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The '''sensamagic''' temperament is the [[rank-3 temperament]] [[tempering out]] [[245/243]], the sensamagic comma. It has a canonical 11-limit [[extension]] adding [[385/384]] and [[896/891]] to the comma list.  
{{Infobox regtemp
| Title = Sensamagic
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Comma basis = [[245/243]] (7-limit); <br>[[245/243]], [[385/384]] (11-limit)
| Edo join 1 = 19 | Edo join 2 = 22 | Edo join 3 = 24
| Mapping = 1; 1 1 2 -2; 0 2 -1 -1
| Generators = 3/2, 9/7
| Generators tuning = 703.8, 440.9
| Optimization method = CWE
| Odd limit 1 = 9 | Mistuning 1 = 4.73 | Complexity 1 = ?
| Odd limit 2 = 11-limit 21 | Mistuning 2 = 5.11 | Complexity 2 = ?
}}
'''Sensamagic''' is a [[rank-3 temperament|rank-3]] [[regular temperament|temperament]] with the same [[lattice]] structure as the [[2.3.7 subgroup]], while identifying the [[5/3]] major sixth as a stack of two [[9/7]] supermajor thirds, [[tempering out]] [[245/243]]. It is the head of the [[sensamagic family]].
 
It favors a sharp supermajor third, and subtracting such a third from a [[4/3|perfect fourth]] produces a narrow subminor second of [[28/27]][[~]][[36/35]]. The canonical [[11-limit]] [[extension]] identifies this interval with [[33/32]], so that [[11/8]] is that plus a perfect fourth. This adds [[385/384]] and [[896/891]] to the comma list and makes it a member of both [[keenanismic temperaments]] and [[pentacircle clan]].  


The temperament was named after the corresponding comma, which was named by [[Gene Ward Smith]] in 2010. See [[245/243 #Etymology]].  
The temperament was named after the corresponding comma, which was named by [[Gene Ward Smith]] in 2010. See [[245/243 #Etymology]].  
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== Interval lattice ==
== Interval lattice ==
[[File:Lattice_Sensamagic.png]]
<gallery>
File:Lattice_Sensamagic.png|11-limit sensamagic
</gallery>


== Notation ==
== Notation ==
Sensamagic can be notated the same as 2.3.7 [[just intonation]] since they share the same lattice structure.  
Sensamagic can be notated the same as 2.3.7 [[just intonation]] since they share the same lattice structure. One way to do this is to take the conventional [[circle-of-fifths notation]] with an additional module of accidentals for the [[64/63]] comma. In this system, 7/4 is a minor seventh, 5/4 an augmented second, and 11/8 a diminished fifth.  
 
One way to do this is to take the conventional [[circle-of-fifths notation]] with an additional module of accidentals for the [[64/63]] comma. In this system, 7/4 is a minor seventh, 5/4 an augmented second, and 11/8 a diminished fifth.  


{| class="wikitable center-all"
{| class="wikitable center-1 center-3"
|+Sensamagic nomenclature<br>for selected intervals
|+Sensamagic nomenclature<br>for selected intervals
! Ratio
! Ratio
Line 24: Line 38:
|-
|-
| 5/4
| 5/4
| Double up augmented 2nd
| Double-up augmented 2nd
| C-^^D#
| C-^^D#
|-
|-
Line 36: Line 50:
|}
|}


Alternatively, it can be notated the same as full prime-limit just intonation, with a distinct accidental module for each prime harmonic. That makes some intervals more intuitive, at the cost of hiding the structure of sensamagic tempering. For example, it is customary of the 5/4 to be a major third, and 7/4 to be a minor seventh. As a result, the fact that the 5/3 is a stack of two 9/7's is not revealed, and the related chords can be confusing.
Alternatively, it can be notated the same as full prime-limit just intonation, with a distinct accidental module for each prime harmonic. That makes some intervals more intuitive, at the cost of hiding the structure of sensamagic tempering. For example, it is customary of the 5/4 to be a major third, and 7/4 to be a minor seventh. As a result, the fact that the 5/3 is a stack of two 9/7's is not revealed, and the related chords can be less convenient.


== Chords ==
== Chords and harmony ==
Sensamagic enables [[essentially tempered chord]]s of [[Sensamagic chords|sensamagic]], [[Keenanismic chords|keenanismic]], [[Pentacircle chords|pentacircle]], and [[Undecimal sensamagic chords|undecimal sensamagic]].  
Sensamagic enables [[essentially tempered chord]]s of [[Sensamagic chords|sensamagic]], [[Keenanismic chords|keenanismic]], [[Pentacircle chords|pentacircle]], and [[Undecimal sensamagic chords|undecimal sensamagic]].  


