User:Eufalesio/Ultimate: Difference between revisions
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is 41&53&217, with mapping {{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]]'' (technical info inside), also [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|here]]; but | is 41&53&217, with mapping {{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]]'' (technical info inside), also [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|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. | Special thanks for [[Kite Giedraitis]] for feedback and edits. | ||
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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. | 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 [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|here]]. | |||
=== Interval list === | === Interval list === | ||
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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. | 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 === | === Precision levels and usability === | ||
{{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) | {{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) | ||
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==== 41 ==== | ==== 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 | 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 ==== | ==== 53 ==== | ||
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=== Non-cassandra Ultimate === | === Non-cassandra Ultimate === | ||
They are very fine and likely impossible to implement into real instruments. They can be written with sharps and flats, ↑/'''↓''' for the pomas, ⇑/'''⇓''' for doubled pomas, and the addition of ^/v for the MC, 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". | They are very fine and likely impossible to implement into real instruments with an Ultimate layout. They can be written with sharps and flats, ↑/'''↓''' for the pomas, ⇑/'''⇓''' for doubled pomas, and the addition of ^/v for the MC, 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". | ||
==== 217 ==== | ==== 217 ==== | ||
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==== 270 ==== | ==== 270 ==== | ||
Peak. Just like 94edo, it allows half pomas ¡/! for easier navigation. 270edo could be approached with a 2D layout using [[ | 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 ==== | ==== 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! | 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'' === | ==== 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. | 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 94edo and 270edo == | == 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]]. | |||
It's possible to use Gariwizmic wholesale, though it only slightly improves 270edo in precision, | 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. | ||