Xen concepts for beginners: Difference between revisions
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== Interval math == | == Interval math == | ||
Xen discussion uses two kinds of units: | Xen discussion uses two kinds of units: | ||
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* ''Logarithmic units'' such as [[cents]] and [[edo]] steps that treat intervals we hear as equal as the same additive unit | * ''Logarithmic units'' such as [[cents]] and [[edo]] steps that treat intervals we hear as equal as the same additive unit | ||
To stack two intervals, we use different types of operations for the two kinds of units. To stack two intervals written as ratios, we *multiply*, whereas to stack two intervals written as cents or edo steps, we | To stack two intervals, we use different types of operations for the two kinds of units. To stack two intervals written as ratios, we *multiply*, whereas to stack two intervals written as cents or edo steps, we ''add'' the intuitive way. To "unstack" an interval from another interval, we ''divide'' the respective ratios and ''subtract'' logarithmic units. To convert between cents and ratios we use the following formulas: | ||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
\text{cents} &= 1200 | \text{cents} &= 1200 \cdot \log_{2} \left( \text{ratio} \right) \\ | ||
\text{ratio} &= 2^\left( | \text{ratio} &= 2^{\left( \text{cents}/1200 \right)} | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
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The unison has frequency ratio 1/1 and is 0 cents. The octave has frequency ratio 2/1 and is exactly 1200 cents. A standard semitone (in 12edo) has frequency ratio <math>\sqrt[12]{2}</math> and is exactly 100 cents (by definition). | The unison has frequency ratio 1/1 and is 0 cents. The octave has frequency ratio 2/1 and is exactly 1200 cents. A standard semitone (in 12edo) has frequency ratio <math>\sqrt[12]{2}</math> and is exactly 100 cents (by definition). | ||
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+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. | 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. | 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. | ||
== Basic JI == | == Basic JI == | ||
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As stacking JI ratios involves multiplying, primes are important as the simplest building blocks of arbitrary JI ratios. So we can write every ratio as a vector called a ''monzo'', a list of powers for primes. We can visualize each ratio as living in some JI lattice (the set of all intervals built by stacking a finite set of basic intervals). | As stacking JI ratios involves multiplying, primes are important as the simplest building blocks of arbitrary JI ratios. So we can write every ratio as a vector called a ''monzo'', a list of powers for primes. We can visualize each ratio as living in some JI lattice (the set of all intervals built by stacking a finite set of basic intervals). | ||
There are many approaches to JI music: lattice-based JI, constant structure scales, free JI, primodality, tonality diamonds, combination product | There are many approaches to JI music: lattice-based JI, constant structure scales, free JI, primodality, tonality diamonds, combination product sets… | ||
JI is usually less mathy than RTT. | JI is usually less mathy than RTT. | ||
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There are two ways the term ''[[limit]]'' is used. | There are two ways the term ''[[limit]]'' is used. | ||
* The ''p- | * 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 ''n- | * The ''[[odd limit|n-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 ''n''. For example, 7/6, 13/5, 11/10, and 16/15 are all in the 15-odd-limit. | ||
== Basic RTT == | == Basic RTT == | ||
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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. | 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. 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 }}). | |||
* The patent val for 12edo in the 5-limit is {{val| 12 19 28}}. | 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 }}. | * 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 | ** 12 × (-4) = -48, corresponding to going down 4 octaves. | ||
** 19 | ** 19 × 4 = 76, corresponding to going up 4 perfect twelfths (or, to going up 4 octaves and 4 fifths). | ||
** 28 | ** 28 × (-1) = -28, corresponding to dividing by 5 (going down two octaves and a major third). | ||
** (76 - 48) - 28 = 0 | ** (76 - 48) - 28 = 0 | ||
* Since the result is 0, 12edo supports | * 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 | <math>\vmp{12 & 19 & 28}{-4 & 4 & -1} = 12 \times \left(-4\right) + 19 \times 4 + 28 \times \left(-1\right) = -48 + 76 - 28 = 0.</math> | ||
== MOS scales == | == MOS scales == | ||
[[ | [[Mos]] (moment of symmetry) scales are one way to generalize the diatonic scale; the diatonic scale is a mos scale. They are scales with two step sizes (large (L) and small (s)) with a uniquely elegant combination of properties: | ||