[[Category:Temperaments]]
The [[sensamagic dominant chord]] is a dominant seventh chord useful for tonal harmony in this temperament.
 
== 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: ~3/2 = 703.7424{{c}}, ~9/7 = 440.9020{{c}}
| CWE: ~3/2 = 703.7411{{c}}, ~9/7 = 440.9017{{c}}
| POTE: ~3/2 = 703.7424{{c}}, ~9/7 = 440.9020{{c}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 703.7737{{c}}, ~9/7 = 440.9186{{c}}
| CWE: ~3/2 = 703.7948{{c}}, ~9/7 = 440.9180{{c}}
| POTE: ~3/2 = 703.8004{{c}}, ~9/7 = 440.9178{{c}}
|}
 
=== Tuning spectrum ===
{{Todo|complete section|inline=1}}
 
[[Category:Sensamagic| ]] <!-- main article -->
[[Category:Sensamagic| ]] <!-- main article -->
[[Category:Rank-3 temperaments]]
[[Category:Sensamagic family]]
[[Category:Sensamagic family]]
[[Category:Keenanismic temperaments]]
[[Category:Keenanismic temperaments]]
[[Category:Pentacircle clan]]
[[Category:Pentacircle clan]]

Latest revision as of 19:12, 12 March 2026

Sensamagic
Subgroups 2.3.5.7, 2.3.5.7.11
Comma basis 245/243 (7-limit);
245/243, 385/384 (11-limit)
Reduced mapping ⟨1; 1 1 2 -2; 0 2 -1 -1]
ET join 19 & 22 & 24
Generators (CWE) ~3/2 = 703.8 ¢, ~9/7 = 440.9 ¢
MOS scales n/a
Ploidacot n/a
Minimax error 9-odd-limit: 4.73 ¢;
11-limit 21-odd-limit: 5.11 ¢
Target scale size 9-odd-limit: ? notes;
11-limit 21-odd-limit: ? notes

Sensamagic is a rank-3 temperament with the same lattice structure as the 2.3.7 subgroup, while identifying the 5/3 major sixth as a stack of two 9/7 supermajor thirds, tempering out 245/243. It is the head of the sensamagic family.

It favors a sharp supermajor third, and subtracting such a third from a perfect fourth produces a narrow subminor second of 28/27~36/35. The canonical 11-limit extension identifies this interval with 33/32, so that 11/8 is that plus a perfect fourth. This adds 385/384 and 896/891 to the comma list and makes it a member of both keenanismic temperaments and pentacircle clan.

The temperament was named after the corresponding comma, which was named by Gene Ward Smith in 2010. See 245/243 #Etymology.

See Sensamagic family #Sensamagic for technical data.

Interval lattice

  • 11-limit sensamagic
    11-limit sensamagic

Notation

Sensamagic can be notated the same as 2.3.7 just intonation since they share the same lattice structure. One way to do this is to take the conventional circle-of-fifths notation with an additional module of accidentals for the 64/63 comma. In this system, 7/4 is a minor seventh, 5/4 an augmented second, and 11/8 a diminished fifth.

Sensamagic nomenclature
for selected intervals
Ratio Nominal Example
3/2 Perfect 5th C-G
5/4 Double-up augmented 2nd C-^^D#
7/4 Down minor 7th C-vBb
11/8 Down diminished 5th C-vGb

Alternatively, it can be notated the same as full prime-limit just intonation, with a distinct accidental module for each prime harmonic. That makes some intervals more intuitive, at the cost of hiding the structure of sensamagic tempering. For example, it is customary of the 5/4 to be a major third, and 7/4 to be a minor seventh. As a result, the fact that the 5/3 is a stack of two 9/7's is not revealed, and the related chords can be less convenient.

Chords and harmony

Sensamagic enables essentially tempered chords of sensamagic, keenanismic, pentacircle, and undecimal sensamagic.

The sensamagic dominant chord is a dominant seventh chord useful for tonal harmony in this temperament.

Tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 703.7424 ¢, ~9/7 = 440.9020 ¢ CWE: ~3/2 = 703.7411 ¢, ~9/7 = 440.9017 ¢ POTE: ~3/2 = 703.7424 ¢, ~9/7 = 440.9020 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 703.7737 ¢, ~9/7 = 440.9186 ¢ CWE: ~3/2 = 703.7948 ¢, ~9/7 = 440.9180 ¢ POTE: ~3/2 = 703.8004 ¢, ~9/7 = 440.9178 ¢

Tuning spectrum

Todo: complete section
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