Diamond tradeoff: Difference between revisions

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A tuning for a [[rank]]-''r'' [[Prime limit|''p''-limit]] [[regular temperament]] is '''diamond tradeoff''', or '''diamond strict''', if it fits the following definition: we may define the diamond tradeoff 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 (this definition is based on Gene Ward Smith's one).

A tuning for a [[rank]]-''r'' [[Prime limit|''p''-limit]] [[regular temperament]] is '''diamond tradeoff''', or '''diamond strict''', if it fits the following definition: we may define the diamond tradeoff 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 (unchanged-intervals) 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 (this definition is based on Gene Ward Smith's one).

== Original name ==

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This is a 5-limit temperament, so we consider the 5-limit tonality diamond: {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}. Of these seven pitches, there are only three we care about. We don't care about the unison, and half of the remaining pitches are octave-complements of the others are thus irrelevant. So, we'll only look at {4/3, 5/4, 6/5}.

For each of these three diamond consonances, we want to find what generator is required in order that this pitch remains pure after tempering, or in other words, that it is an unchanged interval (sometimes called an [[eigenmonzo]]). And we want to know this for the situation where octaves are pure.

For each of these three diamond consonances, we want to find what generator is required in order that this pitch remains pure after tempering, or in other words, that it is an unchanged interval (sometimes called an [[eigenmonzo]], or unchanged-interval). And we want to know this for the situation where octaves are pure.

==== First diamond extrema ====

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-A tuning for a [[rank]]-''r'' [[Prime limit|''p''-limit]] [[regular temperament]] is '''diamond tradeoff''', or '''diamond strict''', if it fits the following definition: we may define the diamond tradeoff 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 (this definition is based on Gene Ward Smith's one).
+A tuning for a [[rank]]-''r'' [[Prime limit|''p''-limit]] [[regular temperament]] is '''diamond tradeoff''', or '''diamond strict''', if it fits the following definition: we may define the diamond tradeoff 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 (unchanged-intervals) 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 (this definition is based on Gene Ward Smith's one).
 == Original name ==
@@ Line 40: / Line 40: @@
 This is a 5-limit temperament, so we consider the 5-limit tonality diamond: {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}. Of these seven pitches, there are only three we care about. We don't care about the unison, and half of the remaining pitches are octave-complements of the others are thus irrelevant. So, we'll only look at {4/3, 5/4, 6/5}.
-For each of these three diamond consonances, we want to find what generator is required in order that this pitch remains pure after tempering, or in other words, that it is an unchanged interval (sometimes called an [[eigenmonzo]]). And we want to know this for the situation where octaves are pure.
+For each of these three diamond consonances, we want to find what generator is required in order that this pitch remains pure after tempering, or in other words, that it is an unchanged interval (sometimes called an [[eigenmonzo]], or unchanged-interval). And we want to know this for the situation where octaves are pure.
 ==== First diamond extrema ====