4L 7s
| ↖3L 6s | ↑4L 6s | 5L 6s↗ |
| ←3L 7s | 4L 7s | 5L 7s→ |
| ↙3L 8s | ↓4L 8s | 5L 8s↘ |
4L 7s refers to the structure of MOS scales with generators ranging from 1\4edo (one degree of 4edo, 300¢) to 3\11edo (three degrees of 11edo, 327.27¢), representing approximate diatonic minor thirds (6/5). One of the harmonic entropy minimums in this range is Kleismic/Hanson.
4L 7s has a heptatonic subset, which is the hard end of the spectrum of the smitonic scale (4L 3s).
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 | ||||||