | diminished dodecave | -11 | А# | augmented unison (chroma) |
| 12 | Дb | diminished kleifourth | -12 | Ѳ# | augmented kleininth |
| 13 | Зbb | diminished kleiseventh | -13 | Ѕ# | augmented kleisixth |
| 14 | Іbb | diminished kleitenth | -14 | Г# | augmented kleithird |
Genchain
The generator chain for this scale is as follows:
| Дb | Аb | Ѳb | Ѕb | Гb | Ѧb | Иb | Еb | Вb | Іb | Зb | Д | А | Ѳ | Ѕ | Г | Ѧ | И | Е | В | І | З | Д# | А# | Ѳ# | Ѕ# | Г# | Ѧ# | И# | Е# | В# | І# | З# |
| d4 | d12 | d9 | m6 | m3 | m11 | m8 | m5 | m2 | m10 | m7 | P4 | P1 | P9 | M6 | M3 | M11 | M8 | M5 | M2 | M10 | M7 | A4 | A1 | A9 | A6 | A3 | A11 | A8 | A5 | A2 | A10 | A7 |
Tuning ranges
Soft range
The soft range for tunings of p-chro smitonic encompasses parasoft and hyposoft tunings. This implies step ratios smaller than 2/1, meaning a generator sharper than 4\15 = 320¢.
This is the range associated with extensions of Orgone[7]. The small step is recognizable as a near diatonic semitone, while the large step is in the ambiguous area of neutral seconds.
Soft p-chro smitonic edos include 15edo and 26edo. The sizes of the generator, large step and small step of p-chro smitonic are as follows in various soft tunings:
| 15edo (basic) | 26edo (soft) | Some JI approximations | |
|---|---|---|---|
| generator (g) | 4\15, 320.00 | 7\26, 323.08 | 77/64, 6/5 |
| L (octave - 3g) | 2\15, 160.00 | 3\26, 138.46 | 12/11, 13/12 |
| s (4g - octave) | 1\15, 80.00 | 2\19, 92.31 | 21/20, 22/21, 20/19 |
Hypohard
Hypohard tunings of p-chro smitonic have step ratios between 2/1 and 3/1, implying a generator sharper than 5\19 = 315.79¢ and flatter than 4\15 = 320¢.
This range represents one of the harmonic entropy minimums, where 6 generators make a just diatonic fifth (3/2), an octave above. This is the range associated with the eponymous Kleismic (aka Hanson) temperament and its extensions.
Hypohard p-chro smitonic edos include 15edo, 19edo, and 34edo. The sizes of the generator, large step and small step of p-chro smitonic are as follows in various hypohard p-chro smitonic tunings:
| 15edo (basic) | 19edo (hard) | 34edo (semihard) | Some JI approximations | |
|---|---|---|---|---|
| generator (g) | 4\15, 320.00 | 5\19, 315.79 | 9\34, 317.65 | 6/5 |
| L (octave - 3g) | 2\15, 160.00 | 3\19, 189.47 | 5\34, 176.47 | 10/9, 11/10 (in 15edo) |
| s (4g - octave) | 1\15, 80.00 | 1\19, 63.16 | 2\34, 70.59 | 25/24, 26/25 (in better kleismic tunings) |
Parahard
Parahard tunings of p-chro smitonic have step ratios between 3/1 and 4/1, implying a generator sharper than 6\23 = 313.04¢ and flatter than 5\19 = 315.79¢.
The minor third is at its purest here, but the resulting scales tend to approximate intervals that employ a much higher limit harmony, especially in the case of the superhard 23edo. However, the large step is recognizable as a regular diatonic whole step, approximating both 10/9 and 9/8, while the small step is a slightly sharp of a quarter tone.
Parahard p-chro smitonic edos include 19edo, 23edo, and 42edo. The sizes of the generator, large step and small step of p-chro smitonic are as follows in various parahard p-chro smitonic tunings:
| 19edo (hard) | 23edo (superhard) | 42edo (parahard) | Some JI approximations | |
|---|---|---|---|---|
| generator (g) | 5\19, 315.79 | 6\23, 313.04 | 11\42, 314.29 | 6/5 |
| L (octave - 3g) | 3\19, 189.47 | 4\23, 208.70 | 7\42, 200.00 | 10/9, 9/8 |
| s (4g - octave) | 1\19, 63.16 | 1\23, 52.17 | 2\42, 57.14 | 28/27, 33/32 |
Hyperhard
Hyperhard tunings of p-chro smitonic have step ratios between 4/1 and 6/1, implying a generator sharper than 8\31 = 309.68¢ and flatter than 6\23 = 313.04¢.
The temperament known as Myna (a pun on "minor third") resides here, as this is the range where 10 generators make a just diatonic fifth (3/2), two octaves above. These scales are stacked with simple intervals, but are melodically difficult due to the extreme step size disparity, where the small step is generally flat of a quarter tone.
