Sensi

Sensi is a regular temperament for the 2.3.5.7.13 subgroup 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 (soft sensoid), 27edo (supersoft sensoid), and 46edo (more optimized for sensi temperament).

In the language of regular temperaments, it is the rank-2 temperament defined by tempering out the zozoyo (245/243), zotrigu (126/125), and thozogu (91/90) commas.

See Sensipent family #Sensi or Starling temperaments #Sensi for more technical data. For full 13-limit extensions of sensi, see Sensi extensions.

Intervals

Interval chain

In the following table, odd harmonics 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 4:5:6:7:13. However, the full chord isn't available in the 8-note MOS.

The most common consonant triad 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].

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].

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 (one octave)

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