Juggernaut: Difference between revisions
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'''Juggernaut''' is a 5.7.11 [[nonoctave]] [[regular temperament]], first documented by [[User:CompactStar]], tempering out 125/121. Its subgroup does not contain harmonics 2 and 3 and so it uses the [[5/1|pentave]] (5/1) as its equivalence instead of the more common [[2/1|octave]] or even [[3/1|tritave]]. It has a period of 1\[[2ed5]] (1393 cents) representing [[11/5]], and a generator representing [[7/5]] (in fact, in the [[CTE tuning]] it is exactly 7/5). This gives juggernaut an extremely low [[complexity]] with 5th, 7th, and 11th harmonics all reachable within just 1 generator, while still having only a moderately high error. It is one of the lowest-[[badness]] 5/1-equivalent or "no-twos-or-threes" temperaments, similar to [[meantone]] and [[BPS]]/lambda in their respective spheres. [[14ed5]] (practically the same as [[6edo]]) is the first ed5 offering a workable tuning of juggernaut with the generator as 3\14ed5, while [[24ed5]] offers a more accurate tuning with the generator of 5\24ed5. | '''Juggernaut''' is a 5.7.11 [[nonoctave]] [[regular temperament]], first documented by [[User:CompactStar]], tempering out [[125/121]]. Its subgroup does not contain harmonics 2 and 3 and so it uses the [[5/1|pentave]] (5/1) as its equivalence instead of the more common [[2/1|octave]] or even [[3/1|tritave]]. It has a period of 1\[[2ed5]] (1393 cents) representing [[11/5]], and a generator representing [[7/5]] (in fact, in the [[CTE tuning]] it is exactly 7/5). This gives juggernaut an extremely low [[complexity]] with 5th, 7th, and 11th harmonics all reachable within just 1 generator, while still having only a moderately high error. It is one of the lowest-[[badness]] 5/1-equivalent or "no-twos-or-threes" temperaments, similar to [[meantone]] and [[BPS]]/lambda in their respective spheres. [[14ed5]] (practically the same as [[6edo]]) is the first ed5 offering a workable tuning of juggernaut with the generator as 3\14ed5, while [[24ed5]] offers a more accurate tuning with the generator of 5\24ed5. | ||
The best extension of juggernaut to the no-twos-or-threes 13-limit, named "cuthbernaut", splits the 7/5 generator into two [[13/11]] by tempering out [[847/845]]. The next best extension has been named "tridecimal juggernaut" since it preserves the original 7/5 generator, mapping [[13/5]] to -2 generators by tempering out 637/625. Tridecimal juggernaut favors a flatter 7/5 (in the vicinity of 570 cents) for the least error. | The best extension of juggernaut to the no-twos-or-threes 13-limit, named "cuthbernaut", splits the 7/5 generator into two [[13/11]] by tempering out [[847/845]]. The next best extension has been named "tridecimal juggernaut" since it preserves the original 7/5 generator, mapping [[13/5]] to -2 generators by tempering out 637/625. Tridecimal juggernaut favors a flatter 7/5 (in the vicinity of 570 cents) for the least error. | ||
Juggernaut contains multi-[[MOS scale]]s of the families [[4L 2s (5/1-equivalent)|4L 2s]], [[4L 6s (5/1-equivalent)|4L 6s]], [[10L 4s (5/1-equivalent)|10L 4s]], [[14L 10s (5/1-equivalent)|14L 10s]], and [[24L 14s (5/1-equivalent)|24L 14s]]. The 6-note MOS is rendered unusable because it has very large melodic steps (it corresponds to to 6*log(2)/log(5) ≈ 2.6 note octave-repeating scale) and contains too little 5:7:11 chords for the usage in no-twos-or-threes harmony. | Juggernaut contains multi-[[MOS scale]]s of the families [[4L 2s (5/1-equivalent)|4L 2s]], [[4L 6s (5/1-equivalent)|4L 6s]], [[10L 4s (5/1-equivalent)|10L 4s]], [[14L 10s (5/1-equivalent)|14L 10s]], and [[24L 14s (5/1-equivalent)|24L 14s]]. The 6-note MOS is rendered unusable because it has very large melodic steps (it corresponds to to 6*log(2)/log(5) ≈ 2.6 note octave-repeating scale) and contains too little 5:7:11 chords for the usage in no-twos-or-threes harmony. | ||
Technical data: [[No-twos subgroup temperaments#Juggernaut]]. | |||
