Zeus: Difference between revisions

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{{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.

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</gallery>

== Chords ==

== Chords and harmony ==

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

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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== 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.

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

~~[[Category:Temperaments]]~~

[[Category:Zeus| ]]

[[Category:Rank-3 temperaments]]

@@ Line 1: / Line 1: @@
+{{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.
@@ Line 11: / Line 23: @@
 </gallery>
-== Chords ==
+== Chords and harmony ==
-Zeus enables [[essentially tempered chord]]s of [[biyatismic chords|biyatismic]], [[valinorsmic chords|valinorsmic]], and [[keenanismic chords|keenanismic]].
+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 27: / Line 39: @@
 == 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.
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 * [https://www.youtube.com/watch?v=zo1IYJW43II ''Cloudtop Reverie''] (2021) – in Zeus[7], 99edo tuning
-[[Category:Temperaments]]
 [[Category:Zeus| ]] <!-- main article -->
 [[Category:Rank-3 temperaments]]