Zeus: Difference between revisions

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'''Zeus''' is ~~the~~ [[rank-3 temperament]] tempering out [[121/120]], [[176/175]], and [[385/384]]. Notice 121/120 = (176/175)(385/384)~~, it thus equates the septimal neutral second of [[35/32]] with both [[11/10]] and [[12/11]]~~. That makes 11-limit harmony particularly efficient and deeply entangled with 7-limit harmony. ~~The~~ temperament is, in fact, one of the best ways to further temper on top of tempering out 121/120. That said, 121/120 is such a comma that, on tempering out, will neutralize the distinction between many otonal- and utonal-11 chords, so it might not fit those who wish to explore the subtlety of tones in the 11-limit.

{{Infobox regtemp

| Title = Zeus

| Subgroups = 2.3.5.7.11; 2.3.5.7.11.13

| Comma basis = [[121/120]], [[176/175]] (11-limit); [[121/120]], [[176/175]], [[351/350]] (13-limit)

| Edo join 1 = 31 | Edo join 2 = 46 | Edo join 3 = 53

| Mapping = 1; 1 1 -1 1 -2; 0 -2 3 -1 -1

| Generators = 3/2, 12/11

| Generators tuning = 701.9, 157.0

| Optimization method = CWE

| Odd limit 1 = 11 | Mistuning 1 = 7.48 | Complexity 1 = ?

| Odd limit 2 = 13-limit 21 | Mistuning 2 = 7.72 | Complexity 2 = ?

}}

'''Zeus''' is a [[rank-3 temperament|rank-3]] [[regular temperament|temperament]] generated by an [[2/1|octave]], a [[3/2|fifth]], and a neutral second which stands in for both undecimal neutral seconds of [[11/10]] and [[12/11]] as well as the septimal neutral second of [[35/32]], [[tempering out]] [[121/120]], [[176/175]], and [[385/384]]. Notice {{nowrap| 121/120 {{=}} (176/175)⋅(385/384)}}. That makes 11-limit harmony particularly efficient and deeply entangled with 7-limit harmony. This temperament is, in fact, one of the best ways to further temper on top of tempering out 121/120. That said, 121/120 is such a comma that, on tempering out, will neutralize the distinction between many otonal- and utonal-11 chords, so it might not fit those who wish to explore the subtlety of tones in the 11-limit.

~~See~~ [[~~Porwell family #Zeus~~]] or [[~~Biyatismic clan #Zeus~~]] ~~for technical data~~.

It also has an obvious extension to the 13-limit tempering out [[351/350]] and [[352/351]].

== Chords ==

See [[Biyatismic clan #Zeus]] for technical data.

Zeus enables [[essentially tempered chord]]s of [[~~Biyatismic~~ chords|biyatismic]], [[~~Valinorsmic~~ chords|valinorsmic]], and [[~~Keenanismic~~ chords|~~keenanismic~~]].

== Interval lattice ==

File:Lattice Zeus.png|11-limit zeus

File:Lattice Zeus13.png|13-limit zeus

</gallery>

== Chords and harmony ==

Zeus enables [[essentially tempered chord]]s of [[biyatismic chords|biyatismic]], [[valinorsmic chords|valinorsmic]], [[keenanismic chords|keenanismic]], and [[zeus chords|zeus]]. In the 13-limit, it enables chords of [[ratwolfsmic chords|ratwolfsmic]], [[major minthmic chords|major minthmic]], and [[catadictmic chords|catadictmic]].

== Scales ==

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* [[Genus1155zeus]]

== ~~Tuning spectrum~~ ==

== Tunings ==

=== Norm-based tunings ===

{| 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 = 702.5294{{c}}, ~12/11 = 157.2407{{c}}

| CWE: ~3/2 = 702.2478{{c}}, ~12/11 = 157.1265{{c}}

| POTE: ~3/2 = 702.1530{{c}}, ~12/11 = 157.0881{{c}}

|}

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings

|-

! rowspan="2" |

! colspan="3" | Euclidean

|-

! Constrained

! Constrained & skewed

! Destretched

|-

! Tenney

| CTE: ~3/2 = 701.9638{{c}}, ~12/11 = 156.9485{{c}}

| CWE: ~3/2 = 701.8818{{c}}, ~12/11 = 156.9568{{c}}

| POTE: ~3/2 = 701.8679{{c}}, ~12/11 = 156.9582{{c}}

|}

=== Tuning spectrum ===

This spectrum assumes pure 2 and 7/5.

~~Gencom: [2 3 11/10; 121/120 176/175 351/350]~~

~~Gencom mapping: {{mapping| 1 0 1 4 2 7 | 0 1 1 -1 1 -2 | 0 0 -2 3 -1 -1 }}~~

{| class="wikitable center-1 center-2 center-3"

|-

! [[Eigenmonzo|Eigenmonzo (Unchanged-interval]])

! [[Eigenmonzo|Eigenmonzo (Unchanged-interval)]]

! Fifth (¢)

! Major Third (¢)

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|}

[[~~Category~~:~~Temperaments~~]]

== Music ==

; [[Mundoworld]]

* [https://www.youtube.com/watch?v=zo1IYJW43II ''Cloudtop Reverie''] (2021) – in Zeus[7], 99edo tuning

[[Category:Zeus| ]]

[[Category:Rank-3 temperaments]]

[[Category:Porwell family]]

[[Category:Biyatismic clan]]

[[Category:Valinorsmic clan]]

[[Category:Keenanismic temperaments]]

