User:Eufalesio/Ultimate: Difference between revisions
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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 remains 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. | 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 remains 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. Behold, ''the'' '' | 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. Behold, ''the'' ''Ultimate'' ''sequence''. | ||
* '''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... | * '''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... | ||
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* No other edo in their vicinity is as good as them. | * No other edo in their vicinity is as good as them. | ||
Beyond that, I see no reason to use edos. The sequence still continues beyond Ultimate | Beyond that, I see no reason to use edos. The sequence still continues beyond Ultimate, which I dub the [[User:Eufalesio/Ultimate#Beyond Ultimate - the ULTRAOLYMPIC sequence|ULTRAOLYMPIC sequence]]. | ||
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 the best 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 the best equal tuning. | ||
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==== 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 | It is possible to forgo edos altogether and use Ultimate as is, providing the absolute best tuning possible. However, it's a rank-3 temperament. 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 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 == | == 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 | 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; while more complex it can be extended to the 23-odd-limit; which could be useful to some. Not me. | ||
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]]. | 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 | 41edo is particularly interesting because joining it with 270edo results in [[newt]], an extremely accurate rank 2 microtemperament of Ultimate that is practically indistinguishable from it. Instead of halving the poma, it halves the fifth, finding the MC "generator" at -41 generators, which firmly places this as a 41edo [[well temperament]]. | ||
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94edo and 270edo | 94edo and 270edo are 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, though optimization ends here. | ||
It's possible to use | 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 good structure, not necessarily the best tuning. | ||
== Beyond Ultimate - the ULTRAOLYMPIC sequence == | == Beyond Ultimate - the ULTRAOLYMPIC sequence == | ||
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I'm aware temps can't have NSFW names. "Insanismic" is the SFW, clean term. The one for the front page. The ''correct'' mask. | I'm aware temps can't have NSFW names. "Insanismic" is the SFW, clean term. The one for the front page. The ''correct'' mask. | ||
But know: the ''real'' name is '''batshit.''' Because that's what this temperament is. Batshit insane! Yes, I like crude, evocative names. The English language has a beautiful expression for this special obsession that I will not '''not''' use. The unfathomable precision it bestows. The madness that ensues upon the search of perfection. | But know: the ''real'' name is '''batshit.''' Because that's what this temperament is. Batshit insane! Yes, I like crude, evocative names. The English language has a beautiful expression for this special obsession that I will not '''not''' use. The unfathomable precision it bestows. The madness that ensues upon the search of perfection. HERE is perfection! GO CRAZY!{{Databox|1=Temperament data|2=[[Subgroup]]: 2.3.5.7.11.13 | ||
[[Comma list]]: [[5767168/5767125]] | [[Comma list]]: [[5767168/5767125]] | ||
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==== Insatinismic AKA Batshittin ==== | ==== Insatinismic AKA Batshittin ==== | ||
The "Tinismic" part comes from it being a weak extension that notation-wise results in the addition of a tina accidental.{{Databox|1=Temperament data|2=[[Subgroup]]: 2.3.5.7.11.13.17.19 | The "Tinismic" part comes from it being a weak extension that notation-wise results in the addition of a tina accidental. {{Databox|1=Temperament data|2=[[Subgroup]]: 2.3.5.7.11.13.17.19 | ||
[[Comma list]]: 5767168/5767125, [[11413376/11413325]], [[633556/633555]] | [[Comma list]]: 5767168/5767125, [[11413376/11413325]], [[633556/633555]] | ||