Hyperhard p-chro smitonic edos include 23edo, 31edo, and 27edo. The sizes of the generator, large step and small step of p-chro smitonic are as follows in various hyperhard p-chro smitonic tunings:
| 23edo (superhard) | 31edo (extrahard) | 27edo (pentahard) | Some JI approximations | |
|---|---|---|---|---|
| generator (g) | 6\23, 313.04 | 8\31, 309.68 | 7\27, 311.11 | 6/5 |
| L (octave - 3g) | 4\23, 208.70 | 6\31, 232.26 | 5\27, 222.22 | 8/7, 9/8 |
| s (4g - octave) | 1\23, 52.17 | 1\31, 38.71 | 1\27, 44.44 | 36/35, 45/44 |
Modes
The names are based on smitonic modes, modified with the "super-" prefix, with thematic additions, as there are an extra 4 modes available.
| Mode | UDP | Name |
|---|---|---|
| LsLssLssLss | 10|0 | Supernerevarine |
| LssLsLssLss | 9|1 | Supervivecan |
| LssLssLsLss | 8|2 | Superbaardauan |
| LssLssLssLs | 7|3 | Superlorkhanic |
| sLsLssLssLs | 6|4 | Supervvardenic |
| sLssLsLssLs | 5|5 | Supersothic |
| sLssLssLsLs | 4|6 | Supernumidian |
| sLssLssLssL | 3|7 | Superkagrenacan |
| ssLsLssLssL | 2|8 | Supernecromic |
| ssLssLsLssL | 1|9 | Superalmalexian |
| ssLssLssLsL | 0|10 | Superdagothic |
Temperaments
Scales
Scale tree
The spectrum looks like this:
| Generator | Cents | L | s | L/s | Comments | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Chroma-positive | Chroma-negative | ||||||||||
| 8\11 | 872.727 | 327.273 | 1 | 1 | 1.000 | ||||||
| 43\59 | 874.576 | 325.424 | 6 | 5 | 1.200 | Oregon | |||||
| 35\48 | 875.000 | 325.000 | 5 | 4 | 1.250 | ||||||
| 62\85 | 875.294 | 324.706 | 9 | 7 | 1.286 | ||||||
| 27\37 | 875.676 | 324.324 | 4 | 3 | 1.333 | ||||||
| 73\100 | 876.000 | 324.000 | 11 | 8 | 1.375 | ||||||
| 46\63 | 876.190 | 323.810 | 7 | 5 | 1.400 | ||||||
| 65\89 | 876.404 | 323.596 | 10 | 7 | 1.428 | Orgone | |||||
| 19\26 | 876.923 | 323.077 | 3 | 2 | 1.500 | L/s = 3/2 | |||||
| 68\93 | 877.419 | 322.581 | 11 | 7 | 1.571 | Magicaltet | |||||
| 49\67 | 877.612 | 322.388 | 8 | 5 | 1.600 | ||||||
| 79\108 | 877.778 | 322.222 | 13 | 8 | 1.625 | Golden superkleismic | |||||
| 30\41 | 878.049 | 321.951 | 5 | 3 | 1.667 | Superkleismic | |||||
| 71\97 | 878.351 | 321.649 | 12 | 7 | 1.714 | ||||||
| 41\56 | 878.571 | 321.429 | 7 | 4 | 1.750 | ||||||
| 52\71 | 878.873 | 321.127 | 9 | 5 | 1.800 | ||||||
| 11\15 | 880.000 | 320.000 | 2 | 1 | 2.000 | Basic p-chro smitonic (Generators smaller than this are proper) | |||||
| 47\64 | 881.250 | 318.750 | 9 | 4 | 2.250 | ||||||
| 36\49 | 881.633 | 318.367 | 7 | 3 | 2.333 | Catalan | |||||
| 61\83 | 881.928 | 318.072 | 12 | 5 | 2.400 | ||||||
| 25\34 | 882.353 | 317.647 | 5 | 2 | 2.500 | ||||||
| 64\87 | 882.759 | 317.241 | 13 | 5 | 2.600 | Countercata | |||||
| 39\53 | 883.019 | 316.981 | 8 | 3 | 2.667 | Hanson/cata | |||||
| 53\72 | 883.333 | 316.667 | 11 | 4 | 2.750 | Catakleismic | |||||
| 14\19 | 884.211 | 315.789 | 3 | 1 | 3.000 | L/s = 3/1 | |||||
| 45\61 | 885.246 | 314.754 | 10 | 3 | 3.333 | Parakleismic | |||||
| 31\42 | 885.714 | 314.286 | 7 | 2 | 3.500 | ||||||
| 48\65 | 886.154 | 313.846 | 11 | 3 | 3.667 | ||||||
| 17\23 | 886.957 | 313.043 | 4 | 1 | 4.000 | ||||||
| 37\50 | 888.000 | 312.000 | 9 | 2 | 4.500 | Oolong | |||||
| 20\27 | 888.889 | 311.111 | 5 | 1 | 5.000 | Starlingtet | |||||
| 23\31 | 890.323 | 309.677 | 6 | 1 | 6.000 | Myna | |||||
| 3\4 | 900.000 | 300.000 | 1 | 0 | → inf | ||||||