@@ Line 1: / Line 1: @@
-'''Zeus''' is the [[rank-3 temperament]] tempering out [[121/120]], [[176/175]], and [[385/384]]. Notice 121/120 = (176/175)(385/384), it thus equates the septimal neutral second of [[35/32]] with both [[11/10]] and [[12/11]]. That makes 11-limit harmony particularly efficient and deeply entangled with 7-limit harmony. The temperament is, in fact, one of the best ways to further temper on top of tempering out 121/120. That said, 121/120 is such a comma that, on tempering out, will neutralize the distinction between many otonal- and utonal-11 chords, so it might not fit those who wish to explore the subtlety of tones in the 11-limit.
+{{Infobox regtemp
+| Title = Zeus
+| Subgroups = 2.3.5.7.11; 2.3.5.7.11.13
+| Comma basis = [[121/120]], [[176/175]] (11-limit); <br>[[121/120]], [[176/175]], [[351/350]] (13-limit)
+| Edo join 1 = 31 | Edo join 2 = 46 | Edo join 3 = 53
+| Mapping = 1; 1 1 -1 1 -2; 0 -2 3 -1 -1
+| Generators = 3/2, 12/11
+| Generators tuning = 701.9, 157.0
+| Optimization method = CWE
+| Odd limit 1 = 11 | Mistuning 1 = 7.48 | Complexity 1 = ?
+| Odd limit 2 = 13-limit 21 | Mistuning 2 = 7.72 | Complexity 2 = ?
+}}
+'''Zeus''' is a [[rank-3 temperament|rank-3]] [[regular temperament|temperament]] generated by an [[2/1|octave]], a [[3/2|fifth]], and a neutral second which stands in for both undecimal neutral seconds of [[11/10]] and [[12/11]] as well as the septimal neutral second of [[35/32]], [[tempering out]] [[121/120]], [[176/175]], and [[385/384]]. Notice {{nowrap| 121/120 {{=}} (176/175)⋅(385/384)}}. That makes 11-limit harmony particularly efficient and deeply entangled with 7-limit harmony. This temperament is, in fact, one of the best ways to further temper on top of tempering out 121/120. That said, 121/120 is such a comma that, on tempering out, will neutralize the distinction between many otonal- and utonal-11 chords, so it might not fit those who wish to explore the subtlety of tones in the 11-limit.
-See [[Porwell family #Zeus]] or [[Biyatismic clan #Zeus]] for technical data.
+It also has an obvious extension to the 13-limit tempering out [[351/350]] and [[352/351]].
-== Chords ==
+See [[Biyatismic clan #Zeus]] for technical data.
-Zeus enables [[essentially tempered chord]]s of [[Biyatismic chords|biyatismic]], [[Valinorsmic chords|valinorsmic]], and [[Keenanismic chords|keenanismic]].
+== Interval lattice ==
+<gallery>
+File:Lattice Zeus.png|11-limit zeus
+File:Lattice Zeus13.png|13-limit zeus
+</gallery>
+== Chords and harmony ==
+Zeus enables [[essentially tempered chord]]s of [[biyatismic chords|biyatismic]], [[valinorsmic chords|valinorsmic]], [[keenanismic chords|keenanismic]], and [[zeus chords|zeus]]. In the 13-limit, it enables chords of [[ratwolfsmic chords|ratwolfsmic]], [[major minthmic chords|major minthmic]], and [[catadictmic chords|catadictmic]].
 == Scales ==
@@ Line 18: / Line 38: @@
 * [[Genus1155zeus]]
-== Tuning spectrum ==
+== Tunings ==
+=== Norm-based tunings ===
+{| 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 = 702.5294{{c}}, ~12/11 = 157.2407{{c}}
+| CWE: ~3/2 = 702.2478{{c}}, ~12/11 = 157.1265{{c}}
+| POTE: ~3/2 = 702.1530{{c}}, ~12/11 = 157.0881{{c}}
+|}
+{| class="wikitable mw-collapsible mw-collapsed"
+|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings
+|-
+! rowspan="2" |
+! colspan="3" | Euclidean
+|-
+! Constrained
+! Constrained & skewed
+! Destretched
+|-
+! Tenney
+| CTE: ~3/2 = 701.9638{{c}}, ~12/11 = 156.9485{{c}}
+| CWE: ~3/2 = 701.8818{{c}}, ~12/11 = 156.9568{{c}}
+| POTE: ~3/2 = 701.8679{{c}}, ~12/11 = 156.9582{{c}}
+|}
+=== Tuning spectrum ===
 This spectrum assumes pure 2 and 7/5.
-Gencom: [2 3 11/10; 121/120 176/175 351/350]
-Gencom mapping: {{mapping| 1 0 1 4 2 7 | 0 1 1 -1 1 -2 | 0 0 -2 3 -1 -1 }}
 {| class="wikitable center-1 center-2 center-3"
 |-
-! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval]])
+! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
 ! Fifth<br>(¢)
 ! Major<br>Third (¢)
@@ Line 114: / Line 163: @@
 |}
-[[Category:Temperaments]]
+== Music ==
+; [[Mundoworld]]
+* [https://www.youtube.com/watch?v=zo1IYJW43II ''Cloudtop Reverie''] (2021) – in Zeus[7], 99edo tuning
 [[Category:Zeus| ]] <!-- main article -->
+[[Category:Rank-3 temperaments]]
 [[Category:Porwell family]]
 [[Category:Biyatismic clan]]
 [[Category:Valinorsmic clan]]
 [[Category:Keenanismic temperaments]]