User:Eufalesio/Ultimate
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is 41&53&217, otherwise known by in the wiki as cassaschismic (technical info inside), but I will simply call it Ultimate. My reasoning of this will become clear. Or at least, I expect you to understand why it's clear in my mind.
Quick definition
Ultimate can be easily defined in the 13-limit as tempering out the sinaisma, minisma, and eufalesma. This indirectly means that the salozo, tribilo, sathoyo, salururu, sasaru, lolotrizo commas, among an infinitude more, are all tempered out too.
Ultimate has the following notable equations:
- Apotome = 77/72 (salozoma t.o)
- 77/72 = 27/26 * 36/35 (eufalesma t.o)
- Poma = 64/63 (sasaruma t.o)
- 2 pomas = 33/32 (salururuma t.o)
- 36/35 = 1053/1024 (minisma t.o)
- 32/27 = 11/10 * 14/13 (sinaisma t.o)
et cetera...
Interval list
Here is a quick compressed cheat sheet of octave-reduced intervals. This is a MASSIVE simplification with many (infinitely many) intervals left out for the sake of brevity. Bolded ones are octave reduced harmonics or subharmonics.
| Fifths | Gary seq. | -comma |
|---|---|---|
| 0 | 1/1 | 720/361 |
| 1 | 3/2 | 256/171 |
| 2 | 9/8 | 64/57 |
| 3 | 27/16 | 32/19 |
| 4 | 81/64 | 24/19 |
| 5 | 243/128 | 36/19 |
| 6 | 729/512 | 64/45 |
| 7 | 77/72 | 16/15 |
| 8 | 77/48 | 8/5 |
| 9 | 77/64 | 6/5 |
| 10 | 65/36 | 9/5 |
| 11 | 65/48 | 27/20 |
| 12 | 64/63 | 81/80 |
| 13 | 32/21 | 243/160 |
| 14 | 8/7 | 729/640 |
| 15 | 12/7 | 416/243 |
| 16 | 9/7 | 104/81 |
| 17 | 27/14 | 52/27 |
| 18 | 81/56 | 13/9 |
| 19 | 88/81 | 13/12 |
| 20 | 44/27 | 13/8 |
| 21 | 11/9 | 39/32 |
| 22 | 11/6 | 64/35 |
| 23 | 11/8 | 48/35 |
| 24 | 33/32 | 36/35 |
Justification
The chain of fifths is a very important framework historically. It's been in Western music THE way to think about everything all the way from plainchant to Renaissance meantone temperaments to the modern day; where the 12-pitch-class circle of fifths is taught; 12edo, a massively overrepresented tuning. It has a bit of a bad reputation in the xen circles, but the more I researched, the more I realized it is a paragon, and that its position nowadays is very much well earned.
My main aim is to expand tonality with JI, and there is no better way to do so than to also extend the fundamental tuning framework to its logical conclusion.
12edo introduces the compton framework, which closes the chain of fifths with 12 flat, but proportionally very good fifths. Compton sensu stricto uses an independent generator to each all the different primes, and is generally a very good temperament. However, if the fifths are tuned sharper to become closer to just, the chain goes on for longer...
41edo introduces the cassandra framework, which thanks to its incredibly accurate fifths, introduces the poma as an accidental. In cassandra, the poma acts as 81/80~64/63 at around ~29c, and two of them add to 1053/1024~33/32~36/35. 41edo is a bit overtempered but notable as a proportionally coarse but good system, 53edo has practically pure fifths and very good p5 and p13, but p7 and p11 are worsely tuned.
94edo is in my opinion the optimal cassandra equal tuning with a poma of 25.5c, tempered just enough for the 13-limit to reach <4 cents of error and very easy to use.
However, if you forego p5 and p13 for the chain of fifths, you end up with gary. Gary is a serendipitious temperament, the same as cassandra but optimized for the zala. 135edo is an essentially perfect tuning, reaching sub-cent levels of error in the subgroup, with a poma of 26.6 c.
Ultimate is not just an extension of the concept, but what I believe to be the end of that extension. Ultimate adds an independent generator for p5 and p13, which acts as 385/384~352/351~5120/5103... etc, being around 4.4c; so I call it the "minicomma". It doesn't begin to make sense up until 217edo, but 270edo and 311edo are arguably the best tunings. 217edo is alright, but not the best, which would make it a waste of time if it weren't a multiple of 31edo.
