Xen concepts for beginners: Difference between revisions
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The notation ''m''\''n'' means ''m'' steps of ''n''-edo. For example, 12edo's perfect fifth can be denoted as 7\12, meaning "7 steps of 12-tone equal temperament". | The notation ''m''\''n'' means ''m'' steps of ''n''-edo. For example, 12edo's perfect fifth can be denoted as 7\12, meaning "7 steps of 12-tone equal temperament". | ||
A common operation in xen math is the [[mediant]]. The mediant of two fractions, ''a''/''b'' and ''c''/''d'', is the "freshman sum" | A common operation in xen math is the [[mediant]]. The mediant of two fractions, ''a''/''b'' and ''c''/''d'', is the "freshman sum" {{sfrac|''a'' + ''b''|''c'' + ''d''}}, which is always between ''a''/''b'' and ''c''/''d''. For example, the mediant of 4/3, the just perfect fourth, and 5/4, the just major third, is 9/7, the supermajor third. If two fractions are in lowest terms, their mediant is the simplest fraction that is strictly between both. The mediant is commonly used for both JI ratios and edo intervals. | ||
Another important operation is [[octave reduction|reduction]]. To reduce an interval a by an interval b means to stack or "unstack" ''b'' from ''a'' until ''a'' is at least the unison and less than ''b''. For example, 3/1 reduced by 2/1 is 3/2. | Another important operation is [[octave reduction|reduction]]. To reduce an interval a by an interval b means to stack or "unstack" ''b'' from ''a'' until ''a'' is at least the unison and less than ''b''. For example, 3/1 reduced by 2/1 is 3/2. | ||
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There are two ways the term ''[[limit]]'' is used. | There are two ways the term ''[[limit]]'' is used. | ||
* The ''[[harmonic limit|p-prime-limit]]'' is the lattice built by multiplying the primes at most ''p'', possibly multiple times. We write a JI lattice by writing the basic intervals separated by periods. For example, 6/5 = 2 × 3 / 5 is in the 5-limit, or the 2.3.5 subgroup, and so is 45/32 = (3 × 3 × 5) / (2 × 2 × 2 × 2 × 2). | * The ''[[harmonic limit|p-prime-limit]]'' is the lattice built by multiplying the primes at most ''p'', possibly multiple times. We write a JI lattice by writing the basic intervals separated by periods. For example, {{nowrap|6/5 {{=}} 2 × 3 / 5}} is in the 5-limit, or the 2.3.5 subgroup, and so is {{nowrap|45/32 {{=}} (3 × 3 × 5) / (2 × 2 × 2 × 2 × 2)}}. | ||
* The ''[[odd limit|q-odd-limit]]'' is the set of all JI ratios where the larger of the numerator and denominator after removing factors of 2 from the JI ratios is at most the odd number ''q''. For example, 7/6, 13/5, 11/10, and 16/15 are all in the 15-odd-limit. | * The ''[[odd limit|q-odd-limit]]'' is the set of all JI ratios where the larger of the numerator and denominator after removing factors of 2 from the JI ratios is at most the odd number ''q''. For example, 7/6, 13/5, 11/10, and 16/15 are all in the 15-odd-limit. | ||
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RTT views edos as regular temperaments. Under this view, edos simplify the infinite JI space to a finite set, deforming the intervals so that certain chosen intervals vanish. We can also approach simplifying JI ratios from edos themselves, namely how edos approximate each prime. This is a vector called a [[val]]. Vals map primes to a set number of edo steps and thus tell us how many edo steps each interval in JI is mapped to. The usual 12edo val (called the 12edo [[patent val]]) in the 5-limit is {{val| 12 19 28 }}, as the 12edo intervals that are closest to 2/1, 3/1 and 5/1 are 12, 19 and 28 steps respectively. | RTT views edos as regular temperaments. Under this view, edos simplify the infinite JI space to a finite set, deforming the intervals so that certain chosen intervals vanish. We can also approach simplifying JI ratios from edos themselves, namely how edos approximate each prime. This is a vector called a [[val]]. Vals map primes to a set number of edo steps and thus tell us how many edo steps each interval in JI is mapped to. The usual 12edo val (called the 12edo [[patent val]]) in the 5-limit is {{val| 12 19 28 }}, as the 12edo intervals that are closest to 2/1, 3/1 and 5/1 are 12, 19 and 28 steps respectively. | ||
