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'''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.  
{{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]] 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.  


It also has an obvious extension to the 13-limit tempering out [[351/350]] and [[352/351]].  
It also has an obvious extension to the 13-limit tempering out [[351/350]] and [[352/351]].  
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== Tunings ==
== 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 ===
=== Tuning spectrum ===
This spectrum assumes pure 2 and 7/5.  
This spectrum assumes pure 2 and 7/5.  

Latest revision as of 08:36, 8 June 2026

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)
Reduced mapping ⟨1; 1 1 -1 1 -2; 0 -2 3 -1 -1]
ET join 31 & 46 & 53
Generators (CWE) ~3/2 = 701.9 ¢, ~12/11 = 157.0 ¢
MOS scales n/a
Ploidacot n/a
Minimax error 11-odd-limit: 7.48 ¢;
13-limit 21-odd-limit: 7.72 ¢
Target scale size 11-odd-limit: ? notes;
13-limit 21-odd-limit: ? notes

Zeus is a rank-3 temperament generated by an octave, a 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 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.

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

See Biyatismic clan #Zeus for technical data.

Interval lattice

  • 11-limit zeus
    11-limit zeus
  • 13-limit zeus
    13-limit zeus

Chords and harmony

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

Scales

Tunings

Norm-based tunings

11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 702.5294 ¢, ~12/11 = 157.2407 ¢ CWE: ~3/2 = 702.2478 ¢, ~12/11 = 157.1265 ¢ POTE: ~3/2 = 702.1530 ¢, ~12/11 = 157.0881 ¢
13-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 701.9638 ¢, ~12/11 = 156.9485 ¢ CWE: ~3/2 = 701.8818 ¢, ~12/11 = 156.9568 ¢ POTE: ~3/2 = 701.8679 ¢, ~12/11 = 156.9582 ¢

Tuning spectrum

This spectrum assumes pure 2 and 7/5.

Eigenmonzo
(Unchanged-interval) 		Fifth
(¢) 		Major
Third (¢) 		Comments
12/11 685.337 384.062
15/11 689.089 384.813
5/4 696.593 386.314
11/9 697.207 386.436
14/13 700.880 387.171
16/15 701.061 387.207
15/13 701.179 387.231
16/13 701.237 387.243
13/12 701.449 387.285
18/13 701.564 387.308
4/3 701.955 387.386 7-, 9- and 15-odd-limit minimax
10/9 702.551 387.505 11- and 13-odd-limit minimax
6/5 703.296 387.654
13/11 703.597 387.714
11/8 713.034 389.602
11/10 721.254 391.246

Music

Mundoworld
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