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".

A ~~very important~~ operation in xen math is the [[mediant]]. The mediant of two fractions, a/b and c/d, is the "freshman sum" (a+b)/(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.

A common operation in xen math is the [[mediant]]. The mediant of two fractions, a/b and c/d, is the "freshman sum" (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.

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From the perspective of an edo user, another problem RTT solves is that there are very few small edos and they do not constitute that wide a palette. Especially in larger edos, RTT provides a way of not being overwhelmed with dozens of notes.

RTT views edos as regular temperaments. To simplify the infinite JI space to a finite set, ~~we need to deform~~ 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)^4 = 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 }}).

A val tempers out a comma if the ~~dot product of~~ the val ~~and~~ the ~~monzo of the comma~~ is 0. 12edo is a Meantone edo because the ~~dot product of the vectors~~ {{val| 12 19 28 }} ~~and~~ {{monzo| -4 4 -1 }} is 0:

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:

* The patent val for 12edo in the 5-limit is {{val| 12 19 28}}.

* 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".

** 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).

** 28*-1 = -28, corresponding to dividing by 5 (going down two octaves and a major third).

** (76 - 48) - 28 = 0

* Since the result is 0, 12edo supports Meantone. In RTT math, this can be written as:

<math>\vmp{12 & 19 & 28}{-4 & 4 & -1} = 12 * \left(-4\right) + 19 * 4 + 28 * \left(-1\right) = -48 + 76 - 28 = 0.</math>

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* [[5edo]]: Equalized pentatonic ("Equipentatonic").

* [[7edo]]: Equalized diatonic ("Equiheptatonic").

* [[9edo]]: The simplest edo with a [[2L5s]] MOS (sssLssL).

* [[9edo]]: The simplest edo with a [[2L5s]] 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.

* [[11edo]]: Stretched 12edo, has [[4L3s]] MOS (LLsLsLs) which is a stretched diatonic.

* [[13edo]]: Compressed 12edo having the [[5L3s]] MOS (LLsLLsLs) which is a compressed version of the diatonic scale.

* [[15edo]]: The smallest edo with a [[5L5s]] MOS (LsLsLsLsLs) commonly called the Blackwood scale.

* [[15edo]]: The smallest edo with a [[5L5s]] 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 2L5s (sssLssL) and [[7L2s]] (LLLsLLLLs).

* [[17edo]]: The smallest edo after 12edo with a diatonic scale, which ~~can be harmonically very different from~~ 12edo ~~diatonic depending on how you use~~ it. 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's often said to have a dramatic sound. First neutral diatonic edo (providing neutral seconds, thirds, sixths, and sevenths).

* [[18edo]]: Has two fifths, 733c and 667c, 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. 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. First [[interordinal]] diatonic edo (interordinals are semifourths, semisixths, semitenths, and semitwelfths).

* [[22edo]]: Diatonic MOS with a fifth so sharp that it has supermajor and subminor thirds (approximately [[9/7]] and [[7/6]]) for its major and minor thirds. Has a 5-limit major third (approximate [[5/4]]) which *cannot* be reached by stacking four fifths. Supports [[Superpyth]] ~~like [[27edo]]~~.

* [[22edo]]: Diatonic MOS with a fifth so sharp that it has supermajor and subminor thirds (approximately [[9/7]] and [[7/6]]) for its major and minor thirds. Has a 5-limit major third (approximate [[5/4]]) which *cannot* be reached by stacking four fifths. Supports [[Superpyth]] and 7L 1s.

* [[23edo]]: The largest edo without a diatonic, 5edo, or 7edo 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.

* [[26edo]]: 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; the fifth is approximately about as sharp (by 9.2c) as 26edo's is flat (by 9.6c). It has 12edo's [[5/4]], a near-exact [[7/6]], and an approximate [[16/13]] neutral third.

* [[29edo]]: ~~Weird flat neogothic~~ edo.

* [[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]]: ~~Often considered~~ the ~~best~~ Meantone ~~edo~~. 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~~, also approximating other JI ratios like [[6/5]] (just minor third), [[7/6]] (septimal subminor third), and [[25/16]] (classical augmented fifth)~~.

* [[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.

* [[34edo]]: Good for the 5-limit (2.3.5), as it doesn't temper out 81/80 and has a good 5/4.

* [[34edo]]: Good for the 5-limit (2.3.5), as it doesn't 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]].

* [[37edo]]: Good for primes [[5/4|5]], [[7/4|7]], [[11/8|11]] and [[13/8|13]], ~~in return for a sharp~~ 3/2.

* [[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]]: 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]].

* [[46edo]]: Neogothic 3; sharp 5; flat 7, 11, and 13; good 17. Supports [[Parapyth]]. ~~Some~~ favor ~~it over 41edo~~.

* [[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.

* [[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. The final boss of RTT edos.