There are various temperaments in xen with varying levels of practicality. The most important one to know is probably [[meantone]] temperament, which equates four fifths ((3/2)<sup>4</sup> = 81/16) with a major third plus two octaves ((5/4) × 4 = 5 = 80/16), which is encoded by tempering out the syntonic comma [[81/80]] (monzo {{monzo| -4 4 -1 }}). | There are various temperaments in xen with varying levels of practicality. The most important one to know is probably [[meantone]] temperament, which equates four fifths ({{nowrap|(3/2)<sup>4</sup> {{=}} 81/16}}) with a major third plus two octaves ({{nowrap|(5/4) × 4 = 5 {{=}} 80/16}}), which is encoded by tempering out the syntonic comma [[81/80]] (monzo {{monzo| -4 4 -1 }}). | ||
A val tempers out a comma if, when you construct the comma from primes according to their tunings in the val, the result is 0 cents or the unison. For example, 12edo is a meantone edo because: | A val tempers out a comma if, when you construct the comma from primes according to their tunings in the val, the result is 0 cents or the unison. For example, 12edo is a meantone edo because: | ||
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* The comma 81/80 has monzo {{monzo| -4 4 -1 }}. | * The comma 81/80 has monzo {{monzo| -4 4 -1 }}. | ||
* Constructing the tuning of a comma from mappings of primes involves multiplying each entry in the val to a corresponding entry in the comma's monzo, and then adding the resulting numbers together; this operation is called a "dot product". | * Constructing the tuning of a comma from mappings of primes involves multiplying each entry in the val to a corresponding entry in the comma's monzo, and then adding the resulting numbers together; this operation is called a "dot product". | ||
** 12 × ( | ** {{nowrap|12 × (−4) {{=}} −48}}, corresponding to going down 4 octaves. | ||
** 19 × 4 = 76, corresponding to going up 4 perfect twelfths (or, to going up 4 octaves and 4 fifths). | ** {{nowrap|19 × 4 {{=}} 76}}, corresponding to going up 4 perfect twelfths (or, to going up 4 octaves and 4 fifths). | ||
** 28 × ( | ** {{nowrap|28 × (−1) = −28}}, corresponding to dividing by 5 (going down two octaves and a major third). | ||
** (76 | ** {{nowrap|(76 − 48) − 28 {{=}} 0}} | ||
* Since the result is 0, 12edo supports meantone. In RTT math, this can be written as: | * Since the result is 0, 12edo supports meantone. In RTT math, this can be written as: | ||
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An important way that mos scales vary is [[hardness]], defined as the size (in cents) of the L divided by the size (in cents) of the s step. Hardness can range from 1 to infinity. The larger the hardness, the harder the mos tuning; the smaller (closer to 1) the hardness, the softer the tuning. The two extremes are where the mos pattern no longer holds; 1 is where L and s steps are equal, and infinity is where s is so small that it disappears. | An important way that mos scales vary is [[hardness]], defined as the size (in cents) of the L divided by the size (in cents) of the s step. Hardness can range from 1 to infinity. The larger the hardness, the harder the mos tuning; the smaller (closer to 1) the hardness, the softer the tuning. The two extremes are where the mos pattern no longer holds; 1 is where L and s steps are equal, and infinity is where s is so small that it disappears. | ||
