Tuning system: Difference between revisions

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== Open and closed systems ==

Another basic distinction among tuning systems, considered by [[Gene Ward Smith]], is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all ~~2097151~~ notes of the {{w|MIDI tuning standard}}. An example of an open system is 12edo, which ~~puts~~ no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12edo, and in fact includes it, as in practice only a finite range of 12edo is used. From a theoretical point of view, 12edo has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12edo semitones.

Another basic distinction among tuning systems, considered by [[Gene Ward Smith]], is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2,097,151 notes of the {{w|MIDI tuning standard}}. An example of an open system is 12edo, which places no limit on how high or low the range of tones extends. From a practical point of view, the MTS is vastly more capable of representing musical intervals than 12edo, and in fact includes it, as in practice only a finite range of 12edo is used. From a theoretical point of view, 12edo has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12edo semitones.

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-limit|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]].

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-limit|3-prime-limit]] JI (also known as [[Pythagorean tuning]]), a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Regular temperaments (other than equal temperaments) are also open systems with infinitely many pitches in a finite range. Another example would be any of the golden horagrams of [[Erv Wilson]].

Among open systems, the most important kinds are [[periodic scale]]s and group systems. The latter refers to "groups" in the mathematical sense of {{w|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]]; all intervals in a [[just intonation subgroup]]; and all intervals in a regular temperament.

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 == Open and closed systems ==
-Another basic distinction among tuning systems, considered by [[Gene Ward Smith]], is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2097151 notes of the {{w|MIDI tuning standard}}. An example of an open system is 12edo, which puts no limit on how high or low the range of tones extends. From a practical point of view MTS is vastly more capable of representing musical intervals than 12edo, and in fact includes it, as in practice only a finite range of 12edo is used. From a theoretical point of view, 12edo has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12edo semitones.
+Another basic distinction among tuning systems, considered by [[Gene Ward Smith]], is between open and closed systems, where a closed system has a finite set of possible musical intervals, and an open system has an infinite set. An example of a closed system would be all 2,097,151 notes of the {{w|MIDI tuning standard}}. An example of an open system is 12edo, which places no limit on how high or low the range of tones extends. From a practical point of view, the MTS is vastly more capable of representing musical intervals than 12edo, and in fact includes it, as in practice only a finite range of 12edo is used. From a theoretical point of view, 12edo has an infinite set of available intervals, since mathematically there is nothing preventing you from calculating frequencies well beyond the range of human hearing (or the ability to produce such frequencies) that are nonetheless related to each other by 12edo semitones.
-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-limit|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]].
+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-limit|3-prime-limit]] JI (also known as [[Pythagorean tuning]]), a musical interval system in which intervals are generated by successive combinations of the 2nd and 3rd harmonics. Regular temperaments (other than equal temperaments) are also open systems with infinitely many pitches in a finite range. Another example would be any of the golden horagrams of [[Erv Wilson]].
 Among open systems, the most important kinds are [[periodic scale]]s and group systems. The latter refers to "groups" in the mathematical sense of {{w|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]]; all intervals in a [[just intonation subgroup]]; and all intervals in a regular temperament.