Xen concepts for beginners: Difference between revisions
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{{Beginner}} | {{Beginner}} | ||
== Interval math == | == Interval math == | ||
Xen discussion uses two kinds of units: | Xen discussion uses two kinds of units: | ||
* '' | * ''Frequency ratios'' | ||
** The frequency is the absolute pitch of any given tone, usually measured in hertz (Hz). The ratio between frequencies is just a number. Equal intervals have the same frequency ratio. | ** The frequency is the absolute pitch of any given tone, usually measured in hertz (Hz). The ratio between frequencies is just a number. Equal intervals have the same frequency ratio. | ||
* '' | * ''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 *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: | 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> | |||
\begin{align} | |||
\text{cents} &= 1200*\log_{2}\left(\text{ratio}\right) \\ | |||
\text{ratio} &= 2^\left(\frac{\text{cents}}{1200}\right) | |||
\end{align} | |||
</math> | |||
The unison has frequency ratio 1/1 and is 0 cents. The octave has frequency ratio 2/1 and is exactly 1200 cents. A 12edo/12tet semitone has frequency ratio 2^(1/12) and is exactly 100 cents. | The unison has frequency ratio 1/1 and is 0 cents. The octave has frequency ratio 2/1 and is exactly 1200 cents. A 12edo/12tet semitone has frequency ratio 2^(1/12) and is exactly 100 cents. | ||
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Assuming several things from common 12edo practice, JI has several disadvantages. To get infinite modulation and have exactly the same chords on every note, we need infinitely many notes unlike the finitely many notes of 12edo. JI also has small intervals that may be undesirable, called commas. This is the problem that [[regular temperament theory]] (RTT) exists to solve. Regular temperaments equate certain intervals by considering the difference between them as a comma and "[[tempering out]]" the difference. | Assuming several things from common 12edo practice, JI has several disadvantages. To get infinite modulation and have exactly the same chords on every note, we need infinitely many notes unlike the finitely many notes of 12edo. JI also has small intervals that may be undesirable, called commas. This is the problem that [[regular temperament theory]] (RTT) exists to solve. Regular temperaments equate certain intervals by considering the difference between them as a comma and "[[tempering out]]" the difference. | ||
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 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 | 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 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 | 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 | 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 {{val| 12 19 28 }} and {{monzo| -4 4 -1 }} is 0. | ||
== MOS scales == | == MOS scales == | ||
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== Edos == | == Edos == | ||
* 5edo: Equalized pentatonic ("Equipentatonic"). | |||
* 5edo: | * 7edo: Equalized diatonic ("Equiheptatonic"). | ||
* 7edo: | * 9edo: The simplest edo with a 2L5s MOS (sssLssL). | ||
* 9edo: | * 11edo: Stretched 12edo, has 4L3s MOS (LLsLsLs) which is a stretched diatonic. | ||
* 11edo: | * 13edo: Compressed 12edo having the 5L3s MOS (LLsLLsLs) which is a compressed version of the diatonic scale. | ||
* 13edo: | * 15edo: The smallest edo with a 5L5s MOS (LsLsLsLsLs) commonly called the Blackwood scale. | ||
* 15edo: | * 16edo: Has 2L5s (sssLssL) and 7L2s (LLLsLLLLs). | ||
* 16edo: | * 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: | * 18edo: Has two fifths, 733c and 667c, that are nearly equally off from 3/2. | ||
* 18edo: | * 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: | * 22edo: Diatonic MOS with a fifth so sharp that it has supermajor and subminor thirds for its major and minor thirds. "Minor sevenths" such as A–G and C–B♭ are approximately 7/4 instead of 16/9 or 9/5. Has a 5-limit major third (approximate 5/4) which *cannot* be reached by stacking four fifths. Supports [[superpyth]] like 27edo. | ||
* 22edo: | * 23edo: The largest edo without a diatonic, 5edo, or 7edo fifth. | ||
* 23edo: | * 24edo: Has both neutral thirds (350c) and semifourths (250c), each of these lending itself to different harmony. Has 12edo MOS scales as well as new ones. | ||
* 24edo: | * 29edo: Weird flat neogothic edo. | ||
* 29edo: | * 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: | * 34edo: Good for the 5-limit (2.3.5), as it doesn't temper out 81/80 and has a good 5/4. | ||
* 34edo: | * 36edo: Good for primes 3 and 7. | ||
* 36edo: | * 37edo: Good for primes 5, 7, 11 and 13, in return for a sharp 3/2. | ||
* 37edo: | * 41edo: Good 3; flat 5 and 7; sharp 11 and 13. Known for [[Kite guitar]]. | ||
* 41edo: | * 46edo: Neogothic 3; sharp 5; flat 7, 11, and 13; good 17. Supports [[Parapyth]]. Some favor it over 41edo. | ||
* 46edo: | * 53edo: Is a stack of near-just 3/2's which also approximates primes 5, 7, 13, and 19. | ||
* 53edo: | |||