Diamond tradeoff: Difference between revisions

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A tuning for a [[rank]]-''r'' [[Prime limit|''p''-limit]] [[regular temperament]] is '''diamond pure''', or '''diamond nice''', if it fits the following definition: we may define the diamond pure tuning range by finding the [[Wikipedia: Convex hull|convex hull]] in [[Vals and Tuning Space|tuning space]] of the set of all tunings with one [[Eigenmonzo subgroup|eigenmonzo]] 2 (pure octaves tunings) and the rest of the eigenmonzos any set of ''r'' - 1 members of the ''p''-odd limit [[tonality diamond]], whenever such a tuning is defined.

A tuning for a [[rank]]-''r'' [[Prime limit|''p''-limit]] [[regular temperament]] is '''diamond pure''', or '''diamond nice''', if it fits the following definition: we may define the diamond pure tuning range by finding the [[Wikipedia: Convex hull|convex hull]] in [[Vals and Tuning Space|tuning space]] of the set of all tunings with ''r'' eigenmonzos chosen as follows: one [[Eigenmonzo subgroup|eigenmonzo]] 2 (pure octaves tunings) and the rest of the eigenmonzos any set of ''r'' - 1 members of the ''p''-odd limit [[tonality diamond]], whenever such a tuning is defined.

In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] — and to some extent on the wiki and in the regular temperament community — this tuning range was referred to simply as the "nice" tuning range.

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	A tuning for a [[rank]]-''r'' [[Prime limit\|''p''-limit]] [[regular temperament]] is '''diamond pure''', or '''diamond nice''', if it fits the following definition: we may define the diamond pure tuning range by finding the [[Wikipedia: Convex hull\|convex hull]] in [[Vals and Tuning Space\|tuning space]] of the set of all tunings with one [[Eigenmonzo subgroup\|eigenmonzo]] 2 (pure octaves tunings) and the rest of the eigenmonzos any set of ''r'' - 1 members of the ''p''-odd limit [[tonality diamond]], whenever such a tuning is defined.		A tuning for a [[rank]]-''r'' [[Prime limit\|''p''-limit]] [[regular temperament]] is '''diamond pure''', or '''diamond nice''', if it fits the following definition: we may define the diamond pure tuning range by finding the [[Wikipedia: Convex hull\|convex hull]] in [[Vals and Tuning Space\|tuning space]] of the set of all tunings with ''r'' eigenmonzos chosen as follows: one [[Eigenmonzo subgroup\|eigenmonzo]] 2 (pure octaves tunings) and the rest of the eigenmonzos any set of ''r'' - 1 members of the ''p''-odd limit [[tonality diamond]], whenever such a tuning is defined.

	In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] — and to some extent on the wiki and in the regular temperament community — this tuning range was referred to simply as the "nice" tuning range.		In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] — and to some extent on the wiki and in the regular temperament community — this tuning range was referred to simply as the "nice" tuning range.