Any given mos pattern is available in more than one edo, and the basic tuning of a mos pattern gives the smallest edo that provides that mos pattern. To adjust the hardness of a mos provided by an edo, we can add two edos, obtaining an edo where the hardness is the mediant of the two original edos'. For a diatonic example, 12edo has basic (L/s = 2/1) diatonic, 17edo has hard (L/s = 3/1) diatonic, and 19edo has soft (L/s = 3/2) diatonic. 12 + 19 = 31, and 31edo diatonic has hardness | Any given mos pattern is available in more than one edo, and the basic tuning of a mos pattern gives the smallest edo that provides that mos pattern. To adjust the hardness of a mos provided by an edo, we can add two edos, obtaining an edo where the hardness is the mediant of the two original edos'. For a diatonic example, 12edo has basic ({{nowrap|L/s {{=}} 2/1}}) diatonic, 17edo has hard ({{nowrap|L/s {{=}} 3/1}}) diatonic, and 19edo has soft ({{nowrap|L/s {{=}} 3/2}}) diatonic. {{nowrap|12 + 19 {{=}} 31}}, and 31edo diatonic has hardness {{nowrap|{{sfrac|2 + 3|1 + 2}} {{=}} 5/3}}. | ||
The generator size and the period thus determine the mos scales that can be obtained. Hardness varies with generator size within a mos's range. | The generator size and the period thus determine the mos scales that can be obtained. Hardness varies with generator size within a mos's range. | ||
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[[TAMNAMS]] is a common method for naming intervals of a mos scale. | [[TAMNAMS]] is a common method for naming intervals of a mos scale. | ||
The table below shows the tuning spectrum for the diatonic scale and the temperaments each subset is associated with | The table below shows the tuning spectrum for the diatonic scale and the temperaments each subset is associated with: | ||
{| class="wikitable" style="margin: auto auto auto auto;" | {| class="wikitable" style="margin: auto auto auto auto;" | ||
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* [[5edo]]: Equalized pentatonic ("equipentatonic"). | * [[5edo]]: Equalized pentatonic ("equipentatonic"). | ||
* [[7edo]]: Equalized diatonic ("equiheptatonic"). | * [[7edo]]: Equalized diatonic ("equiheptatonic"). | ||
* [[9edo]]: The simplest edo with a [[2L 5s]] mos (sssLssL). This mos is of interest because it can be viewed as a tuning of the diatonic scale where whole steps are smaller than half steps. | * [[9edo]]: The simplest edo with a [[2L 5s]] mos (sssLssL). This mos is of interest because it can be viewed as a tuning of the diatonic scale where whole steps are smaller than half steps (an "antidiatonic" scale). The corresponding temperament is [[mavila]], which is llike meantone except major and minor intervals are swapped. Some larger edos like 16edo and 23edo tune it better, though mavila has poor accuracy in general since the fifth is very flat. | ||
* [[11edo]]: Stretched 12edo, has [[4L 3s]] mos (LLsLsLs) which is a stretched diatonic. | * [[11edo]]: Stretched 12edo, has [[4L 3s]] mos (LLsLsLs) which is a stretched diatonic. | ||
* [[13edo]]: Compressed 12edo having the [[5L 3s]] mos (LLsLLsLs) which is a compressed version of the diatonic scale. | * [[13edo]]: Compressed 12edo having the [[5L 3s]] mos (LLsLLsLs) which is a compressed version of the diatonic scale. | ||
* [[15edo]]: The smallest edo with a [[5L 5s]] mos (LsLsLsLsLs) commonly called the blackwood scale. Also the smallest with a [[7L 1s]] mos (LLLLsLLL). Both scales are known for supporting relatively familiar major and minor chords with relatively unfamiliar melodic structures. | * [[15edo]]: The smallest edo with a [[5L 5s]] mos (LsLsLsLsLs) commonly called the blackwood scale. Also the smallest with a [[7L 1s]] mos (LLLLsLLL). Both scales are known for supporting relatively familiar major and minor chords with relatively unfamiliar melodic structures. | ||
