User:2^67-1/Sandbox: Difference between revisions
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add approximation to JIP that looks interesting (its gens are plausible at least despite being fairly complex as a rank 3 temp, and there's an obvious extension to prime 13) |
Why do I keep adding more temperaments?? |
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| 0 || 2.5 = 5/2 || [[Slendric schisma|Slendrismic]] || [[Slendric schisma|68719476736/68641485507]] || {{monzo| 36 -5 0 -10 }} || Y | | 0 || 2.5 = 5/2 || [[Slendric schisma|Slendrismic]] || [[Slendric schisma|68719476736/68641485507]] || {{monzo| 36 -5 0 -10 }} || Y | ||
|- | |- | ||
| 0 || 2.6 = 13/5 || [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=on&weights=weil&target=&edos=&commas=4988891938119860195948410209%2F4951760157141521099596496896&submit_comma=submit 5 & 171] || [[ | | 0 || 2.6 = 13/5 || 2.3.7 [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=on&weights=weil&target=&edos=&commas=4988891938119860195948410209%2F4951760157141521099596496896&submit_comma=submit 5 & 171] || [[4988891938119860195948410209/4951760157141521099596496896|(28 digits)]] || {{monzo| -92 12 0 26 }} || Y | ||
|- | |- | ||
| 0 || 2. | | 0 || 2.{{overline|6}} = 8/3 || 5 & 212 || 72680419155717387/72057594037927936 || {{monzo| -56 7 0 16 }} || n | ||
|- | |- | ||
| 0 || 3 || [[Slendric]] (squared) || [[1029/1024|1058841/1048576]] || {{monzo| -20 2 0 6 }} || n | | 0 || 3 || [[Slendric]] (squared) || [[1029/1024|1058841/1048576]] || {{monzo| -20 2 0 6 }} || n | ||
| Line 52: | Line 52: | ||
| 1.25 = 5/4 || 0 || [[Quintosec]] || 140737488355328/140126044921875 || {{monzo| 47 -15 -10 0 }} || n | | 1.25 = 5/4 || 0 || [[Quintosec]] || 140737488355328/140126044921875 || {{monzo| 47 -15 -10 0 }} || n | ||
|- | |- | ||
| 1. | | 1.3 = 13/10 || 0 || 2.3.5 [https://sintel.pythonanywhere.com/result?subgroup=5&reduce=on&weights=weil&target=&edos=&commas=%5B119+-37+-26%3E&submit_comma=submit 10 & 171] || [[670975306467707996070384979248046875/664613997892457936451903530140172288|(36 digits)]] || {{monzo| -119 37 26 0 }} || n | ||
|- | |||
| 1.{{overline|3}} = 4/3 || 0 || [[Marvel temperaments#Submajor|Submajor]] || 69198046875/68719476736 || {{monzo| -36 11 8 }} || Y | |||
|- | |- | ||
| 1.5 = 3/2 || 0 || 2.3.5 [[Miracle]] || [[34171875/33554432]] || {{monzo| -25 7 6 }} || Y | | 1.5 = 3/2 || 0 || 2.3.5 [[Miracle]] || [[34171875/33554432]] || {{monzo| -25 7 6 }} || Y | ||
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| 1 || 1 || [[Mirwomo temperaments|Mirwomo]] || [[33075/32768]] || {{monzo| 15 -3 -2 -2 }} || n | | 1 || 1 || [[Mirwomo temperaments|Mirwomo]] || [[33075/32768]] || {{monzo| 15 -3 -2 -2 }} || n | ||
|- | |- | ||
| 0.{{overline|857142}} || 6/7 || [https://sintel.pythonanywhere.com/result?subgroup=7&reduce=off&weights=weil&target=&edos=&commas=318130560585842112604248046875%2F316912650057057350374175801344&submit_comma=submit 10 & 77 & 308] || [[318130560585842112604248046875/31691265005705735037417580134|(30 digits)]] || {{monzo| -98 23 12 12 }} || Y | | 0.{{overline|857142}} = 6/7 || 0.{{overline|857142}} = 6/7 || [https://sintel.pythonanywhere.com/result?subgroup=7&reduce=off&weights=weil&target=&edos=&commas=318130560585842112604248046875%2F316912650057057350374175801344&submit_comma=submit 10 & 77 & 308] || [[318130560585842112604248046875/31691265005705735037417580134|(30 digits)]] || {{monzo| -98 23 12 12 }} || Y | ||
|- | |- | ||
| 1 || ∞ || [[Jubilic]] || [[50/49]] || {{monzo| 1 0 2 -2 }} || n | | 1 || ∞ || [[Jubilic]] || [[50/49]] || {{monzo| 1 0 2 -2 }} || n | ||
Revision as of 01:13, 26 July 2024
Testing
The blackwood-dicot-semaphore equivalence continuum is a continuum of 7-limit rank-3 temperaments describing the set of all 7-limit rank-3 temperaments supported by 10edo. Any rank-2 temperament supported by 10edo can thus be represented by a line between two points in this continuum.
