Sensi
Lua error in Module:Infobox_regtemp at line 138: attempt to perform arithmetic on local 'generator_size' (a nil value). Sensi is a rank-2 regular temperament that is generated by an extremely sharp major third of between 442 and 445 ¢, which is taken in the 7-limit to represent a sharpened 9/7. The most important equivalence in sensi (i.e. tempering out the comma 245/243) is known as sensamagic, by which two of these thirds stack to a major sixth which approximates 5/3. Sensi then makes the additional tempering of 126/125, through which three of these major sixths approximate 7/6, two octaves up. The 6th harmonic is therefore split into seven, and 5/4 is divided into three parts, each identified with 15/14. Furthermore, since the supermajor third is tempered so sharply, it makes sense to have it represent both 9/7 and 13/10, which means 91/90 is tempered out in the 2.3.5.7.13 subgroup. There the 15/14 interval also represents 14/13 and 13/12, which results in 169/168 and 196/195 being tempered out.
The structure whereby 5/3 is split into two supermajor thirds is obviously xenharmonic as this cannot occur in 12edo. But particularly, as the simplest EDOs with similar structures are 8edo and 11edo (hence the 8-note (3L 5s, checkertonic) and 11-note (8L 3s, flanatonic) MOS scales), sensi has a very xenmelodic character compared to many other ways of organizing the 7-limit (such as superpyth, which is based on the familiar chain of fifths, and even porcupine, which is fundamentally heptatonic).
Equal temperaments that support sensi include 19edo (generator 7\19; soft checkertonic), 27edo (generator 10\27; supersoft checkertonic), as well as 46edo (generator 17\46; L/s = 7/5, more optimized for sensi temperament) and 65edo (generator 24\65; L/s = 10/7) using the 65f val with a flat 13.
See Sensipent family #Sensi for more technical data, sensi extensions for extensions of sensi to include the 11th and 17th harmonics, and #Related temperaments for alternative interpretations of similar structures to sensi.
Theory
Interval chain
In the following table, odd harmonics and subharmonics 1–21 are in bold.
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 443.4 | 9/7, 13/10 |
| 2 | 886.7 | 5/3, 42/25 |
| 3 | 130.1 | 13/12, 14/13, 15/14, 27/25 |
| 4 | 573.4 | 7/5, 18/13, 25/18 |
| 5 | 1016.8 | 9/5 |
| 6 | 260.1 | 7/6, 15/13 |
| 7 | 703.5 | 3/2 |
| 8 | 1146.9 | 27/14, 35/18 |
| 9 | 390.2 | 5/4 |
| 10 | 833.6 | 13/8, 21/13 |
| 11 | 76.9 | 21/20, 25/24 |
| 12 | 520.3 | 27/20 |
| 13 | 963.7 | 7/4 |
| 14 | 207.0 | 9/8 |
| 15 | 650.4 | 35/24 (sensor 16/11, sensus 22/15) |
| 16 | 1093.7 | 15/8 (sensor 32/17, sensus 17/9) |
| 17 | 337.1 | 39/32 (sensus 17/14) |
| 18 | 780.4 | 25/16 |
| 19 | 23.8 | 49/48, 65/64, 81/80 |
| 20 | 467.2 | 21/16 |
* In 2.3.5.7.13 CWE tuning
Intervals of Sensi[8]
Sensi[8] is a mos scale with a 3L 5s pattern. See 3L 5s #Modes to see which modes have which qualities for each interval size.
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 | # generators up |
|---|---|---|---|---|---|
| Unison | 0\19, 0.0 | 0\27, 0.0 | 0\46, 0.0 | 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 |
Related temperaments
It is worth noting that sensi distinguishes itself from other structures, the sensamagic temperaments, based around 245/243 (whose basic form in the 2.9/7.5/3 subgroup is known as sentry) by virtue of its minor third (6/5) being flattened from just rather than sharpened. This results in the supermajor third being sharpened even more than is typical, so much so that it is tuned interseptimally and may not fulfill all the functions that ~9/7 is intended to have.
One way around this is to eschew the generator's interpretation as 9/7 altogether, and focus on the 5-limit part of sensi, which is known as sensipent (whose comma is 78732/78125). From there, an interpretation of the generator as 31/24~40/31 is apparent. Beyond the 2.3.5.31 subgroup, more accurate interpretations (in comparison to sensi) of sensipent's extended harmony are given by sensible (adding primes 11, 17, and 23) and sendai (adding 23 and 29). There are also alternative mappings of 7, including sensei (+32 generators, with a tuning flat of 65edo) and warrior (−33 generators, with a tuning between 65edo and 46edo); warrior combines well with the mapping of sensible, and sensei with sendai.
BPS
Since the sensamagic comma, 245/243, contains no 2 in its factorization, only primes 3, 5, and 7, it can be tempered out in a tritave (3/1)-repeating, 3.5.7 subgroup context, where the generator (9/7) is now the tritave-reduced 7th subharmonic, two of which give the 5th harmonic. This is known as Bohlen-Pierce-Stearns (BPS) temperament, and it generates a 4L 5s scale against the tritave (sometimes known as Lambda). Where this temperament connects to sensi is that, at 7 generators, BPS reaches an interval that it identifies with 125/63, which is rather close to the octave; sensi is obtained by treating this interval as the mapping of 2/1, which provides the interesting notion of using sensi in a 3/1-periodic 3.5.7.2 setting.
Chords and harmony
The fundamental otonal consonance of sensi is 4:5:6:7:9:13. However, the full chord is only available in the 19-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 ¢ 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
Norm-based tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~9/7 = 443.3166 ¢ | CWE: ~9/7 = 443.3493 ¢ | POTE: ~9/7 = 443.3827 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~9/7 = 443.4016 ¢ | CWE: ~9/7 = 443.3581 ¢ | POTE: ~9/7 = 443.3220 ¢ |
Target tunings
| Target | Minimax | |
|---|---|---|
| Generator | Eigenmonzo* | |
| 7-odd-limit | ~9/7 = 443.756 ¢ | 7/4 |
| 9-odd-limit | ~9/7 = 443.519 ¢ | 9/5 |
| no-11 13-odd-limit | ~9/7 = 443.519 ¢ | 9/5 |
| no-11 15-odd-limit | ~9/7 = 443.136 ¢ | 3/2 |
Tuning spectrum
| Edo generators |
Unchanged interval (eigenmonzo)* |
Generator (¢) | Comments |
|---|---|---|---|
| 9/7 | 435.084 | ||
| 4\11 | 436.364 | 11cdf val | |
| 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 | ||
| 24\65 | 443.077 | 65f val | |
| 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 | |
| 27\73 | 443.836 | ||
| 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 | ||
| 117/70 | 444.649 | Exact geometric mean of 9/7 and 13/10 | |
| 7/5 | 445.628 | ||
| 13/12 | 446.191 | ||
| 3\8 | 450.000 | 8d val, 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
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)