Sensi
Sensi, in this article, is the rank-2 regular temperament for the 2.3.5.7.13 subgroup defined by tempering out 245/243, 126/125, and 91/90.
It can be seen as implying a rank-2 tuning which is generated by an extremely sharp major third of about 443 cents which represents both 9/7 and 13/10. It is so named because the generator is a "semisixth": two generators make a major sixth which approximates 5/3, which cannot occur in 12edo. Equal temperaments that support sensi include 19edo (generator 7\19; soft sensoid), 27edo (generator 10\27; supersoft sensoid), and 46edo (generator 17\46; L/s = 7/5, more optimized for sensi temperament).
See Sensipent family #Sensi or Sensamagic clan #Sensi for more technical data. For full 13-limit extensions of sensi, see Sensi extensions.
Intervals
Interval chain
In the following table, odd harmonics and subharmonics 1–21 are in bold.
| # | Cents* | Approximate Ratios† |
|---|---|---|
| 0 | 0.000 | 1/1 |
| 1 | 443.3 | 9/7, 13/10 |
| 2 | 886.6 | 5/3, 42/25 |
| 3 | 129.9 | 13/12, 14/13, 15/14, 27/25 |
| 4 | 573.3 | 7/5, 18/13, 25/18 |
| 5 | 1016.6 | 9/5 |
| 6 | 259.9 | 7/6, 15/13 |
| 7 | 703.2 | 3/2 |
| 8 | 1146.5 | 27/14, 35/18 |
| 9 | 389.8 | 5/4 |
| 10 | 833.2 | 13/8, 21/13 |
| 11 | 76.5 | 21/20, 25/24 |
| 12 | 519.8 | 27/20 |
| 13 | 963.1 | 7/4 |
| 14 | 206.4 | 9/8 |
| 15 | 649.7 | 35/24 (sensor 16/11, sensus 22/15) |
| 16 | 1093.1 | 15/8 (sensor 32/17, sensus 17/9) |
| 17 | 336.4 | 39/32 (sensus 17/14) |
| 18 | 779.7 | 25/16 |
| 19 | 23.0 | 49/48, 65/64, 81/80 |
| 20 | 466.3 | 21/16 |
- * in 2.3.5.7.13 CTE tuning
- † 2.3.5.7.13 ratio interpretations
In sensi[8]
Sortable table of sensi[8]'s major and minor intervals in various sensi tunings:
| Degree | Size in 19edo (soft) | Size in 27edo (supersoft) | Size in 46edo | Approximate ratios | #Gens up |
|---|---|---|---|---|---|
| unison | 0\19, 0.00 | 0\27, 0.00 | 0\46, 0.00 | 1/1 | 0 |
| min. sen2nd | 2\19, 126.3 | 3\27, 133.3 | 5\46, 130.4 | 14/13 | +3 |
| maj. sen2nd | 3\19, 189.5 | 4\27, 177.8 | 7\46, 182.6 | 10/9 | -5 |
| min. sen3rd | 4\19, 252.6 | 6\27, 266.7 | 10\46, 260.9 | 7/6 | +6 |
| maj. sen3rd | 5\19, 315.8 | 7\27, 311.1 | 12\46, 313.0 | 6/5 | -2 |
| perf. sen4th | 7\19, 442.1 | 10\27, 444.4 | 17\46, 443.5 | 9/7, 13/10 | +1 |
| aug. sen4th | 8\19, 505.3 | 11\27, 488.9 | 19\46, 495.7 | 4/3 | -7 |
| min. sen5th | 9\19, 568.4 | 13\27, 577.8 | 22\46, 573.9 | 7/5, 18/13 | +4 |
| maj. sen5th | 10\19, 631.6 | 14\27, 622.2 | 24\46, 626.1 | 10/7, 13/9 | -4 |
| dim. sen6th | 11\19, 694.7 | 16\27, 711.1 | 27\46, 704.3 | 3/2 | +7 |
| perf. sen6th | 12\19, 757.9 | 17\27, 755.6 | 20\46, 756.5 | 14/9, 20/13 | -1 |
| min. sen7th | 14\19, 884.2 | 20\27, 888.9 | 34\46, 887.0 | 5/3 | +2 |
| maj. sen7th | 15\19, 947.4 | 21\27, 933.3 | 36\46, 939.1 | 12/7 | -6 |
| min. sen8th | 16\19, 1010.5 | 23\27, 1022.2 | 39\46, 1017.4 | 9/5 | +5 |
| maj. sen8th | 17\19, 1073.7 | 24\27, 1066.7 | 41\46, 1069.6 | 13/7 | -3 |
Chords
- Main article: Chords of sensus
The "fundamental otonal consonance" of sensi (in this article's definition of sensi) is 4:5:6:7:9:13. However, the full chord isn't available in the 8-note MOS.
One of the most common consonant triads in sensi is the 6:10:13 triad, which spans 3 generators. Sensi[8] has five 6:10:13 triads, four 7:9:13 triads, three 5:6:7:9 tetrads and one 5:6:7:9:13 pentad. Having many diminished triads, it is similar to the 12edo diminished scale in some ways. Sensi is interesting mainly because it gives new 13-limit interpretations to fairly familiar (in the sense of extended meantone-like) intervals. Restricted to the 8-note MOS, it is essentially a non-over-1 temperament.
