Interval system: Difference between revisions

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Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is 3-prime-limit JI, a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of Erv Wilson.

Among open systems, the most important kinds are [[~~Periodic_scale|~~periodic ~~scales~~]] and group systems. The latter refers to "groups" in the mathematical sense of [http://en.wikipedia.org/wiki/Abelian_group abelian groups], and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.

Among open systems, the most important kinds are [[periodic scale]]s and group systems. The latter refers to "groups" in the mathematical sense of [http://en.wikipedia.org/wiki/Abelian_group abelian groups], and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.

Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen 1/1; all rational numbers in a given [[~~Harmonic_Limit|~~harmonic limit]]; all intervals in a [[~~Just_intonation_subgroups~~|just intonation subgroup]]; and all intervals in a [[~~Regular_Temperaments~~|regular temperament]].

Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen 1/1; all rational numbers in a given [[harmonic limit]]; all intervals in a [[Just intonation subgroups|just intonation subgroup]]; and all intervals in a [[Regular Temperaments|regular temperament]].

[[Category:~~overview~~]]

[[Category:Overview]]

[[Category:Theory]]

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 Another type of open system can be infinite even if its pitches occupy a finite frequency range, because it is defined by a rule for generating successive intervals under which, no matter how many times the generative process is repeated, no new interval is ever identical to a previous interval. An example of this is 3-prime-limit JI, a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Another example would be any of the golden horagrams of Erv Wilson.
-Among open systems, the most important kinds are [[Periodic_scale|periodic scales]] and group systems. The latter refers to "groups" in the mathematical sense of [http://en.wikipedia.org/wiki/Abelian_group abelian groups], and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.
+Among open systems, the most important kinds are [[periodic scale]]s and group systems. The latter refers to "groups" in the mathematical sense of [http://en.wikipedia.org/wiki/Abelian_group abelian groups], and means that you are always allowed to invert intervals, and that given any two intervals, you may combine them.
-Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen 1/1; all rational numbers in a given [[Harmonic_Limit|harmonic limit]]; all intervals in a [[Just_intonation_subgroups|just intonation subgroup]]; and all intervals in a [[Regular_Temperaments|regular temperament]].
+Examples of group systems are all positive real numbers under multiplication, regarded as frequencies in hertz; all real numbers under addition, regarded as intervals in cents; all positive rational numbers, regarded as intervals from a chosen 1/1; all rational numbers in a given [[harmonic limit]]; all intervals in a [[Just intonation subgroups|just intonation subgroup]]; and all intervals in a [[Regular Temperaments|regular temperament]].
-[[Category:overview]]
+[[Category:Overview]]
+[[Category:Theory]]