User:Eufalesio/Ultimate

iis 41&53&217, with mapping [⟨1 0 0 25 -33 -13], ⟨0 1 0 -14 23 12], ⟨0 0 1 0 0 -1]]. It's otherwise known by in the wiki as cassaschismic (tad less rambly), also here; but we will simply call it Ultimate. Our reasoning of this will become clear. Or at least, we expect you to understand why it's clear in our mind.

Special thanks for Kite Giedraitis for feedback and edits.

Quick definition

Ultimate can be easily defined in the 13-limit as nullifying the sinaisma, minisma, and eufalesma. This indirectly means that the salozo, tribilo, sathoyo, salururu, sasaru, lolotrizo commas, among an infinitude more, are all nullified too.

Ultimate has the following notable equations:

• Apotome = 77/72 (salozoma tempered out)
• 77/72 = 27/26 * 36/35 (eufalesma tempered out)
• Poma = 64/63 (sasaruma tempered out)
• 2 pomas = 33/32 (salururuma tempered out)
• 36/35 = 1053/1024 (minisma tempered out)
• 32/27 = 11/10 * 14/13 (sinaisma tempered out)

et cetera...

The pergen is (P8, P5, ^1), where ^1 is the "minicomma" (from this point forward refered to as "MC"); a 3~5c interval that represents 385/384, 352/351, 5120/5103, 513/512, the layoma, etc. 4:5:6:7:9:11:13 is notated as P1 ^↓M3 P5 ↓m7 M9 ↑↑11 v↑↑m13. Using pomas (pythagorean commas, [↑/↓]) improves this notation for reasons that will be exposed later.

Some temperament properties of Ultimate and its subsets are exposed here.

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. For every entry here, ratios here represent pitch-classes and their pitch class inverses; so for instance 8/5 pitch class is mapped to +8 fifths -1 MC, being the octave inverse of 5/4 pitch class negates the mappings so it is found at -8 fifths + 1 MC. There are no octave reduced primes or prime inverses with positive fifth-span and MC-span.

	MC-span
Fifth-span	-1	0
0	351/176	1/1
1	256/171	3/2
2	64/57	9/8
3	32/19	27/16
4	24/19	19/15
5	36/19	19/10
6	27/19	57/40
7	16/15	77/72
8	8/5	77/48
9	6/5	77/64
10	9/5	65/36
11	27/20	65/48
12	81/80	64/63
13	38/25	32/21
14	57/50	8/7
15	77/45	12/7
16	77/60	9/7
17	52/27	27/14
18	13/9	81/56
19	13/12	88/81
20	13/8	44/27
21	39/32	11/9
22	64/35	11/6
23	48/35	11/8
24	36/35	33/32

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 over-represented 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 (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.

However, if you forgo p5 and p13 for the chain of fifths, you end up with gary. Gary is a serendipitous 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 "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.

Precision levels and usability

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 apotomes, since the poma and the MC 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. 24edo can't be used, as it uses a new mapping disjoint from the circle of fitfths and the poma is still tempered out.

Cassandra edos

They are the simplest of the bunch and the easiest to work with. They can be written with apotomes, ↑/↓ for the pomas reaching qualites of p5 and p7, and ⇑/⇓ for doubled pomas reaching qualities of p11 and p13.

41

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

53

Another good cassandra tuning. It is playable and around the extremum possible inside the Lumatone, which despite having a p7 and p11 that are not too well tuned (though its p7 is good enough for some); it has good 13-limit capabilities, and a pure fifth that's a gift to many. The cassandra layout can used in a guitar with the skip freting system 53 4 9.

94

Best cassandra tuning. Since the chain takes much longer to close, ¡ and ! may be used to raise or lower by a 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.

Non-cassandra Ultimate

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, ⇑/⇓ for doubled pomas, and the addition of ^/v for the MC, taken directly from the 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".

217

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 2.76 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.

270

Peak. Just like 94edo, it allows half pomas ¡/! for easier navigation. 270edo could be approached with a 2D layout using decoid layout, though this is hard to do on the Lumatone, having octaves that are very far apart. Just as 217edo, it can be approached using a subset, namely 27edo, but that's not Ultimate. That's ennealimmal.

311

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.

Ultimate sensu stricto

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.

