Xen concepts for beginners - Revision history

Overthink: /* Edos */ grammar

2026-02-25T03:54:07Z

Edos: grammar

← Older revision		Revision as of 03:54, 25 February 2026
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	* [[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.		* [[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]].

Overthink: /* Edos */ fix typo

2026-02-25T03:51:03Z

Edos: fix typo

← Older revision		Revision as of 03:51, 25 February 2026
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	* [[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]].		* [[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]]: 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]].		* [[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; 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 ~~thirdand~~ three fourths give a subminor third, just like in 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 historical [[quarter-comma meantone]]. Not only is its major third close to just [[5/4]], it also matches the harmonic seventh [[7/4]] well.

ArrowHead294: /* Edos */

2026-02-25T03:39:55Z

Edos

ArrowHead294 at 13:15, 19 February 2026

2026-02-19T13:15:21Z

Show changes

Overthink: /* Edos */ mention mavila

2026-02-15T05:19:43Z

Edos: mention mavila

Overthink: /* Basic JI */ order these better

2026-02-15T05:10:32Z

Basic JI: order these better

← Older revision		Revision as of 05:10, 15 February 2026
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	[[Just intonation]] (JI) is the set of intervals that are tuned to rational frequency ratios, ones can be written as fractions of whole numbers.		[[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.		The easiest way to get concordance (smoothness, blending and buzzing) is to use low-numbered JI ratios in your interval or chord, for example the just perfect fifth [[3/2]], the just major third [[5/4]], and the lesser septimal tritone [[7/5]]. 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.		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.
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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, 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 ''[[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.		* 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.

	== Basic RTT ==		== Basic RTT ==

Inthar: /* Basic RTT /The criticism itself uses assumptions that are explicit*

2026-01-30T13:39:42Z

Basic RTT: The criticism itself uses assumptions that are *explicit*

← Older revision		Revision as of 13:39, 30 January 2026
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	== Basic RTT ==		== 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.		''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. However, note 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.		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.

Inthar: /* Interval math */

2026-01-01T15:59:49Z

Interval math

← Older revision		Revision as of 15:59, 1 January 2026
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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 ''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>		<math>

FloraC: Correct/improve associated temps with fifth ranges

2026-01-01T13:39:44Z

Correct/improve associated temps with fifth ranges

← Older revision		Revision as of 13:39, 1 January 2026
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	\| 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]], [[flattone]]		\| 33edo to 19edo \|\| 690.909 to 694.737 \|\| 5/4 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 700.000 \|\| 3/2 to 2/1 \|\| [[Septimal meantone]]
	\|-		\|-
	\| 12edo to 29edo \|\| 700 to 703.448 \|\| 2/1 to 5/2 \|\| colspan="2" \| [[~~Schismatic~~]]		\| 12edo to 29edo \|\| 700.000 to 703.448 \|\| 2/1 to 5/2 \|\| colspan="2" \| [[Schismic]]
	\|-		\|-
	\| 29edo to 17edo \|\| 703.448 to 705.882 \|\| 5/2 to 3/1 \|\| colspan="2" \| [[~~Neogothic]], [[parapyth~~]], [[leapday]]		\| 29edo to 17edo \|\| 703.448 to 705.882 \|\| 5/2 to 3/1 \|\| colspan="2" \| [[Pepperoni]], [[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 \|\| [[Quasisuper]]
	\|-		\|-
	\| 22edo to 27edo \|\| 709.091 to 711.111 \|\| 4/1 to 5/1 \|\| [[Superpyth]]		\| 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]]		\| 27edo to 5edo \|\| 711.111 to 720.000 \|\| 5/1 to ∞ \|\| [[Ultrapyth]]
	\|}		\|}

FloraC: Style

2026-01-01T13:29:36Z

Style