Sensi: Difference between revisions

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{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|-
|-
! #
! #
! Cents*
! Cents*
! class="unsortable" | Approximate ratios**
! class="unsortable" | Approximate ratios
|-
|-
| 0
| 0
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|-
|-
| 1
| 1
| 443.3
| 443.4
| 9/7, 13/10
| 9/7, 13/10
|-
|-
| 2
| 2
| 886.6
| 886.7
| 5/3, 42/25
| 5/3, 42/25
|-
|-
| 3
| 3
| 129.9
| 130.1
| 13/12, 14/13, 15/14, 27/25
| 13/12, 14/13, 15/14, 27/25
|-
|-
| 4
| 4
| 573.3
| 573.4
| 7/5, 18/13, 25/18
| 7/5, 18/13, 25/18
|-
|-
| 5
| 5
| 1016.6
| 1016.8
| 9/5
| 9/5
|-
|-
| 6
| 6
| 259.9
| 260.1
| 7/6, 15/13
| 7/6, 15/13
|-
|-
| 7
| 7
| 703.2
| 703.5
| '''3/2'''
| '''3/2'''
|-
|-
| 8
| 8
| 1146.5
| 1146.9
| 27/14, 35/18
| 27/14, 35/18
|-
|-
| 9
| 9
| 389.8
| 390.2
| '''5/4'''
| '''5/4'''
|-
|-
| 10
| 10
| 833.2
| 833.6
| '''13/8''', 21/13
| '''13/8''', 21/13
|-
|-
| 11
| 11
| 76.5
| 76.9
| 21/20, 25/24
| 21/20, 25/24
|-
|-
| 12
| 12
| 519.8
| 520.3
| 27/20
| 27/20
|-
|-
| 13
| 13
| 963.1
| 963.7
| '''7/4'''
| '''7/4'''
|-
|-
| 14
| 14
| 206.4
| 207.0
| '''9/8'''
| '''9/8'''
|-
|-
| 15
| 15
| 649.7
| 650.4
| 35/24 (sensor '''16/11''', sensus 22/15)
| 35/24 (sensor '''16/11''', sensus 22/15)
|-
|-
| 16
| 16
| 1093.1
| 1093.7
| '''15/8''' (sensor '''32/17''', sensus 17/9)
| '''15/8''' (sensor '''32/17''', sensus 17/9)
|-
|-
| 17
| 17
| 336.4
| 337.1
| 39/32 (sensus 17/14)
| 39/32 (sensus 17/14)
|-
|-
| 18
| 18
| 779.7
| 780.4
| 25/16
| 25/16
|-
|-
| 19
| 19
| 23.0
| 23.8
| 49/48, 65/64, 81/80
| 49/48, 65/64, 81/80
|-
|-
| 20
| 20
| 466.3
| 467.2
| '''21/16'''
| '''21/16'''
|}
|}
<nowiki/>* In 2.3.5.7.13 CTE tuning
<nowiki/>* In 2.3.5.7.13 CWE tuning
 
<nowiki/>** 2.3.5.7.13 ratio interpretations


=== In Sensi[8] ===
=== In Sensi[8] ===
Sensi[8] is a [[mos scale]] with a [[3L&nbsp;5s]] pattern (or [[5L&nbsp;3s]] in extreme cases where the generator is larger than 450{{c}}). See [[3L&nbsp;5s#Modes]] (resp. [[5L&nbsp;3s#Modes]]) to see which modes have which qualities for each interval size.
Sensi[8] is a [[mos scale]] with a [[3L&nbsp;5s]] pattern (or [[5L&nbsp;3s]] in extreme cases where the generator is larger than 450{{c}}). See [[3L&nbsp;5s #Modes]] (resp. [[5L&nbsp;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:
Sortable table of Sensi[8]'s major and minor intervals in various sensi tunings:
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{{Main| Chords of sensus }}
{{Main| 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.
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]].
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{{c}} 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.
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{{c}} 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):
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
* Q M P = ssLs sLs L ≈ 5:7:9
* J N Q = sLss LsL s is the odd one out
* J N Q = sLss LsL s is the odd one out
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* P L O = LssL ssL s ≈ 7:10:13
* P L O = LssL ssL s ≈ 7:10:13


Other otonal chords approximated in the 8-note MOS include:
Other otonal chords approximated in the 8-note mos include:


* {{dash|Root, maj. sen7th, maj. sen8th ≈ 7:12:13|s=space}}
* {{dash|Root, maj. sen7th, maj. sen8th ≈ 7:12:13|s=space}}
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== Visualizations ==
== Visualizations ==
=== Steps of sensi ===
=== 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].  
This diagram shows Sensi[5], [8], [11], and [19] with intervals named in relation to the L and s of Sensi[8].  


[[File:steps_of_sensi.png|Steps of sensi|alt=steps_of_sensi.png]]
[[File:steps_of_sensi.png|Steps of sensi|alt=steps_of_sensi.png]]
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=== Map of sensi ===
=== 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].  
These diagrams relate the sensi generator chain (horizontal axis) to the steps within the octave (vertical axis) for Sensi[8] and [11].  


[[File:map_of_sensi-8-.png|Map of sensi[8]|alt=map_of_sensi[8].png]]
[[File:map_of_sensi-8-.png|Map of Sensi[8]|alt=map_of_sensi[8].png]]
[[File:map_of_sensi-11-_correction2.png|map of sensi[11]|alt=map_of_sensi[11]_correction2.png]]
[[File:map_of_sensi-11-_correction2.png|Map of Sensi[11]|alt=map_of_sensi[11]_correction2.png]]


=== Isomorphic layout ===
=== Isomorphic layout ===

Revision as of 15:13, 7 October 2025

Sensi is a rank-2 regular temperament in the 2.3.5.7.13 subgroup generated by an extremely sharp major third of between 440 and 446 ¢ which represents both 9/7 and 13/10, such that two of these thirds stack to a major sixth which approximates 5/3, which cannot occur in 12edo. This results in 91/90, 126/125, and 245/243 being tempered out, making it an extension of both sensipent and BPS.

Equal temperaments that support sensi include 19edo (generator 7\19; soft checkertonic), 27edo (generator 10\27; supersoft checkertonic), 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, and Sensi extensions for extensions of sensi that include the 11th harmonic.

Intervals

Interval chain

In the following table, odd harmonics and subharmonics 1–21 are in bold.

# Cents* Approximate ratios
0 0.0 1/1
1 443.4 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

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

Main article: Chords of sensus

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

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~9/7 = 443.3166 ¢ CWE: ~9/7 = 443.3493 ¢ POTE: ~9/7 = 443.3827 ¢
2.3.5.7.13-subgroup norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~9/7 = 443.4016 ¢ CWE: ~9/7 = 443.3581 ¢ POTE: ~9/7 = 443.3220 ¢

Target tunings

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
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
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].

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

See also: Lumatone mapping for sensi

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
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