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.

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 are in bold.

Generators Cents* Approximate ratios
0 0.000 1/1
1 443.322 9/7, 13/10
2 886.644 5/3, 42/25
3 129.966 13/12, 14/13, 15/14, 27/25
4 573.288 7/5, 18/13, 25/18
5 1016.610 9/5, 70/39
6 259.932 7/6, 15/13
7 703.253 3/2
8 1146.576 27/14, 35/18
9 389.896 5/4
10 833.220 13/8, 21/13
11 76.542 21/20, 25/24
12 519.864 27/20
13 963.185 7/4
14 206.507 9/8
15 649.829 35/24 (close to 16/11)
16 1093.151 15/8
17 336.473 39/32 (close to 17/14)
18 779.795 25/16
19 23.117 49/48, 65/64, 81/80
20 466.439 21/16
* in 2.3.5.7.13 POTE 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

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 (one octave)

sensi[19]in46.jpg

SOMEONE PLEASE MAKE ONE OF THESE AND SEND IT TO DUSTIN SCHALLERT!

Music