Sensamagic: Difference between revisions

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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]].

See [[Sensamagic family #Sensamagic]] for technical data.

== Interval lattice ==

[[File:Lattice_Sensamagic.png]]

File:Lattice_Sensamagic.png|11-limit sensamagic

</gallery>

== 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

! Ratio

Line 22:

Line 38:

|-

| 5/4

| Double up augmented 2nd

| Double-up augmented 2nd

| C-^^D#

|-

Line 34:

Line 50:

|}

== Chords ==

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 chord]]s of [[Sensamagic chords|sensamagic]], [[Keenanismic chords|keenanismic]], [[Pentacircle chords|pentacircle]], and [[Undecimal sensamagic chords|undecimal sensamagic]].

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 ===

[[Category:Sensamagic| ]]

[[Category:Rank-3 temperaments]]

[[Category:Sensamagic family]]

[[Category:Keenanismic temperaments]]

[[Category:Pentacircle ~~temperaments~~]]

[[Category:Pentacircle clan]]

~~{{IoT}}~~

@@ Line 1: / Line 1: @@
-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]].
 See [[Sensamagic family #Sensamagic]] for technical data.
 == Interval lattice ==
-[[File:Lattice_Sensamagic.png]]
+<gallery>
+File:Lattice_Sensamagic.png|11-limit sensamagic
+</gallery>
 == 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
 ! Ratio
@@ Line 22: / Line 38: @@
 |-
 | 5/4
-| Double up augmented 2nd
+| Double-up augmented 2nd
 | C-^^D#
 |-
@@ Line 34: / Line 50: @@
 |}
-== Chords ==
+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.
-{{main| Sensamagic 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]].
+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:Rank-3 temperaments]]
 [[Category:Sensamagic family]]
 [[Category:Keenanismic temperaments]]
-[[Category:Pentacircle temperaments]]
+[[Category:Pentacircle clan]]
-{{IoT}}