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{{interwiki
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'''Sensi''' is a [[regular temperament]] for the 2.3.5.7.13 subgroup. EDOs that support sensi include [[19edo]], [[27edo]], and [[46edo]].
{{Infobox regtemp
| Title = Sensi
| Subgroups = 2.3.5.7, 2.3.5.7.13
| Comma basis = [[126/125]], [[245/243]] (7-limit); <br>[[91/90]], [[126/125]], [[169/168]] (2.3.5.7.13)
| Edo join 1 = 19 | Edo join 2 = 27
| Mapping = 1; 7 9 13 10
| Generators = 9/7 | Generators tuning = 443.3 | Optimization method = CWE
| MOS scales = [[3L&nbsp;2s]], [[3L&nbsp;5s]], [[8L&nbsp;3s]], [[8L&nbsp;11s]]
| Pergen = (P8, ccP5/7)
| Odd limit 1 = 7 | Mistuning 1 = 7.5 | Complexity 1 = 19
| Odd limit 2 = 2.3.5.7.13 21 | Mistuning 2 = 11.1 | Complexity 2 = 27
}}
'''Sensi''' is a [[rank-2 temperament|rank-2]] [[regular temperament]] that is [[generator|generated]] by an extremely sharp major third of between 442 and 445{{cent}}, 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 [[6/1|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 [[EDO]]s with similar structures are [[8edo]] and [[11edo]] (hence the 8-note ([[3L&nbsp;5s]], checkertonic) and 11-note ([[8L&nbsp;3s]], flanatonic) [[MOS scale]]s), 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).


See [[Sensipent family #Sensi]] for more information.
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; {{nowrap| L/s {{=}} 7/5 }}, more optimized for sensi temperament) and [[65edo]] (generator 24\65; {{nowrap| L/s {{=}} 10/7 }}) using the 65f [[val]] with a flat 13.


== Interval chain ==
See [[Sensipent family #Sensi]] for more technical data, [[sensi extensions]] for extensions of sensi to include the [[11/1|11th]] and [[17/1|17th]] harmonics, and [[#Related temperaments]] for alternative interpretations of similar structures to sensi.


{| class="wikitable"
== Theory ==
=== Interval chain ===
In the following table, odd harmonics and subharmonics 1–21 are in '''bold'''.
 
{| class="wikitable right-1 right-2 sortable"
|-
|-
! style="text-align:right" | Generators
! #
! | Cents<sup>*</sup>
! Cents*
! | Approximate ratios<sup>†</sup>
! class="unsortable" | Approximate ratios
|-
|-
| | 0
| 0
| style="text-align:right;" | 0.000
| 0.0
| | 1/1
| '''1/1'''
|-
|-
| | 1
| 1
| style="text-align:right;" | 443.322
| 443.4
| | 13/10~9/7
| 9/7, 13/10
|-
|-
| | 2
| 2
| style="text-align:right;" | 886.644
| 886.7
| | 42/25~5/3
| 5/3, 42/25
|-
|-
| | 3
| 3
| style="text-align:right;" | 129.966
| 130.1
| | 13/12~14/13~15/14
| 13/12, 14/13, 15/14, 27/25
|-
|-
| | 4
| 4
| style="text-align:right;" | 573.288
| 573.4
| | 7/5~25/18~18/13
| 7/5, 18/13, 25/18
|-
|-
| | 5
| 5
| style="text-align:right;" | 1016.610
| 1016.8
| | 9/5~70/39
| 9/5
|-
|-
| | 6
| 6
| style="text-align:right;" | 259.932
| 260.1
| | 7/6~15/13
| 7/6, 15/13
|-
|-
| | 7
| 7
| style="text-align:right;" | 703.253
| 703.5
| | 3/2
| '''3/2'''
|-
|-
| | 8
| 8
| style="text-align:right;" | 1146.576
| 1146.9
| | 35/18~27/14
| 27/14, 35/18
|-
|-
| | 9
| 9
| style="text-align:right;" | 389.896
| 390.2
| | 5/4
| '''5/4'''
|-
|-
| | 10
| 10
| style="text-align:right;" | 833.220
| 833.6
| | 13/8~21/13
| '''13/8''', 21/13
|-
|-
| | 11
| 11
| style="text-align:right;" | 76.542
| 76.9
| | 21/20~25/24
| 21/20, 25/24
|-
|-
| | 12
| 12
| style="text-align:right;" | 519.864
| 520.3
| | 27/20
| 27/20
|-
|-
| | 13
| 13
| style="text-align:right;" | 963.185
| 963.7
| | 7/4
| '''7/4'''
|-
|-
| | 14
| 14
| style="text-align:right;" | 206.507
| 207.0
| | 9/8
| '''9/8'''
|-
|-
| | 15
| 15
| style="text-align:right;" | 649.829
| 650.4
| | 35/24 (close to 16/11)
| 35/24 (sensor '''16/11''', sensus 22/15)
|-
|-
| | 16
| 16
| style="text-align:right;" | 1093.151
| 1093.7
| | 15/8
| '''15/8''' (sensor '''32/17''', sensus 17/9)
|-
|-
| | 17
| 17
| style="text-align:right;" | 336.473
| 337.1
| | 39/32 (close to 17/14)
| 39/32 (sensus 17/14)
|-
|-
| | 18
| 18
| style="text-align:right;" | 779.795
| 780.4
| | 25/16
| 25/16
|-
|-
| | 19
| 19
| style="text-align:right;" | 23.117
| 23.8
| | 49/48~65/64~81/80
| 49/48, 65/64, 81/80
|-
|-
| | 20
| 20
| style="text-align:right;" | 466.439
| 467.2
| | 21/16
| '''21/16'''
|}
|}
<nowiki/>* In 2.3.5.7.13 CWE tuning