* [[16edo]]: Has 2L 5s (sssLssL) and [[7L 2s]] (LLLsLLLLs). | * [[16edo]]: Has 2L 5s (sssLssL) and [[7L 2s]] (LLLsLLLLs), generated by the mavila temperament, for which it is a more accurate tuning than 9edo. | ||
* [[17edo]]: The smallest edo after 12edo with a diatonic scale, and the smallest after 12edo to provide perfect fifths which are consonant for most purposes. Its major intervals are sharper and its minor intervals flatter than in 12edo, so it is often said to have a dramatic sound. First neutral diatonic edo (providing neutral seconds, thirds, sixths, and sevenths). | * [[17edo]]: The smallest edo after 12edo with a diatonic scale, and the smallest after 12edo to provide perfect fifths which are consonant for most purposes. Its major intervals are sharper and its minor intervals flatter than in 12edo, so it is often said to have a dramatic sound. First neutral diatonic edo (providing neutral seconds, thirds, sixths, and sevenths). | ||
* [[18edo]]: Has two fifths, 733{{c}} and 667{{c}}, that are nearly equally off from [[3/2]]. | * [[18edo]]: Has two fifths, 733{{c}} and 667{{c}}, that are nearly equally off from [[3/2]]. | ||
* [[19edo]]: The smallest edo after 12edo which supports [[meantone]]. Just major and minor thirds are better approximated than in 12edo, but perfect fifths are represented significantly worse. First [[interordinal]] diatonic edo (interordinals are semifourths, semisixths, semitenths, and semitwelfths). | * [[19edo]]: The smallest edo after 12edo which supports [[meantone]]. Just major and minor thirds are better approximated than in 12edo, but perfect fifths are represented significantly worse, at 7.2{{c}} flat instead of 2{{c}}. First [[interordinal]] diatonic edo (interordinals are semifourths, semisixths, semitenths, and semitwelfths). Diminished and augmented seconds, thirds, sixths, and sevenths are now distinct intervals with entirely new functions, whereas in 12edo they are conflated with simpler intervals. | ||
* [[22edo]]: Diatonic mos with a fifth so | * [[22edo]]: Diatonic mos with a fifth significantly sharper than just, so that four fifths and three fourths give supermajor and subminor thirds (approximately [[9/7]] and [[7/6]]) instead of major and minor thirds. Has a 5-limit major third (approximate [[5/4]]) which is ''not'' be reached by stacking four fifths. Supports [[superpyth]] along with [[7L 1s]] and [[7L 8s]] [[Porcupine]] scales. | ||
* [[23edo]]: The largest edo without a diatonic, 5edo, or 7edo fifth. | * [[23edo]]: The largest edo without a diatonic, 5edo, or 7edo fifth. Supports mavila, just like 9edo and 16edo, with the flat fifth. | ||
* [[24edo]]: Has both neutral thirds (and other neutral intervals) and semifourths (and other interordinals), each of these lending itself to different harmony. Has 12edo mos scales as well as new ones. | * [[24edo]]: Has both neutral thirds (and other neutral intervals) and semifourths (and other interordinals), each of these lending itself to different harmony. Has 12edo mos scales as well as new ones, such as [[5L 4s]] and [[4L 3s]]. | ||
* [[26edo]]: | * [[26edo]]: Has a fifth even flatter than that of 19edo, at 9.6{{c}} flat, and an even softer diatonic mos than 19edo, so much that the diatonic major third is nearly exactly [[26/21]] and the diatonic minor second is nearly exactly [[13/12]]. The [[7/4]] is also nearly exact, and the edo also has a good [[10/9]], [[14/11]], and [[11/8]]. | ||
