Xen concepts for beginners
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This is a beginner page. It is written to allow new readers to learn about the basics of the topic easily. |
Interval math
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.
- 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:
[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 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".
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.
Another important operation is 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
Just intonation (JI) is the set of intervals that are tuned to rational frequency ratios, ones can be written as fractions of whole numbers.
The easiest way to get concordance (smoothness, blending and buzzing) is to use low-numbered JI ratios in your interval or chord, for example 3/2 the just perfect fifth, 5/4 the just major third, and 7/5 the lesser septimal tritone. When pure JI ratios are used, a psychoacoustic effect called JI buzz occurs. When the overall chord is low number JI, such as 8:9:10:11:12:13:14, the result is very concordant.
No edo interval except for the octave (2/1) and stacks of it is exact JI. A JI ratio might be far from a 12edo interval; for example 7/4 is 969 cents. This is another reason why JI is a common approach to xen.
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 sets...
JI is usually less mathy than RTT.
The approach that RTT cares about the most is lattice-based JI. A JI lattice, or a subgroup, is built by stacking a finite set of JI intervals, usually primes such as 2, 3, 5, and 7.
There are two ways the term limit is used.
- The 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-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
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 with such infinite modulation and regularity 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. One issue with this criticism is that one need not treat JI like one would an edo, and that some regular temperament tunings are infinite and don't provide the advantages of finiteness.
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 ⟨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 [-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 ⟨12 19 28] and [-4 4 -1⟩ is 0:
[math]\vmp{12 & 19 & 28}{-4 & 4 & -1} = 12 * \left(-4\right) + 19 * 4 + 28 * \left(-1\right) = -48 + 76 - 28 = 0.[/math]
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. The scale can be made by stacking a certain fixed interval called the generator (and reducing by an interval called the 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 MOS scale with m large steps and n small steps is a mode of some pattern. This is why you only need to write mL ns for an octave-equivalent MOS scale and specify the mode (using UDP for example). For example, every 5L3s MOS scale is a mode of the pattern LLsLLsLs.
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 (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.
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 (686c) and 3\5 (720c), the resulting pattern is the diatonic MOS.
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
| Range | Temperaments encompassed | |||
|---|---|---|---|---|
| EDO | Cents | Hardness | ||
| 7edo to 33edo | 685.714 to 690.909 | 1/1 to 5/4 | Deeptone | |
| 33edo to 19edo | 690.909 to 694.737 | 4/3 to 3/2 | Meantone | Flattertone, Flattone |
| 19edo to 12edo | 694.737 to 700 | 3/2 to 2/1 | Septimal meantone | |
| 12edo to 29edo | 700 to 703.448 | 2/1 to 5/2 | Schismatic | |
| 29edo to 17edo | 703.448 to 705.882 | 5/2 to 3/1 | Neogothic, Parapyth, Leapday | |
| 17edo to 22edo | 705.882 to 709.091 | 3/1 to 4/1 | Superpyth | Suprapyth |
| 22edo to 27edo | 709.091 to 711.111 | 4/1 to 5/1 | Superpyth | |
| 27edo to 5edo | 711.111 to 720 | 5/1 to ∞ | Ultrapyth | |
Edos
- 5edo: Equalized pentatonic ("Equipentatonic").
- 7edo: Equalized diatonic ("Equiheptatonic").
- 9edo: The simplest edo with a 2L5s MOS (sssLssL).
- 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.
- 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).
- 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).
- 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.
- 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.
- 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).
- 34edo: Good for the 5-limit (2.3.5), as it doesn't temper out 81/80 and has a good 5/4.
- 36edo: Good for primes 3 and 7.
- 37edo: Good for primes 5, 7, 11 and 13, in return for a sharp 3/2.
- 41edo: 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.
- 53edo: Is a stack of near-just 3/2's which also approximates primes 5, 7, 13, and 19.
- 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.