The key reasons on why Ultimate is ultimate, is ultimately due to the fact that 270edo and 311edo are inside the supported equal tunings, because going any further would make the edosteps not discernible, and because no other edo in their vicinity is as good as them. Beyond that, I see no reason to use edos.
270edo and 311edo inherit a chain of fifths that is consistent with cassandra, which itself is an extension of the circle of fifths. The only adition is a single edostep, and respectively, the entire 13-limit is tuned to unfathomable precision, and the 41-limit is fully accessible and very well tuned. However, I prefer sticking to the 13-limit, so 270edo is an optimal equal tuning.
Precision levels
12e, 41, 53, 94, 217, 270, 311 are all part of the same rank-3 tuning, so it allows a piece or a production to be written using the notation, which encodes the same mappings. Of course, using the notation to its fullest extent only makes sense for the finer 217, 270, 311. This necessarily means that there are levels of precision to ultimate. (The notation ideas are heavily WIP)
12e
The coarsest tuning that makes sense. It can be written just with sharps/flats, since the poma and the minicomma are tempered out in all its possible expressions. 12e because patent val tunes 11/8 as a tritone, not fourth. The cassandra mapping is based on 11/8 as a kind of fourth, not tritone. Either way, p11 is NOT there. Consider it an extremely coarse yazatha tuning.
41
The coarsest cassandra tuning. It can be written with sharps/flats, plus ↑/↓ for the pomas. In the case of 41edo, there is no need for double pomas, because the apotome can be split in half. Thus, half sharps and half flats can be used instead of two pomas. ONLY in 41edo. Ideal for 11-limit pieces with acoustic instruments, like the well known Kite guitar, albeit, it is not a cassandra layout, but Skip fretting system 41 2 13. The cassandra layout is skip fretting system 41 3 7.
53
Another good cassandra tuning. Just like 41edo, It can be written with sharps/flats, plus ↑/↓ for the pomas. The poma can be doubled into ⇑/⇓ to reach p11 and p13. It is playable and around the extremum possible inside the Lumatone, which despite having a p7, p11 that are not too well tuned; it has good 13-limit capabilities. It can be used in a guitar with the skip freting system 53 4 9.
94
Best cassandra tuning. Just like 53edo, it can be written with sharps/flats, plus ↑/↓ for the pomas. The poma can be doubled into ⇑/⇓ to reach p11 and p13. Since the chain takes much longer to close, ¡/! may be used to raise by half-pomas. This tuning is optimal and technically usable in the Lumatone, but only as a subset, requiring more than one preset to reach within the Standard Lumatone mapping for Pythagorean. The cassandra layout can be used in a guitar with the skip fretting system 94 7 16. However, in a 6-string guitar there will be no other unisons.
217, 270, 311
They all work much the same way. They can be written with sharps/flats, ↑/↓ for the pomas, ⇑/⇓ for doubled pomas, and the adition of ^/v for the minicomma, taken directly from the ups-and-downs notation. This is completely unfeasible to use with a Lumatone, with any acoustic instrument, isomorphically; though, it can still be used in a DAW without much problem. Because ultimate is rank-3, the layout is 3D and thus it is impossible to play on a flat surface, requiring some sort of eldritch holographic "keyspace".
217edo could in theory be used with binary valve or key systems in woodwinds, granted they have the intonation precision to reliably hit pitches within a maximum error of 2.7649 cents. Which I know won't happen. Best course would be to tune the instrument to 31edo plus a slide to nudge everything into the right place, but that's not ultimate. That's birds.
Ultimate sensu stricto
It is possible to forego edos altogether and use Ultimate as is, providing the absolute best tuning possible. However, it's a rank-3 system. It has no extra unisons unlike the equal tunings. This is reserved for when 270edo is not just enough, and beating is something to avoid as much as possible.
The special place of 94edo and 270edo
Of all the equal tunings supported by Ultimate, the best ones are 94edo and 270edo. They have the key property of being even, and thus also tempers out the kalisma, allowing the poma to be split in halves. Using them this way is reminiscent of Gariwizmic, a very similar temperament to Ultimate, but with the minicomma found deep in the generator chain, not independent. This is useful for easier navigation within a DAW.
It's possible to use Gariwizmic wholesale, but I wouldn't recommend it. For that, Ultimate is a much better choice overall. Gariwizmic would only provide the structure, not the tuning.