* [[27edo]]: Even harder diatonic mos than 22edo; | * [[27edo]]: Even harder diatonic mos than 22edo; at 9.2{{c}} sharp of just, the fifth is approximately about as sharp as 26edo's is flat. It has 12edo's [[5/4]], a near-exact [[7/6]], and an approximate [[16/13]] neutral third. Four fifths give a supermajor third and three fourths give a subminor third, just like in 22edo. | ||
* [[29edo]]: First edo with a perfect fifth closer to just intonation than 12edo. The minor third is extremely close to just [[13/11]]. It offers a tuning of 7L 1s with more consonant fifths than 15edo or 22edo before it. Its diatonic scale has similar melodic properties to 17edo, although subtler. | * [[29edo]]: First edo with a perfect fifth closer to just intonation than 12edo. The minor third is extremely close to just [[13/11]]. It offers a tuning of 7L 1s with more consonant fifths than 15edo or 22edo before it. Its diatonic scale has similar melodic properties to 17edo, although subtler. | ||
* [[31edo]]: One of the most popular meantone edos. Close to historical [[quarter-comma meantone]]. Not only is its major third close to just [[5/4]], it also matches the harmonic seventh [[7/4]] well. | * [[31edo]]: One of the most popular meantone edos. Close to the historical [[quarter-comma meantone]]. Not only is its major third close to just [[5/4]], it also matches the harmonic seventh [[7/4]] well. | ||
* [[34edo]]: Good for the 5-limit (2.3.5), as it does not temper out 81/80 and has a good 5/4. Also contains all notes of 17edo. | * [[34edo]]: Good for the 5-limit (2.3.5), as it does not temper out 81/80 and has a good 5/4. Also contains all notes of 17edo. | ||
* [[36edo]]: Good for primes [[3/2|3]] and [[7/4|7]]. | * [[36edo]]: Good for primes [[3/2|3]] and [[7/4|7]]. | ||
* [[37edo]]: Good for primes [[5/4|5]], [[7/4|7]], [[11/8|11]] and [[13/8|13]], but renders 3/2 sharp, even more so than 27edo. | * [[37edo]]: Good for primes [[5/4|5]], [[7/4|7]], [[11/8|11]] and [[13/8|13]], but renders 3/2 sharp, even more so than 27edo. | ||
* [[41edo]]: Often considered remarkably good for the primes up to 11. Good 3; flat 5 and 7; sharp 11 and 13. Known for the [[Kite guitar]]. | * [[41edo]]: Often considered remarkably good for the primes up to 11. Good 3; flat 5 and 7; sharp 11 and 13. Known for the [[Kite guitar]]. | ||
* [[43edo]]: Possibly the most optimal tuning for meantone, with 5 tuned sharp and 3 tuned flat by almost exactly the same amount. Has better approximations of 11 and 13 than 19edo and 31edo, though the 7/4 is noticeably sharp. | |||
* [[46edo]]: Neogothic 3; sharp 5; flat 7, 11, and 13; good 17. Supports [[parapyth]]. Often compared to 41edo; some favor one, some the other. | * [[46edo]]: Neogothic 3; sharp 5; flat 7, 11, and 13; good 17. Supports [[parapyth]]. Often compared to 41edo; some favor one, some the other. | ||
* [[53edo]]: Is a stack of near-just 3/2's which also approximates primes 5, 7, 13, and 19. | * [[53edo]]: Is a stack of near-just 3/2's which also approximates primes 5, 7, 13, and 19. | ||
* [[72edo]]: A notable subdivision of 12edo that is a very strong 11-limit (primes 2, 3, 5, 7, 11) temperament for its size. | * [[72edo]]: A notable subdivision of 12edo that is a very strong 11-limit (primes 2, 3, 5, 7, 11) temperament for its size. | ||
* [[87edo]]: Even better in 2.3.5.11.13 than 72edo is in the 11-limit, and a consistent and precise edo for approximating harmonics 8 to 16, but ratios with 7 suffer due to the 7 being flat and the 3 being sharp. | * [[87edo]]: Even better in 2.3.5.11.13 than 72edo is in the 11-limit, and a consistent and precise edo for approximating harmonics 8 to 16, but ratios with 7 suffer due to the 7 being flat and the 3 being sharp. | ||
* [[311edo]]: An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41. | * [[311edo]]: An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41. Essentially, the final boss of RTT edos. | ||
[[Category:Overview]] | [[Category:Overview]] | ||