Sensi

From Xenharmonic Wiki
Jump to navigation Jump to search

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. See sensipent for alternative extensions of 5-limit sensi.

It can be seen as implying a rank-2 tuning which is generated by an extremely sharp major third of 440–445 ¢ 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). More obscure but significantly more accurate interpretations of its generator are given by sensible and especially sensipent's extension to the 2.3.5.31 subgroup.

See Sensipent family #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 0.0 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]

Sensi[8] is a mos scale with a 3L 5s pattern (or 5L 3s in extreme cases where the generator is larger than 450 ¢). See 3L 5s #Modes (resp. 5L 3s #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

Chords

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 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
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 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].

steps_of_sensi.png

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 map_of_sensi[11]_correction2.png

Isomorphic layout

This diagram shows a layout for playing sensi temperament on an isomorphic keyboard.

sensi_isomorphic_layout.png

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

Music

Andrew Heathwaite
Budjarn Lambeth