: <sup>*</sup> in 2.3.5.7.13 POTE tuning
=== Intervals of Sensi[8] ===
: <sup>†</sup> 2.3.5.7.13 ratio interpretations
Sensi[8] is a [[mos scale]] with a [[3L&nbsp;5s]] pattern. See [[3L&nbsp;5s #Modes]] to see which modes have which qualities for each interval size.


== Harmony in sensi ==
Sortable table of Sensi[8]'s major and minor intervals in various sensi tunings:
The most common consonant triad in sensi is the 5:6:13 triad, which spans 3 generators. Sensi[8] has five 5:6:13 triads, four 7:9:13 triads, three 5:6:7:9 tetrads and one 5:6:7:9:13 pentad.


== Sensi Visualizations ==
{| class="wikitable right-2 right-3 right-4 sortable"
|-
! class="unsortable" | Degree
! Size in [[19edo]] (soft)
! Size in [[27edo]] (supersoft)
! Size in [[46edo]]
! class="unsortable" | Approximate ratios
! &#35; 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
|}


=== A Diagram of Sensi[5], [8], [11], and [19] with intervals named in relation to the L and s of Sensi[8]: ===
=== Related temperaments ===
[[File:steps_of_sensi.png|alt=steps_of_sensi.png|steps_of_sensi.png]]
It is worth noting that sensi distinguishes itself from other structures, the [[sensamagic clan|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 [[interseptimal]]ly 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 {{nowrap|[[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 ====
: ''Main article: [[Relationship between Bohlen–Pierce and octave-ful temperaments#Relationship of rank-2 Bohlen.E2.80.93Pierce.E2.80.93Stearns temperament to octave-ful temperaments|Relationship between Bohlen–Pierce and octave-ful temperaments]].
 
Since the sensamagic comma, 245/243, contains no 2 in its [[monzo|factorization]], only primes 3, 5, and 7, it can be tempered out in a [[3/1|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 [[BPS|Bohlen–Pierce–Stearns (BPS)]] temperament, and it generates a [[4L 5s (3/1-equivalent)|4L&nbsp;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 ==
{{See also| 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{{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):
* 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:
 
* {{dash|Root, maj. sen7th, maj. sen8th ≈ 7:12:13|s=space}}
* {{dash|Root, maj. sen2nd, maj. sen5th ≈ 9:10:13|s=space}}
* {{dash|Root, min. sen3rd, dim. sen6th ≈ 6:7:9|s=space}}
* {{dash|Root, perf. sen4th, dim. sen6th ≈ 10:13:15 (ultramajor triad)|s=space}}
* {{dash|Root, perf. sen4th, maj. sen7th ≈ 7:9:13|s=space}}
* {{dash|Root, perf. sen4th, maj. sen5th, maj. sen7th ≈ 7:9:10:13|s=space}}
* {{dash|Root, perf. sen4th, min. sen7th ≈ 10:13:18|s=space}}
* {{dash|Root, perf. sen4th, min. sen5th, min. sen7th ≈ 10:13:14:18|s=space}}
* {{dash|Root, min. sen7th, min. sen3rd (+ octave) ≈ 3:5:7|s=space}}
* {{dash|Root, min. sen7th, min. sen2nd (+ octave) ≈ 6:10:13|s=space}}
* {{dash|Root, dim. sen6th, min. sen7th ≈ 6:9:10|s=space}}
* {{dash|Root, dim. sen6th, min. sen2nd (+octave) ≈ 6:9:13|s=space}}
 
== Scales ==
* [[Sensi5]]
* [[Sensi8]]
* [[Sensi11]]
* [[Sensi19]]
* [[Sensi27]]
 
== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~9/7 = 443.3166{{c}}
| CWE: ~9/7 = 443.3493{{c}}
| POTE: ~9/7 = 443.3827{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.7.13-subgroup norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~9/7 = 443.4016{{c}}
| CWE: ~9/7 = 443.3581{{c}}
| POTE: ~9/7 = 443.3220{{c}}
|}
 
