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 ''r'' [[eigenmonzo subgroup|eigenmonzo]]s chosen as follows: one 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 purer''', or '''diamond strict''', if it fits the following definition: we may define the diamond purer 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'' [[eigenmonzo subgroup|eigenmonzo]]s chosen as follows: one 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 (this is not the same thing as what is now called [[diamond nice]], which is the combination of diamond purer and [[diamond monotone]]).

Diamond ~~pure~~ tunings are always guaranteed to occur, but diamond monotone tunings are not.

Diamond purer tunings are always guaranteed to occur, but diamond monotone tunings are not.

For examples and other information, see the topic page [[Tuning_ranges_of_regular_temperaments|Tuning ranges of regular temperaments]].

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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 ''r'' [[eigenmonzo subgroup|eigenmonzo]]s chosen as follows: one 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 purer''', or '''diamond strict''', if it fits the following definition: we may define the diamond purer 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'' [[eigenmonzo subgroup|eigenmonzo]]s chosen as follows: one 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 (this is not the same thing as what is now called [[diamond nice]], which is the combination of diamond purer and [[diamond monotone]]).
-Diamond pure tunings are always guaranteed to occur, but diamond monotone tunings are not.
+Diamond purer tunings are always guaranteed to occur, but diamond monotone tunings are not.
 For examples and other information, see the topic page [[Tuning_ranges_of_regular_temperaments|Tuning ranges of regular temperaments]].