All temperaments in the continuum satisfy (25/24)p(49/48)q ~ 256/243, equating a stack of dicot commas (25/24) and semaphore commas (49/48) with the blackwood comma (256/243).
The blackwood comma is the characteristic 3-limit comma tempered out in 10edo.
User:Godtone notes the following JIP's, each one corresponding to a 1D continua contained therein, for which increasingly efficient approximations generally represents increasingly efficient 7-limit temperaments:
- log2(256/243) / log2(25/24 * 49/48) = 0.8482245109 ; this is the JIP of p=1, q=1 (equiv. to p=-1, q=-1)
- log2(256/243) / log2(25/24) = 1.2766647429 ; this is the JIP of p=1, q=0 (equiv. to p=-1, q=0)
- log2(256/243) / log2(50/49) = 2.5796543166 ; this is the JIP of p=1, q=-1 (equiv. to p=-1, q=1)
- log2(256/243) / log2(2401/2400) = 125.1... ; this is the JIP of p=-1, q=2 (equiv. to p=1, q=-2)
- log2(256/243) / log2(49/48) = 2.5275365063 ; this is the JIP of p=0, q=1 (equiv. to p=0, q=-1)
- log2(256/243) / log2(25/24 * 50/49) = 0.8540148427 ; this is the JIP of p=2, q=-1 (equiv. to p=-2, q=1)
Importantly, each JIP corresponds to a rational, so that, for example, (p, q) = (1, -2) is equivalent to (p, q) = (2, -4) and to (p, q) = (-1, 2) but not to (1, 2). Note that all these JIPs lie on the JIL (just intonation line).
Also note that continua separated by 2401/2400 are meaningfully different, but due to the efficiency of 2401/2400, one may want to examine the continuum of all 7-limit temperaments supported by 10edo for which 2401/2400 is tempered.
| p | q | Temperament | Comma | Added by someone else? | |
|---|---|---|---|---|---|
| Ratio | Monzo | ||||
| 0 | 0 | Blackwood | 256/243 | [8 -5 0 0⟩ | n |
| 0 | 1 | Archytas (squared) | 4096/3969 | [12 -4 0 -2⟩ | n |
| 0 | 2 | Buzzard | 65536/64827 | [16 -3 0 -4⟩ | n |
| 0 | 2.5 = 5/2 | Slendrismic | 68719476736/68641485507 | [36 -5 0 -10⟩ | Y |
| 0 | 2.6 = 13/5 | 2.3.7 5 & 171 | (28 digits) | [-92 12 0 26⟩ | Y |
| 0 | 2.6 = 8/3 | 5 & 212 | 72680419155717387/72057594037927936 | [-56 7 0 16⟩ | n |
| 0 | 3 | Slendric (squared) | 1058841/1048576 | [-20 2 0 6⟩ | n |
| 0 | ∞ | Semaphore | 49/48 | [-4 -1 0 2⟩ | n |
| 1 | 0 | Srutal | 2048/2025 | [11 -4 -2 0⟩ | n |
| 1.25 = 5/4 | 0 | Quintosec | 140737488355328/140126044921875 | [47 -15 -10 0⟩ | n |
| 1.3 = 13/10 | 0 | 2.3.5 10 & 171 | (36 digits) | [-119 37 26 0⟩ | n |
| 1.3 = 4/3 | 0 | Submajor | 69198046875/68719476736 | [-36 11 8⟩ | Y |
| 1.5 = 3/2 | 0 | 2.3.5 Miracle | 34171875/33554432 | [-25 7 6⟩ | Y |
| 2 | 0 | Negri | 16875/16384 | [-14 3 4 0⟩ | n |
| ∞ | 0 | Dicot | 25/24 | [-3 -1 2 0⟩ | n |
| 1 | 1 | Mirwomo | 33075/32768 | [15 -3 -2 -2⟩ | n |
| 0.857142 = 6/7 | 0.857142 = 6/7 | 10 & 77 & 308 | (30 digits) | [-98 23 12 12⟩ | Y |
| 1 | ∞ | Jubilic | 50/49 | [1 0 2 -2⟩ | n |
| ∞ | 1 | Jubilic | 50/49 | [1 0 2 -2⟩ | n |
| ∞ | 2 | Breedsmic | 2401/2400 | [-5 -1 -2 4⟩ | n |