User:Eufalesio/Ultimate: Difference between revisions

* '''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 (prime 5) and p13 (prime 13), but p7 and p11 are tuned worse.

* '''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.

* '''Ultimate''' is not just an extension of the concept, but what I believe to be the '''end''' of that extension. Ultimate adds an independent "minicomma" generator for p5 and p13, which acts as 385/384, ~352/351, ~5120/5103, layoma, ... etc, being around 4.4c. It doesn't begin to be fully useful 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. [[176edo]] also supports it earlier at full structure, but it kinda blows as an equal tuning because the MC is too wide.

'''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 addition 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.

{{EDOs|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)

The coarsest true cassandra tuning. 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. This can '''ONLY''' be done in 41edo. Ideal for 11-limit pieces with acoustic instruments, like the well known [[Kite guitar]], albeit, it follows a [[magic]] layout: [[Skip fretting system 41 2 13]]. The cassandra layout is [[skip fretting system 41 3 7]]. Kite has expressed great passion on this tuning, thanks to its very manageable grain and still decently

Best cassandra tuning. Since the chain takes much longer to close, ¡ and ! may be used to raise or lower by a [[2835/2816|fwiwisma]] (a half-poma, as a result of tempering out kalisma and thus an even-numbered edo). 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.

Note that 94edo is already quite fine for most real instruments, and though its step is very much discernible, human error can begin to slip pitches into the wrong edostep even in very skilled musicians.

They are very fine and likely impossible to implement into real instruments with an Ultimate layout. They can be written with apotomes, ↑/'''↓''' for the pomas (qualities of p7), ⇈/'''⇊''' for doubled pomas (qualities of p11), and the addition of /vv for the MC (for qualities of p5, p13, p17, p19 aside from pomas) taken directly from the [[Kite's ups and downs notation|ups-and-downs notation.]] This is completely unfeasible to use with a Lumatone or with any acoustic instrument. 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".

311edo is well known for its 41-odd-limit consistency, though it is right at the edge of practicality. Using equal tunings finer than this is hard to justify. Its yazalathana is a smidge worse than 270edo, but its natural improvement in all other primes and most importantly prime 23 could be of some use to someone. Not me! Because it is prime, there are no subsets, though a [[vavoom]] layout can be used to approach it in 2D.

It is possible to forgo 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 most error you'll get with this system resides in the chain of fifths (~+0.25c), having all other primes accurate to hundreds of a cent. This is in a sense is reminiscent of [[septimal meantone]], which can tune p5 and p7 near-pure by adding error to the fifth chain.

41edo is the coarsest cassandra edo, with a high ratio of accuracy to simplicity, and being the first ever edo to be distinctly consistent in the 9-odd-limit, making the most out of the next convergent chain of fifths.

 

94edo is arguably the best cassandra edo, making the most out of the chain of fifths, which though more complex can be extended to the entire 23-odd-limit; which could be useful to some.

94edo and 270edo have the key property of being even, tempering out the [[kalisma]] and allowing the poma to be halved. Using them this way is reminiscent of [[Gariwizmic]], a very similar subset of Ultimate, but with the MC found deep in the generator chain, not independent. This is useful for easier navigation within a DAW. It's possible to use Gariwizmic wholesale, though it only slightly improves 270edo in precision, Newt is a much better choice for accuracy's sake, though 94edo does ''not'' support it. Gariwizmic provides structure, not the tuning.

t's something I don't think I'll ever see myself doing because this accuracy is enough to represent highly fine edos such as 494, 764, or 935edo, which is already too much for me.

 

Extensions to the 19-limit are bad: the strong one keeps 1729/1728 and 1216/1215, [[Catalog of rank-4 temperaments#Metaolympic|'''Meta'''olympic]] works, and though it's a weak extension; it can still be written retrocompatibly. Acknowledgements to [[Flora Canou]] for this great insight. 

{S64/S65} is at a level of precision comparable to 8539edo and much, MUCH finer. The people at sagittal.org had already declared its own version of this notation to be of "Insane" precision. Personally, ''insane'' is way too nice of a descriptor. This is is beyond your grasp of perfection.  

Wouldn't I be also batshit insane by having researched it? No. I know I won't use it. But, knowing about it is important; how much '''JI'''uice I can squeeze out of 12edo... taking it to insanity? Well, here is the answer!

Look how goddamn accurate this temp is! Look, I even made a technical temp data section! [expandable]  

 

Many thanks to Flora Canou for help with all the temp stuff... she's the real heroine here behind the numbers.  

[[File:ULTRAOLYMPIANsymbolset.jpg|center|thumb|960x960px|The name of this notation is henceforth the ULTRAOLYMPIC notation, and the main 4 systems.]]

These are all the accidentals you need to know to write in Ultimate, and even beyond it. For most cases, there is no need to go beyond two of anything, in the case of pomas however, you can end up using three or even more if you don't respell enharmonically with buzzardsmas, or go too far down the chain of fifths. One instance is writing 5/4 above a 16/11, which is 20/11. Above a C, this is '''⇊'''G - ^'''↓⇊'''B. The phonetic coding presented and the alterations are