[[Category:Overview]]

@@ Line 20: / Line 20: @@
 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 very important operation in xen math is the [[mediant]]. The mediant of two fractions, a/b and c/d, is the "freshman sum" (a+b)/(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.
+A common operation in xen math is the [[mediant]]. The mediant of two fractions, a/b and c/d, is the "freshman sum" (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.
@@ Line 48: / Line 48: @@
 From the perspective of an edo user, another problem RTT solves is that there are very few small edos and they do not constitute that wide a palette. Especially in larger edos, RTT provides a way of not being overwhelmed with dozens of notes.
-RTT views edos as regular temperaments. To simplify the infinite JI space to a finite set, we need to deform 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)^4 = 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 }}).
-A val tempers out a comma if the dot product of the val and the monzo of the comma is 0. 12edo is a Meantone edo because the dot product of the vectors {{val| 12 19 28 }} and {{monzo| -4 4 -1 }} is 0:
+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:
+* The patent val for 12edo in the 5-limit is {{val| 12 19 28}}.
+* 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".
+** 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).
+** 28*-1 = -28, corresponding to dividing by 5 (going down two octaves and a major third).
+** (76 - 48) - 28 = 0
+* Since the result is 0, 12edo supports Meantone. In RTT math, this can be written as:
 <math>\vmp{12 & 19 & 28}{-4 & 4 & -1} = 12 * \left(-4\right) + 19 * 4 + 28 * \left(-1\right) = -48 + 76 - 28 = 0.</math>
@@ Line 103: / Line 112: @@
 * [[5edo]]: Equalized pentatonic ("Equipentatonic").
 * [[7edo]]: Equalized diatonic ("Equiheptatonic").
-* [[9edo]]: The simplest edo with a [[2L5s]] MOS (sssLssL).
+* [[9edo]]: The simplest edo with a [[2L5s]] 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.
 * [[11edo]]: Stretched 12edo, has [[4L3s]] MOS (LLsLsLs) which is a stretched diatonic.
 * [[13edo]]: Compressed 12edo having the [[5L3s]] MOS (LLsLLsLs) which is a compressed version of the diatonic scale.
-* [[15edo]]: The smallest edo with a [[5L5s]] MOS (LsLsLsLsLs) commonly called the Blackwood scale.
+* [[15edo]]: The smallest edo with a [[5L5s]] 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 2L5s (sssLssL) and [[7L2s]] (LLLsLLLLs).
-* [[17edo]]: The smallest edo after 12edo with a diatonic scale, which can be harmonically very different from 12edo diatonic depending on how you use it. 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's often said to have a dramatic sound. First neutral diatonic edo (providing neutral seconds, thirds, sixths, and sevenths).
 * [[18edo]]: Has two fifths, 733c and 667c, 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. 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. First [[interordinal]] diatonic edo (interordinals are semifourths, semisixths, semitenths, and semitwelfths).
-* [[22edo]]: Diatonic MOS with a fifth so sharp that it has supermajor and subminor thirds (approximately [[9/7]] and [[7/6]]) for its major and minor thirds. Has a 5-limit major third (approximate [[5/4]]) which *cannot* be reached by stacking four fifths. Supports [[Superpyth]] like [[27edo]].
+* [[22edo]]: Diatonic MOS with a fifth so sharp that it has supermajor and subminor thirds (approximately [[9/7]] and [[7/6]]) for its major and minor thirds. Has a 5-limit major third (approximate [[5/4]]) which *cannot* be reached by stacking four fifths. Supports [[Superpyth]] and 7L 1s.
 * [[23edo]]: The largest edo without a diatonic, 5edo, or 7edo 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.
 * [[26edo]]: 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; the fifth is approximately about as sharp (by 9.2c) as 26edo's is flat (by 9.6c). It has 12edo's [[5/4]], a near-exact [[7/6]], and an approximate [[16/13]] neutral third.
-* [[29edo]]: Weird flat neogothic edo.
+* [[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]]: Often considered the best Meantone edo. 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, also approximating other JI ratios like [[6/5]] (just minor third), [[7/6]] (septimal subminor third), and [[25/16]] (classical augmented fifth).
+* [[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.
-* [[34edo]]: Good for the 5-limit (2.3.5), as it doesn't temper out 81/80 and has a good 5/4.
+* [[34edo]]: Good for the 5-limit (2.3.5), as it doesn't 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]].
-* [[37edo]]: Good for primes [[5/4|5]], [[7/4|7]], [[11/8|11]] and [[13/8|13]], in return for a sharp 3/2.
+* [[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]]: 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]].
-* [[46edo]]: Neogothic 3; sharp 5; flat 7, 11, and 13; good 17. Supports [[Parapyth]]. Some favor it over 41edo.
+* [[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.
-* [[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. The final boss of RTT edos.
+[[Category:Overview]]