The special place of 41edo, 94edo and 270edo

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.

270edo is well known for its unbeatable 13-limit, for which, arguably, no other edo finer or coarser comes even close to its ratio of accuracy to "simplicity". It also technically has some useful interpretations for up to the 53-limit which could be even more useful than that of 311edo, as seen by people like Godtone.

41edo is particularly interesting because joining it with 270edo results in newt, an extremely accurate rank 2 subset temperament of Ultimate that is practically indistinguishable from it. Instead of halving the poma, it halves the fifth, finding the MC "generator" at -41 gens, which firmly places this as a 41edo well temperament.

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.

Ultimate still not enough?

There's a reason why I deem Ultimate ultimate. You're supposed to end there and go no further, because Ultimate is right at the limit of practicality. If you're stubborn enough to ignore my warnings and venture into the land of impossible... I still know how you can continue following the same path of reasoning, and add extra pairs of accidentals that are fully retrocompatible.

Olympic

The best course of action for detempering Ultimate is to observe the garischisma, resulting in rank-4 olympic, which is just {S64, S65}. Notationwise, this results in spliting the saruyoma into a schisma (◿/◹) and a garischisma (◺/◸). Unlike in JI, where the schisma is around half the garischisma, here the schisma is ~1.6x LARGER, not smaller.

The simplest choice is to make ↑ 64/63; the poma is ◸↑ now, or if you want to be sleek, p.

Everything else is the same. v↑ for 81/80, ⇑ for 33/32, v⇑ for 1053/1024. It'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.

📝Note:

Find good extensions to the 19-limit.

Insanic

Olympic STILL not enough? Split your losses and use {S64/S65}. Now you observe the olympia and get a schismina accidental. An olympia is 3 of these schisminas. You could write this as dots above or below the accidentals but this is possibly getting a tad crowded. ↑ is 64/63 and 81/80 is v↑. ⇑̱ is 33/32 whilst ⇑ is 4096/3969. v̈⇑ is 1053/1024 whilst v⇑ is 36/35. {S64/S65} is at a level of precision comparable to 8539edo and much finer. The people at sagittal.org had already declared its own version of this notation to be of "Insane" precision, so if you need anything finer, you are beyond insane.

📝Note:

Find good extensions to the whatever limit.

To recap: 12edo (apotomes), Cassandra (+pomas), Ultimate (+saruyomas), Olympic (+sasarumas), {S64/S65} (+schisminas).

Nomenclature and notation

This notation can be easily spoken as well as written, adapting Kite's color notation and ups and downs into a nice collage. At least, the part of Ultimate. What's beyond Ultimate, not my problem.

Easy tables

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 (if possible), 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.

Main accidentals	Ultimate							Beyond Ultimate
	Natural	Apotome		Poma/Ruma		Saruyoma		Sasaruma		Schismina
Symbols				↑	↓	^	v	◺	◸	Ȯ Ö Ō	Ọ O̤ O̱
				⇑	⇓	M	W
Spoken	natural	sharp	flat	po	qu	up	down	Color notation, or some other JI phonetic coding
		–	–	pop(o)	quq(u)	dup	dud	7-limit JI		11-limit JI

The following accidentals are not part of the main bunch because they are either only part of rank-2s and edos or ligatures.

Extra accidentals	Fwiwisma		Half-apotome		Non-gary poma*
Symbols	¡	!			p	q
					pp	qq
Spoken	halfpo	halfqu	halfsharp	halfflat	po	qu
			sesquisharp	sesquiflat	pop(o)	quq(u)

*Not necessary in Ultimate. This can prevent filling the page with garischismas. Logically, here the arrow is spoken ru/zo like in color notation, while the slashed arrow is po/qu.

Syntax

• For an accidental: (schismina) + (sasaruma) + [saruyoma] + [poma] + [apotome] v# = downsharp, M⇓ = dupquq
• For intervals: [accidental] + [5L 7s 6|5 nominal] ↓M2, ^↓M3, v⇑m6; qumajor second, upqumajor third, downpopominor sixth
• For pitches: [5L 7s 6|5 nominal] + [accidental] + [octave number] Example: D4, E⇑b4, F^↓#5; dee four, e popoflat four, ef upqusharp five