Xenharmonic Wiki

Sensi: Difference between revisions

← Older edit
VisualWikitext
mNo edit summary
→Music: copypaste one from 46edo
 
(36 intermediate revisions by 5 users not shown)
Line 1: Line 1:
{{interwiki
{{interwiki
| en = Sensi
| de = Sensi
| de = Sensi
| en = Sensi
}}
}}
'''Sensi''' is a [[rank-2 temperament|rank-2]] [[regular temperament]] in the 2.3.5.7.13 [[subgroup]] [[generator|generated]] by an extremely sharp major third of between 440 and 446{{cent}} 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]].  
{{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.


Equal temperaments that support sensi include [[19edo]] (generator 7\19; [[soft]] [[checkertonic]]), [[27edo]] (generator 10\27; [[supersoft]] checkertonic), and [[46edo]] (generator 17\46; {{nowrap| 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.
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 technical data, and [[Sensi extensions]] for extensions of sensi that include the [[11/1|11th harmonic]].
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.


== Intervals ==
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.
 
== Theory ==
=== Interval chain ===
=== Interval chain ===
In the following table, odd harmonics and subharmonics 1–21 are in '''bold'''.  
In the following table, odd harmonics and subharmonics 1–21 are in '''bold'''.  
Line 15: Line 29:
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|-
|-
! &#35;
! #
! Cents*
! Cents*
! class="unsortable" | Approximate ratios**
! class="unsortable" | Approximate ratios
|-
|-
| 0
| 0
Line 24: Line 38:
|-
|-
| 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] ===
=== Intervals of 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. See [[3L&nbsp;5s #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:
Line 227: Line 239:
|}
|}


== Chords ==
=== Related temperaments ===
{{Main| Chords of sensus }}
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.


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 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]].
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
Line 246: Line 268:
* 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}}
Line 269: Line 291:


== Tunings ==
== 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 ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
Line 282: Line 364:
|  
|  
|-
|-
| 4\11
| [[11edo|4\11]]
|  
|  
| 436.364
| 436.364
Line 302: Line 384:
|  
|  
|-
|-
| 7\19
| [[19edo|7\19]]
|  
|  
| 442.105
| 442.105
Line 331: Line 413:
| 443.025
| 443.025
|  
|  
|-
| [[65edo|24\65]]
|
| 443.077
| 65f val
|-
|-
|  
|  
Line 337: Line 424:
| 2.3.5.7.13-subgroup 15- and 21-odd-limit minimax
| 2.3.5.7.13-subgroup 15- and 21-odd-limit minimax
|-
|-
| 17\46
| [[46edo|17\46]]
|  
|  
| 443.478
| 443.478
Line 356: Line 443:
| 443.756
| 443.756
| 7-odd-limit minimax
| 7-odd-limit minimax
|-
| [[73edo|27\73]]
|
| 443.836
|
|-
|-
|  
|  
Line 367: Line 459:
|
|
|-
|-
| 10\27
| [[27edo|10\27]]
|  
|  
| 444.444
| 444.444
Line 392: Line 484:
|
|
|-
|-
| 3\8
| [[8edo|3\8]]
|  
|  
| 450.000
| 450.000
Line 406: Line 498:
== 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]]
Line 413: Line 505:


=== 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 ===
Line 436: Line 528:


; [[Budjarn Lambeth]]
; [[Budjarn Lambeth]]
* [https://www.youtube.com/watch?v=qc0CkUKj7t4 Music in Sensi Temperament (+ Tempered Octaves) – Mar 2024]
* [https://www.youtube.com/watch?v=qc0CkUKj7t4 ''Music in Sensi Temperament (+ Tempered Octaves)''] (2024)
 
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=rmgWC_jruSg ''Sensi Fugue''] (2024) – fugue for two organs, in sensi, 46edo tuning


[[Category:Sensi| ]] <!-- Main article -->
[[Category:Sensi| ]] <!-- Main article -->
Retrieved from "https://en.xen.wiki/w/Sensi"