== Intervals == | == Intervals == | ||
{|class="wikitable" | {|class="wikitable" | ||
|- | |- | ||
!Generator | ! Generator | ||
!Cents* | ! Cents* | ||
!Ratios | ! Ratios | ||
!Ratios<br>(tridecimal juggernaut) | ! Ratios<br>(tridecimal juggernaut) | ||
|- | |- | ||
| -5 | | -5 | ||
|1266.911 | | 1266.911 | ||
| | | | ||
|715/343 | | 715/343 | ||
|- | |- | ||
| -4 | | -4 | ||
|456.266 | | 456.266 | ||
|3025/2401 | | 3025/2401 | ||
|65/49 | | 65/49 | ||
|- | |- | ||
| -3 | | -3 | ||
|1038.778 | | 1038.778 | ||
|605/343, 625/343 | | 605/343, 625/343 | ||
|[[13/7]], 1625/847 | | [[13/7]], 1625/847 | ||
|- | |- | ||
| -2 | | -2 | ||
|228.133 | | 228.133 | ||
|[[55/49]], 625/539 | | [[55/49]], 625/539 | ||
|[[13/11]], 143/125 | | [[13/11]], 143/125 | ||
|- | |- | ||
| -1 | | -1 | ||
|810.645 | | 810.645 | ||
|[[11/7]], 125/77 | | [[11/7]], 125/77 | ||
|91/55, 1001/625 | | 91/55, 1001/625 | ||
|- | |- | ||
|0 | | 0 | ||
|0.000 | | 0.000 | ||
|[[1/1]] | | [[1/1]] | ||
| | | | ||
|- | |- | ||
|1 | | 1 | ||
|582.512 | | 582.512 | ||
|[[7/5]], 847/625 | | [[7/5]], 847/625 | ||
|121/91, 125/91 | | 121/91, 125/91 | ||
|- | |- | ||
|2 | | 2 | ||
|1165.024 | | 1165.024 | ||
|[[49/25]], 5929/3125 | | [[49/25]], 5929/3125 | ||
|[[25/13]], 121/65 | | [[25/13]], 121/65 | ||
|- | |- | ||
|3 | | 3 | ||
|354.379 | | 354.379 | ||
|343/275, 3773/3125 | | 343/275, 3773/3125 | ||
|77/65, 9317/8125 | | 77/65, 9317/8125 | ||
|- | |- | ||
|4 | | 4 | ||
|936.891 | | 936.891 | ||
|2401/1375 | | 2401/1375 | ||
|539/325 | | 539/325 | ||
|- | |- | ||
|5 | | 5 | ||
|126.246 | | 126.246 | ||
| | | | ||
|343/325 | | 343/325 | ||
|} | |} | ||
<nowiki>*</nowiki>In no-twos-or-threes 11-limit CTE tuning | <nowiki>*</nowiki>In no-twos-or-threes 11-limit CTE tuning | ||
| Line 74: | Line 76: | ||
{|class="wikitable" | {|class="wikitable" | ||
|- | |- | ||
!ED5 generator | ! ED5 generator | ||
![[Eigenmonzo]] (unchanged | ! [[Eigenmonzo]] (unchanged interval) | ||
!Cents | ! Cents | ||
|- | |- | ||
|91/25 | | | ||
|549.588 | | 91/25 | ||
| 549.588 | |||
|- | |- | ||
| | | | ||
|[[13/11]] | | [[13/11]] | ||
|551.974 | | 551.974 | ||
|- | |- | ||
| | | | ||
|77/25 | | 77/25 | ||
|554.360 | | 554.360 | ||
|- | |- | ||
|2\[[10ed5]] | | 2\[[10ed5]] | ||
| | | | ||
|557.263 | | 557.263 | ||
|- | |- | ||
| | | | ||
|[[13/5]] | | [[13/5]] | ||
|566.050 | | 566.050 | ||
|- | |- | ||
| | | | ||
|539/125 | | 539/125 | ||
|568.436 | | 568.436 | ||
|- | |- | ||
| | | | ||
|[[13/7]] | | [[13/7]] | ||
|571.538 | | 571.538 | ||
|- | |- | ||
|7\[[34ed5]] | | 7\[[34ed5]] | ||
| | | | ||
|573.653 | | 573.653 | ||
|- | |- | ||
| | | | ||
|49/13 | | 49/13 | ||
|574.281 | | 574.281 | ||
|- | |- | ||
|5\[[24ed5]] | | 5\[[24ed5]] | ||
| | | | ||
|580.482 | | 580.482 | ||
|- | |- | ||
| | | | ||
|143/125 | | 143/125 | ||
|580.126 | | 580.126 | ||
|- | |- | ||
| | | | ||
|[[7/5]] | | [[7/5]] | ||
|582.512 | | 582.512 | ||
|- | |- | ||
|8\[[38ed5]] | | 8\[[38ed5]] | ||
| | | | ||
|586.592 | | 586.592 | ||
|- | |- | ||
|11\[[52ed5]] | | 11\[[52ed5]] | ||
| | | | ||
|589.413 | | 589.413 | ||
|- | |- | ||
| | | | ||
|49/11 | | 49/11 | ||
|596.589 | | 596.589 | ||
|- | |- | ||
|3\[[14ed5]] | | 3\[[14ed5]] | ||
| | | | ||
|597.067 | | 597.067 | ||
|- | |- | ||
| | | | ||
|343/121 | | 343/121 | ||
|601.281 | | 601.281 | ||
|- | |- | ||
|10\[[46ed5]] | | 10\[[46ed5]] | ||
| | | | ||
|605.720 | | 605.720 | ||
|- | |- | ||
|7\[[32ed5]] | | 7\[[32ed5]] | ||
| | | | ||
|609.506 | | 609.506 | ||
|- | |- | ||
| | | | ||
|[[11/7]] | | [[11/7]] | ||
|610.665 | | 610.665 | ||
|- | |- | ||
|4\[[18ed5]] | | 4\[[18ed5]] | ||
| | | | ||
|619.181 | | 619.181 | ||
|- | |- | ||
|5\[[22ed5]] | | 5\[[22ed5]] | ||
| | | | ||
|633.253 | | 633.253 | ||
|- | |- | ||
| | | | ||
|121/35 | | 121/35 | ||
|638.818 | | 638.818 | ||
|} | |} | ||
[[Category: | [[Category:Juggernaut| ]] <!-- main article --> | ||
[[Category: | [[Category:Rank-2 temperaments]] | ||
[[Category: | [[Category:Non-octave temperaments]] | ||