For every number of steps, the scale has at most two interval sizes with that number of steps. | For every number of steps, the scale has at most two interval sizes with that number of steps. The scale can be made by stacking a certain fixed interval called the ''[[periods and generators|generator]]'' (and reducing by an interval called the ''[[periods and generators|period]]'', usually the octave or some equal division of it such as 1\2 or 1\3), over and over, stopping at some point where there are two step sizes distributed as evenly as possible. | ||
The scale can be made by stacking a certain fixed interval called the ''[[periods and generators|generator]]'' (and reducing by an interval called the ''[[periods and generators|period]]'', usually the octave or some equal division of it such as 1\2 or 1\3), over and over, stopping at some point where there are two step sizes distributed as evenly as possible. | |||
Every | Every mos scale with ''m'' large steps and ''n'' small steps is a mode of some pattern. This is why you only need to write ''m''L ''n''s for an octave-equivalent mos scale and specify the mode (using [[UDP]] for example). For example, every 5L 3s mos scale is a mode of the pattern LLsLLsLs. | ||
An important way that | 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 | 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 (2 + 3)/(1 + 2) = 5/3. | ||
The generator size and the period thus determine the | 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. | ||
Every | Every mos scale pattern has a generator range. Since the familiar diatonic scale is a mos 5L 2s, here is an important fact to know: If the period is the octave and the generator is a fifth between 4\7 (686{{c}}) and 3\5 (720{{c}}), the resulting pattern is the diatonic mos. | ||
[[TAMNAMS]] is a common method for naming intervals of a | [[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;" | ||
|+ style="font-size: 105%;" | Tuning ranges of the diatonic | |+ style="font-size: 105%;" | Tuning ranges of the diatonic mos | ||
|- | |- | ||
! colspan="3" | Range !! rowspan="2" colspan="2" | Temperaments encompassed | ! colspan="3" | Range !! rowspan="2" colspan="2" | Temperaments encompassed | ||
|- | |- | ||
! | ! Edo !! Cents !! Hardness | ||
|- | |- | ||
| 7edo to 33edo || 685.714 to 690.909 || 1/1 to 5/4 || colspan="2" | [[Deeptone]] | | 7edo to 33edo || 685.714 to 690.909 || 1/1 to 5/4 || colspan="2" | [[Deeptone]] | ||
|- | |- | ||
| 33edo to 19edo || 690.909 to 694.737 || 4/3 to 3/2 || rowspan="2" | Meantone || [[Flattertone]], [[ | | 33edo to 19edo || 690.909 to 694.737 || 4/3 to 3/2 || rowspan="2" | Meantone || [[Flattertone]], [[flattone]] | ||
|- | |- | ||
| 19edo to 12edo || 694.737 to 700 || 3/2 to 2/1 || [[Septimal meantone]] | | 19edo to 12edo || 694.737 to 700 || 3/2 to 2/1 || [[Septimal meantone]] | ||
| Line 100: | Line 99: | ||
| 12edo to 29edo || 700 to 703.448 || 2/1 to 5/2 || colspan="2" | [[Schismatic]] | | 12edo to 29edo || 700 to 703.448 || 2/1 to 5/2 || colspan="2" | [[Schismatic]] | ||
|- | |- | ||
| 29edo to 17edo || 703.448 to 705.882 || 5/2 to 3/1 || colspan="2" | [[Neogothic]], [[ | | 29edo to 17edo || 703.448 to 705.882 || 5/2 to 3/1 || colspan="2" | [[Neogothic]], [[parapyth]], [[leapday]] | ||
|- | |- | ||
| 17edo to 22edo || 705.882 to 709.091 || 3/1 to 4/1 || rowspan="3" | Superpyth || [[Suprapyth]] | | 17edo to 22edo || 705.882 to 709.091 || 3/1 to 4/1 || rowspan="3" | Superpyth || [[Suprapyth]] | ||
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== Edos == | == Edos == | ||
* [[5edo]]: Equalized pentatonic (" | * [[5edo]]: Equalized pentatonic ("equipentatonic"). | ||
* [[7edo]]: Equalized diatonic (" | * [[7edo]]: Equalized diatonic ("equiheptatonic"). | ||
* [[9edo]]: The simplest edo with a [[ | * [[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. | ||
* [[11edo]]: Stretched 12edo, has [[ | * [[11edo]]: Stretched 12edo, has [[4L 3s]] mos (LLsLsLs) which is a stretched diatonic. | ||
* [[13edo]]: Compressed 12edo having the [[ | * [[13edo]]: Compressed 12edo having the [[5L 3s]] mos (LLsLLsLs) which is a compressed version of the diatonic scale. | ||
* [[15edo]]: The smallest edo with a [[ | * [[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 | * [[16edo]]: Has 2L 5s (sssLssL) and [[7L 2s]] (LLLsLLLLs). | ||
* [[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 | * [[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, | * [[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 [[ | * [[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 | * [[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. | * [[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 | * [[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 | * [[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 | * [[27edo]]: Even harder diatonic mos than 22edo; the fifth is approximately about as sharp (by 9.2{{c}}) as 26edo's is flat (by 9.6{{c}}). It has 12edo's [[5/4]], a near-exact [[7/6]], and an approximate [[16/13]] neutral third. | ||
* [[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 | * [[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 | * [[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]]. | ||
* [[46edo]]: Neogothic 3; sharp 5; flat 7, 11, and 13; good 17. Supports [[ | * [[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. | ||