=== Target tunings ===
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings
|-
! rowspan="2" | Target
! colspan="2" | Minimax
|-
! Generator
! Eigenmonzo*
|-
| 7-odd-limit
| ~9/7 = 443.756{{c}}
| 7/4
|-
| 9-odd-limit
| ~9/7 = 443.519{{c}}
| 9/5
|-
| no-11 13-odd-limit
| ~9/7 = 443.519{{c}}
| 9/5
|-
| no-11 15-odd-limit
| ~9/7 = 443.136{{c}}
| 3/2
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generators
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
|
| 9/7
| 435.084
|
|-
| [[11edo|4\11]]
|
| 436.364
| 11cdf val
|-
|
| 15/14
| 439.814
|
|-
|
| 13/9
| 440.846
|
|-
|
| 15/13
| 441.290
|
|-
| [[19edo|7\19]]
|
| 442.105
| Lower bound of 7- and 9-odd-limit, <br>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
|
|-
| [[65edo|24\65]]
|
| 443.077
| 65f val
|-
|
| 3/2
| 443.136
| 2.3.5.7.13-subgroup 15- and 21-odd-limit minimax
|-
| [[46edo|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
|-
| [[73edo|27\73]]
|
| 443.836
|
|-
|
| 21/20
| 444.042
|
|-
|
| 13/8
| 444.053
|
|-
| [[27edo|10\27]]
|
| 444.444
| Upper bound of 9-odd-limit, <br>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
|
|-
| [[8edo|3\8]]
|
| 450.000
| 8d val, upper bound of 7-odd-limit diamond monotone
|-
|
| 13/10
| 454.214
|
|}
<nowiki/>* 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].
 
[[File:steps_of_sensi.png|Steps of sensi|alt=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".
Note that X, M and Z are not standard, but d and A are; they are short for "diminished" and "augmented".


=== A Diagram relating the Sensi generator chain (horizontal axis) to the steps within the octave (vertical axis) for Sensi[8] and [11]: ===
=== Map of sensi ===
[[File:map_of_sensi-8-.png|alt=map_of_sensi[8].png|map_of_sensi[8].png]][[File:map_of_sensi-11-_correction2.png|alt=map_of_sensi[11]_correction2.png|map_of_sensi[11]_correction2.png]]
These diagrams relate the sensi generator chain (horizontal axis) to the steps within the octave (vertical axis) for Sensi[8] and [11].  


=== A Diagram showing a layout for playing Sensi Temperament on an Isomorphic Keyboard: ===
[[File:map_of_sensi-8-.png|Map of Sensi[8]|alt=map_of_sensi[8].png]]
[[File:sensi_isomorphic_layout.png|alt=sensi_isomorphic_layout.png|sensi_isomorphic_layout.png]]
[[File:map_of_sensi-11-_correction2.png|Map of Sensi[11]|alt=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]].
 
[[File:sensi_isomorphic_layout.png|sensi_isomorphic_layout.png|alt=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.
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.


=== A Sensi[19] Guitar (one octave): ===
=== Sensi[19] guitar ===
[[File:sensi-19-in46.jpg|alt=sensi[19]in46.jpg|sensi[19]in46.jpg]]
[[File:sensi-19-in46.jpg|sensi[19]in46.jpg|alt=sensi[19]in46.jpg]]


SOMEONE PLEASE MAKE ONE OF THESE AND SEND IT TO DUSTIN SCHALLERT!
== Music ==
; [[Andrew Heathwaite]]
* [[Technical Notes for Newbeams #Tumbledown Stew|"Tumbledown Stew" from ''Newbeams'']]
* [[Technical Notes for Newbeams #Hypnocloudsmack 3|"Hypnocloudsmack 3" from ''Newbeams'']]


== Music ==
; [[Budjarn Lambeth]]
* [[Technical Notes for Newbeams #Tumbledown Stew]] by [[Andrew Heathwaite]]
* [https://www.youtube.com/watch?v=qc0CkUKj7t4 ''Music in Sensi Temperament (+ Tempered Octaves)''] (2024)
* [[Technical Notes for Newbeams #Hypnocloudsmack 3]] by [[Andrew Heathwaite]]     
 
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=rmgWC_jruSg ''Sensi Fugue''] (2024) – fugue for two organs, in sensi, 46edo tuning


[[Category:soft_redirect]]
[[Category:Sensi| ]] <!-- Main article -->
[[Category:Rank-2 temperaments]]
[[Category:Sensipent family]]
[[Category:Sensamagic clan]]
[[Category:Starling temperaments]]
[[Category:Sengic temperaments]]
[[Category:Naiadic]]
Retrieved from "https://en.xen.wiki/w/Sensi"