Juggernaut: Difference between revisions

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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.


== Intervals ==
{| class="wikitable center-1 right-2 right-4"
! rowspan="2" | # gens
! colspan="2" | Period 0
! colspan="2" | Period 1
|-
! Cents*
! Approximate Ratios
! Cents
! Approximate Ratios
|-
| 0
| 0.000
| 1/1
| 600.000
| 7/5, 10/7
|-
| 1
| 230.966
| '''8/7'''
| 830.966
| '''8/5''', '''13/8'''
|-
| 2
| 461.932
| 21/16, 13/10
| 1061.932
| 11/6, 15/8, 24/13
|-
| 3
| 692.898
| '''3/2'''
| 92.898
| 15/14
|-
| 4
| 923.864
| 12/7, 22/13
| 323.864
| 6/5, 39/32
|-
| 5
| 1154.830
| 48/25, 63/32, 96/48
| 554.830
| '''11/8'''
|-
| 6
| 185.796
| 9/8
| 785.796
| 11/7
|-
| 7
| 416.762
| 9/7
| 1016.762
| 9/5
|-
| 8
| 647.728
| 36/25
| 47.728
| 33/32, 36/35
|}
<nowiki>*</nowiki> In 13-limit POTE tuning. This tuning is very close to [[26edo]].


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Revision as of 01:01, 4 August 2023

Juggernaut is a 5.7.11 nonoctave regular temperament, first documented by User:CompactStar, tempering out 125/121. 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. 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.

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.

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.

Intervals

# gens Period 0 Period 1
Cents* Approximate Ratios Cents Approximate Ratios
0 0.000 1/1 600.000 7/5, 10/7
1 230.966 8/7 830.966 8/5, 13/8
2 461.932 21/16, 13/10 1061.932 11/6, 15/8, 24/13
3 692.898 3/2 92.898 15/14
4 923.864 12/7, 22/13 323.864 6/5, 39/32
5 1154.830 48/25, 63/32, 96/48 554.830 11/8
6 185.796 9/8 785.796 11/7
7 416.762 9/7 1016.762 9/5
8 647.728 36/25 47.728 33/32, 36/35

* In 13-limit POTE tuning. This tuning is very close to 26edo.

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