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. 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 center-1 right-2 right-4"
{|class="wikitable"
! rowspan="2" | # gens
! colspan="2" | Period 0
! colspan="2" | Period 1
|-
|-
! Generator
! Cents*
! Cents*
! Approximate Ratios
! Ratios
! Cents
! Ratios<br>(tridecimal juggernaut)
! Approximate Ratios
|-
| -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
| 0.000
| 0.000
| 1/1
| [[1/1]]
| 600.000
|
| 7/5, 10/7
|-
|-
| 1
| 1
| 230.966
| 582.512
| '''8/7'''
| [[7/5]], 847/625
| 830.966
| 121/91, 125/91
| '''8/5''', '''13/8'''
|-
|-
| 2
| 2
| 461.932
| 1165.024
| 21/16, 13/10
| [[49/25]], 5929/3125
| 1061.932
| [[25/13]], 121/65
| 11/6, 15/8, 24/13
|-
|-
| 3
| 3
| 692.898
| 354.379
| '''3/2'''
| 343/275, 3773/3125
| 92.898
| 77/65, 9317/8125
| 15/14
|-
|-
| 4
| 4
| 923.864
| 936.891
| 12/7, 22/13
| 2401/1375
| 323.864
| 539/325
| 6/5, 39/32
|-
|-
| 5
| 5
| 1154.830
| 126.246
| 48/25, 63/32, 96/48
|  
| 554.830
| 343/325
| '''11/8'''
|}
<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
|-
|-
| 6
| 4\[[18ed5]]
| 185.796
|  
| 9/8
| 619.181
| 785.796
| 11/7
|-
|-
| 7
| 5\[[22ed5]]
| 416.762
|  
| 9/7
| 633.253
| 1016.762
| 9/5
|-
|-
| 8
|  
| 647.728
| 121/35
| 36/25
| 638.818
| 47.728
| 33/32, 36/35
|}
|}
<nowiki>*</nowiki> In 13-limit POTE tuning. This tuning is very close to [[26edo]].


{{Stub}}
[[Category:Juggernaut| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Non-octave temperaments]]

Latest revision as of 06:59, 21 June 2025

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 pentave (5/1) as its equivalence instead of the more common octave or even 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.

Juggernaut contains multi-MOS scales of the families 4L 2s, 4L 6s, 10L 4s, 14L 10s, and 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

Generator Cents* Ratios Ratios
(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

*In no-twos-or-threes 11-limit CTE tuning

Tuning spectrum

This assume tridecimal juggernaut mapping for no-twos-or-threes 13-limit intervals.

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
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