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~~ (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)~~.

A tuning for a [[regular temperament]] is '''diamond tradeoff''', or '''diamond strict''', if it is in the range where the concerning approximations to simple [[frequency ratio]]s can have their qualities "traded" against each other, that is, if some ratios are made more accurate, the others will be less so. For example, if you make the 3/2 more accurate, the 5/4 will suffer. However, outside this range, the tunings of ''all'' such intervals will be improved 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 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.

== Mathematical definition ==

[[Gene Ward Smith]] gives the following definition. The [[Odd limit|''q''-odd-limit]] '''diamond tradeoff''' range of a [[rank]]-''r'' [[Prime limit|''p''-limit]] temperament is the [[Wikipedia: Convex hull|convex hull]] in [[Vals and tuning Space|tuning space]] of the set of all tunings with ''r'' [[eigenmonzo|eigenmonzos (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 ''q''-odd limit [[tonality diamond]], whenever such a tuning is defined.

== Original name ==

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

== ~~Additional notes from Andrew Milne~~ ==

== Examples ==

=== Explanation adapted from Keenan Pepper ===

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; that is, 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 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.

== Explanation adapted from Keenan Pepper ==

For meantone temperament, there are three specific tunings that are special: one that tunes 4/3 and 3/2 pure, another that tunes 5/4 and 8/5 pure, and the third that tunes 6/5 and 5/3 pure. The tradeoff tuning range consists of these three points in tuning space ''and everything in between''. In this case, the three points fall along a line, where the pure-5/4 tuning is in between the pure-4/3 tuning and the pure-6/5 tuning.

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On the other hand, if you go outside these boundaries - for example, if you make 4/3 even flatter than pure - then you're making some intervals in the 5-limit diamond worse without making ''any'' of them better. You're past the realm of compromises and now you're just damaging things for no reason.

== ~~Example~~ ==

=== Computation ===

Here we will demonstrate the calculation of the diamond tradeoff tuning range for meantone.

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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 (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).
+A tuning for a [[regular temperament]] is '''diamond tradeoff''', or '''diamond strict''', if it is in the range where the concerning approximations to simple [[frequency ratio]]s can have their qualities "traded" against each other, that is, if some ratios are made more accurate, the others will be less so. For example, if you make the 3/2 more accurate, the 5/4 will suffer. However, outside this range, the tunings of ''all'' such intervals will be improved 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 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.
+== Mathematical definition ==
+[[Gene Ward Smith]] gives the following definition. The [[Odd limit|''q''-odd-limit]] '''diamond tradeoff''' range of a [[rank]]-''r'' [[Prime limit|''p''-limit]] temperament is the [[Wikipedia: Convex hull|convex hull]] in [[Vals and tuning Space|tuning space]] of the set of all tunings with ''r'' [[eigenmonzo|eigenmonzos (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 ''q''-odd limit [[tonality diamond]], whenever such a tuning is defined.
 == Original name ==
@@ Line 9: / Line 14: @@
 Diamond tradeoff tunings are always guaranteed to occur, but [[diamond monotone]] tunings are not.
-== Additional notes from Andrew Milne ==
+== Examples ==
+=== Explanation adapted from Keenan Pepper ===
-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; that is, 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 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.
-== Explanation adapted from Keenan Pepper ==
 For meantone temperament, there are three specific tunings that are special: one that tunes 4/3 and 3/2 pure, another that tunes 5/4 and 8/5 pure, and the third that tunes 6/5 and 5/3 pure. The tradeoff tuning range consists of these three points in tuning space ''and everything in between''. In this case, the three points fall along a line, where the pure-5/4 tuning is in between the pure-4/3 tuning and the pure-6/5 tuning.
@@ Line 23: / Line 23: @@
 On the other hand, if you go outside these boundaries - for example, if you make 4/3 even flatter than pure - then you're making some intervals in the 5-limit diamond worse without making ''any'' of them better. You're past the realm of compromises and now you're just damaging things for no reason.
-== Example ==
+=== Computation ===
 Here we will demonstrate the calculation of the diamond tradeoff tuning range for meantone.