Diamond tradeoff: Difference between revisions

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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.

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.

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]])~~.

In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] this tuning range was known as the "purer" range. On the wiki and in the regular temperament community, for many years this tuning range was referred to simply as the "nice" tuning range.

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

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

The diamond ~~purer~~ tuning range marks tuning boundaries inside of which the temperament's approximations to simple low-ratio frequency ratios can be "traded" against each other - i.e., if I make the 3/2 more accurate, the 5/4 will suffer. However, outside this range, you will improve the tunings of *all* such intervals by moving back inside. This range therefore makes sense when one is concerned with approximating JI as closely as possible (without asserting a priori which specific consonances are the most important) because, under that criterion, it makes no logical sense to choose a tuning outside that range.

The diamond tradeoff tuning range marks tuning boundaries inside of which the temperament's approximations to simple low-ratio frequency ratios can be "traded" against each other - i.e., if I make the 3/2 more accurate, the 5/4 will suffer. However, outside this range, you will improve the tunings of *all* such intervals by moving back inside. This range therefore makes sense when one is concerned with approximating JI as closely as possible (without asserting a priori which specific consonances are the most important) because, under that criterion, it makes no logical sense to choose a tuning outside that range.

However, it is quite clear that tunings outside of this ~~purer~~ range can function perfectly well as less accurate (and arguably more characterful) representations of the JI intervals specified by the temperament. That is, they are likely to be correctly recognized (whatever that actually means). For example, a 17-TET rendition of a standard piece of meantone music still makes complete musical sense, and major and minor chords still sound like major and minor chords, even though this tuning is outside the ~~purer~~ tuning range.

However, it is quite clear that tunings outside of this diamond tradeoff range can function perfectly well as less accurate (and arguably more characterful) representations of the JI intervals specified by the temperament. That is, they are likely to be correctly recognized (whatever that actually means). For example, a 17-TET rendition of a standard piece of meantone music still makes complete musical sense, and major and minor chords still sound like major and minor chords, even though this tuning is outside the diamond tradeoff tuning range.

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 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.
+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.
-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]]).
+In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] this tuning range was known as the "purer" range. On the wiki and in the regular temperament community, for many years this tuning range was referred to simply as the "nice" tuning range.
-Diamond purer tunings are always guaranteed to occur, but diamond monotone tunings are not.
+Diamond tradeoff tunings are always guaranteed to occur, but diamond monotone tunings are not.
-The diamond purer tuning range marks tuning boundaries inside of which the temperament's approximations to simple low-ratio frequency ratios can be "traded" against each other - i.e., if I make the 3/2 more accurate, the 5/4 will suffer. However, outside this range, you will improve the tunings of *all* such intervals by moving back inside. This range therefore makes sense when one is concerned with approximating JI as closely as possible (without asserting a priori which specific consonances are the most important) because, under that criterion, it makes no logical sense to choose a tuning outside that range.
+The diamond tradeoff tuning range marks tuning boundaries inside of which the temperament's approximations to simple low-ratio frequency ratios can be "traded" against each other - i.e., if I make the 3/2 more accurate, the 5/4 will suffer. However, outside this range, you will improve the tunings of *all* such intervals by moving back inside. This range therefore makes sense when one is concerned with approximating JI as closely as possible (without asserting a priori which specific consonances are the most important) because, under that criterion, it makes no logical sense to choose a tuning outside that range.
-However, it is quite clear that tunings outside of this purer range can function perfectly well as less accurate (and arguably more characterful) representations of the JI intervals specified by the temperament. That is, they are likely to be correctly recognized (whatever that actually means). For example, a 17-TET rendition of a standard piece of meantone music still makes complete musical sense, and major and minor chords still sound like major and minor chords, even though this tuning is outside the purer tuning range.
+However, it is quite clear that tunings outside of this diamond tradeoff range can function perfectly well as less accurate (and arguably more characterful) representations of the JI intervals specified by the temperament. That is, they are likely to be correctly recognized (whatever that actually means). For example, a 17-TET rendition of a standard piece of meantone music still makes complete musical sense, and major and minor chords still sound like major and minor chords, even though this tuning is outside the diamond tradeoff tuning range.
 For examples and other information, see the topic page [[Tuning_ranges_of_regular_temperaments|Tuning ranges of regular temperaments]].