Melodically, sensi[8] sounds fairly familiar because many intervals are either 5-limit or have familiar categorical interpretations, being represented in the meantone tuning 19edo. For example, the small step of about 130 cents categorizes pretty well as a large semitone (except at places in the scale where two of them make a flat subminor third); the large step is a small whole tone representing 10/9.
The root-sen5th-sen8th chords in sensi[8] usually spell 5:7:9 (root-minor sen5th-minor sen8th) and 7:10:13 (root-major sen5th-major sen8th) chords (shown in the Anti-Dylathian mode QJKLMNOPQ = ssLssLsL):
- Q M P = ssLs sLs L ≈ 5:7:9
- J N Q = sLss LsL s is the odd one out
- K O J = LssL sLs s ≈ 7:10:13
- L P K = ssLs Lss L ≈ 5:7:9
- M Q L = sLsL ssL s ≈ 7:10:13
- N J M = LsLs sLs s ≈ 7:10:13
- O K N = sLss Lss L ≈ 5:7:9
- P L O = LssL ssL s ≈ 7:10:13
Other otonal chords approximated in the 8-note MOS include:
- root - maj. sen7th - maj. sen8th ≈ 7:12:13
- root - maj. sen2nd - maj. sen5th ≈ 9:10:13
- root - min. sen3rd - dim. sen6th ≈ 6:7:9
- root - perf. sen4th - dim. sen6th ≈ 10:13:15 (ultramajor triad)
- root - perf. sen4th - maj. sen7th ≈ 7:9:13
- root - perf. sen4th - maj. sen5th - maj. sen7th ≈ 7:9:10:13
- root - perf. sen4th - min. sen7th ≈ 10:13:18
- root - perf. sen4th - min. sen5th - min. sen7th ≈ 10:13:14:18
- root - min. sen7th - min. sen3rd (+ octave) ≈ 3:5:7
- root - min. sen7th - min. sen2nd (+ octave) ≈ 6:10:13
- root - dim. sen6th - min. sen7th ≈ 6:9:10
- root - dim. sen6th - min. sen2nd (+octave) ≈ 6:9:13
Scales
Tunings
Tuning spectrum
| Edo Generators |
Eigenmonzo (Unchanged-interval)* |
Generator (¢) | Comments |
|---|---|---|---|
| 9/7 | 435.084 | ||
| 4\11 | 436.364 | ||
| 15/14 | 439.814 | ||
| 13/9 | 440.846 | ||
| 15/13 | 441.290 | ||
| 7\19 | 442.105 | Lower bound of 7- and 9-odd-limit, 2.3.5.7.13 subgroup 13-, 15-, and 21-odd-limit diamond monotone | |
| 5/3 | 442.179 | ||
| 13/7 | 442.766 | ||
| 5/4 | 442.924 | 5-odd-limit minimax | |
| 15/8 | 443.017 | ||
| 21/13 | 443.025 | ||
| 3/2 | 443.136 | 2.3.5.7.13 subgroup 15- and 21-odd-limit minimax | |
| 17\46 | 443.478 | ||
| 9/5 | 443.519 | 9-odd-limit and 2.3.5.7.13 subgroup 13-odd-limit minimax | |
| 21/16 | 443.539 | ||
| 7/4 | 443.756 | 7-odd-limit minimax | |
| 21/20 | 444.042 | ||
| 13/8 | 444.053 | ||
| 10\27 | 444.444 | Upper bound of 9-odd-limit, 2.3.5.7.13 subgroup 13-, 15-, and 21-odd-limit diamond monotone | |
| 7/6 | 444.478 | ||
| 7/5 | 445.628 | ||
| 13/12 | 446.191 | ||
| 3\8 | 450.000 | Upper bound of 7-odd-limit diamond monotone | |
| 13/10 | 454.214 |
* besides the octave
Visualizations
Steps of sensi
This diagram shows sensi[5], [8], [11], and [19] with intervals named in relation to the L and s of sensi[8].
Note that X, M and Z are not standard, but d and A are; they are short for "diminished" and "augmented".
Map of sensi
These diagrams relate the sensi generator chain (horizontal axis) to the steps within the octave (vertical axis) for sensi[8] and [11].
![map_of_sensi[8].png](/w/File:Map_of_sensi-8-.png)
![map_of_sensi[11]_correction2.png](/w/File:Map_of_sensi-11-_correction2.png)
Isomorphic layout
- See also: Lumatone mapping for sensi
This diagram shows a layout for playing sensi temperament on an isomorphic keyboard.
The darkest hexagons represent the same note (eg. C), but offset by octaves. The next-darkest hexagons show the notes of Sensi[5]. Imagine stepping from hex to hex as you move across the keyboard from left to right, landing only on the darkest and next-darkest hexes. The light red hexagons show additional notes needed to play Sensi[8]. The Large step of Sensi[8] is represented by a move straight down, so this pattern is a little more zig-zaggy than the pattern for Sensi[5]. Add the white hexes and you have Sensi[11]. The small step of Sensi[11] (indicated in the diagram as "c" for chroma), is represented by a move straight down and down-left. This pattern actually involves moving backward in the horizontal direction, and is therefore more zig-zaggy.
Sensi[19] guitar
![sensi[19]in46.jpg](/w/File:Sensi-19-in46.jpg)