Xenharmonic Wiki talk:Things to do: Difference between revisions
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→13-Limit, 17-Limit and 19-Limit Comma Pages: important note on my old suggestion about temperament organisation |
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More generally, if the IUS is tempered from rank n to rank k < n through tempering (n - k) commas, then: | More generally, if the IUS is tempered from rank n to rank k < n through tempering (n - k) commas, then: | ||
If the tempered result is rank 1, it is an equal temperament. | If the tempered result is rank 1, it is an equal temperament. | ||
If the tempered result is rank 2, it is | If the tempered result is rank 2, it is a temperament. | ||
If it is rank 3, it is a clan or family, alternatively named a planar temperament if using or considering it with no further tempering is desired. | If it is rank 3, it is a clan or family, alternatively named a planar temperament if using or considering it with no further tempering is desired. | ||
If it is rank 4, it is a superfamily. | If it is rank 4, it is a superfamily. | ||
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Rank 8 - the IUS (initial untempered subgroup). | Rank 8 - the IUS (initial untempered subgroup). | ||
</pre> | </pre> | ||
Note: A partial reason I chose "Metafamilies" to correspond to 3 commas (and thus "Metafamily temperaments" to 4) is to suggest focusing on tempering at least 3 or 4 commas in high rank situations so that we don't just get a whole bunch of 1 or 2 comma temperaments which are comparatively uninteresting and of which, at that scale, there will inevitably be (perhaps far too) many. This also has the neat property of meaning that Metafamily temperaments in the 19-prime-limit are "cubic" in analogy to how rank 3 temperaments are "planar", | Note: A partial reason I chose "Metafamilies" to correspond to 3 commas (and thus "Metafamily temperaments" to 4) is to suggest focusing on tempering at least 3 or 4 commas in high rank situations so that we don't just get a whole bunch of 1 or 2 comma temperaments which are comparatively uninteresting and of which, at that scale, there will inevitably be (perhaps far too) many. This also has the neat property of meaning that Metafamily temperaments in the 19-prime-limit are "cubic" in analogy to how rank 3 temperaments are "planar", and that Metafamily temperaments in the 17-prime-limit are "planar temperaments". Plus I like the fact that "Metafamily temperaments" are either equated or closely connected to Superfamilies and/or Ultrafamilies depending on how high of a rank your IUS is, further justifying the aesthetic choice of "meta" (at least IMO).<br/> | ||
'''[ End of suggestion/reply by''' [[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 23:06, 15 January 2021 (UTC)'''. ]'''<br/><br/> | '''[ End of suggestion/reply by''' [[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 23:06, 15 January 2021 (UTC)'''. ]'''<br/> | ||
In hindsight this suggestion of mine is suggesting to redefine the word "temperament" but considering terms such as "planar temperament" and "linear temperament" I think the confusion is understandable, plus I still think the spirit of the suggestion is still relevant and if anything more relevant than before considering temperaments describing complete or near-complete harmonic series maps for the purpose of making constant structure harmonic series scales will be very high limit by default. Also I think the most important/least aged insight here is merging the information for temperaments that are dependent strongly on a single comma into the page for that comma, where it should logically belong; why talk of a comma if you don't talk of tempering it? If you don't temper it, then temperaments are either irrelevant, or you can suggest other commas to equate it with and thereby link to corresponding temperaments. In other words, every interesting small comma should ideally have some information on consequences and relation to temperaments in a standard "Temperaments" subsection. Information about how it relates to other intervals (the structure of JI) would '''not''' go under the Temperaments subsection. | |||
For example, for [[676/675]] it is notable that it is a [[square superparticular]] (its numerator is 26<sup>2</sup> and its denominator one less) and also a ratio of the [[square superparticular]]s S13 = [[169/168]] and S15 = [[225/224]] which gives a mathematical explanation of why tempering [[676/675]] means tempering ([[4/3]])/([[15/13]])<sup>2</sup>. (For an explanation of why check the "semiparticular" section of [[square superparticular]]s.) It suggests that tempering {S13, S15} is an efficient simple temperament that tempers S26. If we then notice another pattern explained on that page, we can temper {S25, S26, S27} which implies tempering both S13 and S15 by multiple coincidences. Specifically, S25*S26*S27 = S15 (a 1/3-square-particular; guaranteed to be superparticular in this case but a coincidence that it is also a square superparticular) and S25*S26*S26*S27 = S13 (note that 26/2 = 13 which is how this general pattern works). This all together suggests that the no-11's 13-limit [[catakleismic]] temperament '''2.3.5.7.13[19 & 53] = {S13, S15, S25, S26, S27}''' is very efficient because many of the commas imply each-other and thus this description's multiple redundancieds shows how naturally these commas pair together. | |||
'''[ End of comment/refinement by''' [[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 21:41, 29 September 2022 (UTC)'''. ]''' | |||
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: Allow me to open a giant can of worms. :) I feel pretty strongly that a rank-2 "family" should only apply to strong extensions (meaning the pergen doesn't change), and weak extensions should not be part of a family. For example, tempering out both 81/80 and 243/242 from 2.3.5.11 gives you a temperament that I wouldn't call a member of the meantone family, because it has a fundamentally different structure. In pergen terms, it's (P8, P5/2) not (P8, P5). I would call this temperament a member of the Neutral/Lulu family instead. In other words, 243/242 is the comma that identifies/defines the family, not 81/80, even though the latter is 5-limit and the former is 11-limit. This suits my personal preference not to elevate prime 5 too much above primes 7, 11, etc. Now this is *not* how family has been used in the past, and much of the xenwiki is based on including weak extensions, and having the comma with the lowest prime limit be the one to identify the family. So maybe this is a futile request. But think about it -- you usually can't translate a piece in Neutral to Meantone. Half the notes simply aren't there. Thoughts? | : Allow me to open a giant can of worms. :) I feel pretty strongly that a rank-2 "family" should only apply to strong extensions (meaning the pergen doesn't change), and weak extensions should not be part of a family. For example, tempering out both 81/80 and 243/242 from 2.3.5.11 gives you a temperament that I wouldn't call a member of the meantone family, because it has a fundamentally different structure. In pergen terms, it's (P8, P5/2) not (P8, P5). I would call this temperament a member of the Neutral/Lulu family instead. In other words, 243/242 is the comma that identifies/defines the family, not 81/80, even though the latter is 5-limit and the former is 11-limit. This suits my personal preference not to elevate prime 5 too much above primes 7, 11, etc. Now this is *not* how family has been used in the past, and much of the xenwiki is based on including weak extensions, and having the comma with the lowest prime limit be the one to identify the family. So maybe this is a futile request. But think about it -- you usually can't translate a piece in Neutral to Meantone. Half the notes simply aren't there. Thoughts? | ||