Xenharmonic Wiki

Juggernaut: Difference between revisions

← Older edit
VisualWikitext
CompactStar (talk | contribs)
autopatrolled
5,084 edits
This was a placeholder, removed for now
Lériendil (talk | contribs)
8,044 edits
m Text replacement - "(unchanged-interval)" to "(unchanged interval)"
 
(28 intermediate revisions by 4 users not shown)
Line 1: Line 1:
'''Juggernaut''' is a 5.7.11 [[nonoctave]] [[regular temperament]], first documented by [[User:CompactStar]], tempering out 125/121. 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]]. 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. [[24ed5]] is the first ed5 offering a workable tuning of juggernaut with the generator as 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 "tridecimal juggernaut" extension preserves the original 7/5 generator, and maps to [[13/5]] to -2 generators by tempering out 637/625.
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"
|-
! Generator
! Cents*
! Ratios
! Ratios<br>(tridecimal juggernaut)
|-
| -5
| 1266.911
|
| 715/343
|-
| -4
| 456.266
| 3025/2401
| 65/49
|-
| -3
| 1038.778
| 605/343, 625/343
| [[13/7]], 1625/847
|-
| -2
| 228.133
| [[55/49]], 625/539
| [[13/11]], 143/125
|-
| -1
| 810.645
| [[11/7]], 125/77
| 91/55, 1001/625
|-
| 0
| 0.000
| [[1/1]]
|
|-
| 1
| 582.512
| [[7/5]], 847/625
| 121/91, 125/91
|-
| 2
| 1165.024
| [[49/25]], 5929/3125
| [[25/13]], 121/65
|-
| 3
| 354.379
| 343/275, 3773/3125
| 77/65, 9317/8125
|-
| 4
| 936.891
| 2401/1375
| 539/325
|-
| 5
| 126.246
|
| 343/325
|}
<nowiki>*</nowiki>In no-twos-or-threes 11-limit CTE tuning
== Tuning spectrum ==
This assume tridecimal juggernaut mapping for no-twos-or-threes 13-limit intervals.
{|class="wikitable"
|-
! ED5 generator
! [[Eigenmonzo]] (unchanged interval)
! Cents
|-
|
| 91/25
| 549.588
|-
|
| [[13/11]]
| 551.974
|-
|
| 77/25
| 554.360
|-
| 2\[[10ed5]]
|
| 557.263
|-
|
| [[13/5]]
| 566.050
|-
|
| 539/125
| 568.436
|-
|
| [[13/7]]
| 571.538
|-
| 7\[[34ed5]]
|
| 573.653
|-
|
| 49/13
| 574.281
|-
| 5\[[24ed5]]
|
| 580.482
|-
|
| 143/125
| 580.126
|-
|
| [[7/5]]
| 582.512
|-
| 8\[[38ed5]]
|
| 586.592
|-
| 11\[[52ed5]]
|
| 589.413
|-
|
| 49/11
| 596.589
|-
| 3\[[14ed5]]
|
| 597.067
|-
|
| 343/121
| 601.281
|-
| 10\[[46ed5]]
|
| 605.720
|-
| 7\[[32ed5]]
|
| 609.506
|-
|
| [[11/7]]
| 610.665
|-
| 4\[[18ed5]]
|
| 619.181
|-
| 5\[[22ed5]]
|
| 633.253
|-
|
| 121/35
| 638.818
|}


{{Stub}}
[[Category:Juggernaut| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Non-octave temperaments]]
Retrieved from "https://en.xen.wiki/w/Juggernaut"