Harmonic limit: Difference between revisions

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== Prime limits as subgroups ==

Prime limits are essentially special cases of [[just intonation subgroup|subgroups]] that include all primes up to the limit rather than skipping any. For any prime number ''p'', the set of all rational numbers in the ''p''-limit defines a ~~{{w|~~Free abelian group|finitely generated free abelian group~~}}.~~ The [[rank]] of this group is equal to π (''p''), the {{w|Prime-counting function|number of prime numbers less than or equal to ''p''}}. Hence, for example, the rank of the [[7-limit]] is 4, as it is generated by 2, 3, 5 and 7. In many cases, it is often more useful to speak of subgroups of the prime-limit, rather than the full limit, and this becomes increasingly true for higher limits as the number of useful temperaments with a good approximation of full limits dwindles, and for that purpose, the term "''p''-horizon" can be used to refer to an entire umbrella of subgroups encompassed by the ''p''-limit.

Prime limits are essentially special cases of [[just intonation subgroup|subgroups]] that include all primes up to the limit rather than skipping any. For any prime number ''p'', the set of all rational numbers in the ''p''-limit defines a [[Free abelian group|finitely generated free abelian group]] The [[rank]] of this group is equal to π (''p''), the {{w|Prime-counting function|number of prime numbers less than or equal to ''p''}}. Hence, for example, the rank of the [[7-limit]] is 4, as it is generated by 2, 3, 5 and 7. In many cases, it is often more useful to speak of subgroups of the prime-limit, rather than the full limit, and this becomes increasingly true for higher limits as the number of useful temperaments with a good approximation of full limits dwindles, and for that purpose, the term "''p''-horizon" can be used to refer to an entire umbrella of subgroups encompassed by the ''p''-limit.

== Individual pages of ''p''-limit JI ==

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 == Prime limits as subgroups ==
-Prime limits are essentially special cases of [[just intonation subgroup|subgroups]] that include all primes up to the limit rather than skipping any. For any prime number ''p'', the set of all rational numbers in the ''p''-limit defines a {{w|Free abelian group|finitely generated free abelian group}}. The [[rank]] of this group is equal to π (''p''), the {{w|Prime-counting function|number of prime numbers less than or equal to ''p''}}. Hence, for example, the rank of the [[7-limit]] is 4, as it is generated by 2, 3, 5 and 7. In many cases, it is often more useful to speak of subgroups of the prime-limit, rather than the full limit, and this becomes increasingly true for higher limits as the number of useful temperaments with a good approximation of full limits dwindles, and for that purpose, the term "''p''-horizon" can be used to refer to an entire umbrella of subgroups encompassed by the ''p''-limit.
+Prime limits are essentially special cases of [[just intonation subgroup|subgroups]] that include all primes up to the limit rather than skipping any. For any prime number ''p'', the set of all rational numbers in the ''p''-limit defines a [[Free abelian group|finitely generated free abelian group]] The [[rank]] of this group is equal to π (''p''), the {{w|Prime-counting function|number of prime numbers less than or equal to ''p''}}. Hence, for example, the rank of the [[7-limit]] is 4, as it is generated by 2, 3, 5 and 7. In many cases, it is often more useful to speak of subgroups of the prime-limit, rather than the full limit, and this becomes increasingly true for higher limits as the number of useful temperaments with a good approximation of full limits dwindles, and for that purpose, the term "''p''-horizon" can be used to refer to an entire umbrella of subgroups encompassed by the ''p''-limit.
 == Individual pages of ''p''-limit JI ==