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| I plan to make pages more accessible to beginners, and also create an introduction page for the wiki and xen in general | | I plan to make pages more accessible to beginners, and also create an introduction page for the wiki and xen in general |
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| I've created a map of regular temperaments, which I might put on the wiki soon
| | My xen compositions are on the Discord |
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| | eventually i will remember to sign my talk page entries with four tildes |
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| DRAFT BELOW
| | == Ongoing projects == |
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| In [[regular temperament theory]], a [[temperament]] (more specifically, its [[mapping]]) - displays '''contorsion''' if there are some interval which no [[just intonation]] interval maps to. If a temperament is contorted, there is some '''contorted generator''' where every just interval's mapping has a multiple of c of that generator, where c, the '''contortion order''', is greater than one. The largest contortion order is called the '''greatest factor'''.
| | [[User:Hkm/Intro page]] |
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| For example, the [[5-limit]] [[36edo|36et]] (with mapping [36 57 84]) uses 12 of its pitches per octave (the ones within [[12edo|12et]]) to map the entire 5-limit gamut. As a result, no 5-limit just intonation interval maps to any of the other 24 pitches, making 36et contorted in the 5-limit. Therefore there is a contorted generator; since there is only one generator of 36et (namely, the 36th-octave), that generator must be contorted. Every pitch is mapped to a number of [[Generator|generators]] that is a multiple of 3 (where the generator of 36et is a 36th of the octave), so this generator has contortion order 3. For a higher-rank example, the 3.5.7 restriction of [[Sensipent family#Septimal%20sensi|septimal sensi]], with mapping [⟨-1 -1 -2], ⟨7 9 13]] and mapping generators "~2", ~9/7, where "~2" is in quotes because it is no longer in the subgroup. This temperament has the interval corresponding to the period which no just intonation maps to. We can thus find a contorted generator; namely, [FIGURE THIS OUT]
| | == Written or rewritten pages == |
| | [[Saturation, torsion, and contorsion]] |
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| A temperament (more specifically, its [[comma basis]]) displays '''torsion''' if there is no mapping which describes it. If a temperament has torsion, it [[temper out|tempers out]] a ''power'' c of some ratio, the '''contorted comma''', but does not temper out that ratio, where c is the '''torsion order'''. For instance, in a temperament with comma basis {[[6561/6250]], [[128/125]]}, (81/80)^2 = (6561/6250)/(128/125) is tempered out but [[81/80]] is not explicitly tempered out. In this temperament, there is no clear way to assign a pitch to 81/80; for this reason, temperaments with torsion are not particularly useful. Similarly to the concept of contorsion order, torsion order can be defined as the lowest power of a generic just intonation interval that is necessarily part of the temperament's lattice.
| | [[Frequency]] |
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| A temperament is '''saturated''' if it shows neither torsion or contortion.
| | [[Harmonic timbre]] |
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| For a more detailed discussion on these issues, see [[Pathology of enfactoring|Pathology of saturation]].
| | [[Just intonation]] |
| ==Saturation algorithms==
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| An unsaturated mapping or comma basis can be made saturated, ensuring our ability to most simply—and thereby uniquely—identify temperaments using only matrices. This need can otherwise be satisfied using [[wedgie]]s. The simplest and fastest algorithm for saturating matrices is called [[column Hermite defactoring]]. For more information on such algorithms, see [[Defactoring algorithms|Saturation algorithms]].
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| ==History and terminology==
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| The term ''saturation'' was coined by {{w|Nicolas Bourbaki}} in 1972<ref>[https://pdfcoffee.com/commutative-algebra-bourbaki-pdf-free.html Nicolas Bourbaki. ''Commutative Algebra'']</ref>, working in the field of commutative algebra. It came to RTT via [[Gene Ward Smith]] and [[Graham Breed]]'s observations of the work of the mathematician {{w|William A. Stein|William Stein}} and his {{w|SageMath|Sage}} software<ref>It may also have come through PARI/GT<!-- typo of PARI/GP? -->.</ref>. The earliest identified terminology for this concept was in 1861 by {{w|Henry John Stephen Smith|H. J. S. Smith}}<ref>H. J. S. Smith is the creator of the {{w|Smith normal form}} used in [[Defactoring algorithms #Precedent: Smith defactoring|Gene Ward Smith's saturation algorithm]].</ref> who called saturated matrices "prime matrices"<ref>Also from ''On Systems of Linear Indeterminate Equations and Congruences'', linked above. Neither ''prime matrix'' nor ''greatest divisor'' seems to have caught on in the mathematical community.</ref>.
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| The term ''torsion'' has been used since at least as early as 1932<ref>[https://scholar.google.com/scholar?q=%22torsion+group%22&hl=en&as_sdt=0%2C5&as_ylo=1900&as_yhi=1940 Google Scholar: Torsion group]</ref><ref>[https://math.stackexchange.com/questions/300586/where-does-the-word-torsion-in-algebra-come-from Stack Exchange | ''Where does the word "torsion" in algebra come from?'']</ref> and came to RTT from the mathematical field of group theory. Historically, a group-theory formalism was used to analyze comma bases with torsion, where the smallest comma displaying torsion was not made to vanish although a power of that comma was, which is musically impossible; using a linear algebra formalism as is preferred now, no such impossibility is suggested. The term ''contorsion'' was invented for RTT in 2002 by [[Paul Erlich]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2456 Yahoo! Tuning Group | ''My top 5--for Paul'']</ref>, as a play on the word "co-torsion", being dual to the situation with "torsion" above.
| | [[List of xenharmonic music by community ratings]] |
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| [[Dave Keenan]] and [[Douglas Blumeyer]] have proposed '''''defactoring''''' as a replacement for ''saturation'', and '''''enfactoring''''' as a replacement for both ''torsion'' and ''contorsion''. So, a mapping or comma basis is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion). These terms are used in their writings. See [[Defactoring terminology proposal]] for details. | | [[Reduced mapping]] |
| ==References and footnotes==
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| <references />
| | [[11/10]] (mostly) |
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| | [[Path-based goodness]] |
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| | [[Fokker block]] (mostly) |
fan of 72edo and contorted edos
I plan to make pages more accessible to beginners, and also create an introduction page for the wiki and xen in general
My xen compositions are on the Discord
eventually i will remember to sign my talk page entries with four tildes
Ongoing projects
User:Hkm/Intro page
Written or rewritten pages
Saturation, torsion, and contorsion
Frequency
Harmonic timbre
Just intonation
List of xenharmonic music by community ratings
Reduced mapping
11/10 (mostly)
Path-based goodness
Fokker block (mostly)