User:CompactStar/Ordinal interval notation: Difference between revisions

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'''Lefts and rights notation''' is a notation for [[just intonation]] primarily developed by [[User:CompactStar|CompactStar]]~~. It represents every just interval as a sequence of lefts/rights (referred to as its direction) and a diatonic degree~~.

'''Lefts and rights notation''' is a notation for [[just intonation]] primarily developed by [[User:CompactStar|CompactStar]].

== Explanation ==

All intervals are ~~given~~ a ~~diatonic~~ degree, calculated ~~from their~~ [[7edo]] [[patent val]] [[mapping]]~~. If k is~~ a ~~degree, the central k-th is the simplest (with respect to [[Tenney height]]) k-th which is within 20 cents of (k-1)\7edo~~. The left k-th is the simplest k-th which is flatter than central, and the right k-th is the simplest k-th which is sharper than central. Central, left and right are abbreviated as C, L, and R respectively.

All just intervals are indicated by a direction, which is a sequence of lefts and rights, and a degree. The degree is calculated by adding 1 to the [[7edo]] [[patent val]] [[mapping]] of a ratio. The "default" direction, with no lefts or rights is C

Left and right can be used multiple times to produce more complex directions. For example, leftleft is flatter than left, leftright is between left and central, rightleft is between central and right, and rightright is sharper than right. Adding a left always means to go flatter, and adding a right always means to go sharper, with ~~each new left/right having less and less of an impact. Formally, if k is a degree, X and Y are any sequence of~~ lefts/rights:

* Xleft k-th = the simplest just k-th whose direction starts with X and is ~~flatter than the X k-th~~

* Xright k-th = the simplest just k-th whose direction starts with X and is sharper than the X k-th

== Examples ==

Below are some examples of lefts and rights notation for [[11-odd-limit]] intervals:

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-'''Lefts and rights notation''' is a notation for [[just intonation]] primarily developed by [[User:CompactStar|CompactStar]]. It represents every just interval as a sequence of lefts/rights (referred to as its direction) and a diatonic degree.
+'''Lefts and rights notation''' is a notation for [[just intonation]] primarily developed by [[User:CompactStar|CompactStar]].
 == Explanation ==
-All intervals are given a diatonic degree, calculated from their [[7edo]] [[patent val]] [[mapping]]. If k is a degree, the central k-th is the simplest (with respect to [[Tenney height]])  k-th which is within 20 cents of (k-1)\7edo. The left k-th is the simplest  k-th which is flatter than central, and the right k-th is the simplest k-th which is sharper than central. Central, left and right are abbreviated as C, L, and R respectively.
+All just intervals are indicated by a direction, which is a sequence of lefts and rights, and a degree. The degree is calculated by adding 1 to the [[7edo]] [[patent val]] [[mapping]] of a ratio. The "default" direction, with no lefts or rights is C
-Left and right can be used multiple times to produce more complex directions. For example, leftleft is flatter than left, leftright is between left and central, rightleft is between central and right, and rightright is sharper than right. Adding a left always means to go flatter, and adding a right always means to go sharper, with each new left/right having less and less of an impact. Formally, if k is a degree, X and Y are any sequence of lefts/rights:
-* Xleft k-th = the simplest just k-th whose direction starts with X and is flatter than the X k-th
-* Xright k-th = the simplest just k-th whose direction starts with X and is sharper than the X k-th
 == Examples ==
 Below are some examples of lefts and rights notation for [[11-odd-limit]] intervals: