Diamond tradeoff: Difference between revisions

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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.

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 {{w|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.

[[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 {{w|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 (2/1, octaves) and the rest of the eigenmonzos any set of {{nowrap|''r'' − 1}} members of the ''q''-odd limit [[tonality diamond]], whenever such a tuning is defined.

== Original name ==

In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]], which defined this tuning range, it was known as the "purer" tuning range.<ref>Milne, A. J., Sethares, W. A., and Plamondon, J. (2006). X System. Technical report, Thumtronics, Inc. https://oro.open.ac.uk/21510/1/X_System.pdf</ref> On the wiki and in the regular temperament community, for many years this tuning range was referred to simply as the "nice" tuning range.

== Vs. diamond monotone ==

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

== Examples ==

=== 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.

The next thing to understand is that tunings in the middle between the pure-4/3 tuning and the pure-6/5 tuning (including the pure-5/4 tuning, because that's in the middle) are in a sense "reasonable compromises" because whenever you're making one interval of the diamond worse, you're always making another one better. You're in the realm of tradeoffs.

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.

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.

=== Computation ===

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==== First diamond extrema ====

Let's do it for 4/3 first, {{vector|2 -1 0}}. So, we prepare a matrix out of these two unchanged intervals, 2/1 and 4/3, and call it <math>U</math>:

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Next, we find the generators matrix <math>G</math> corresponding to the tuning where these two intervals are unchanged. The formula for this generators matrix is <math>G = U(MU)^{-1}</math> (for a full explanation of this formula~~, see [[Dave Keenan and Douglas Blumeyer's guide to RTT tuning#Unchanged interval tuning strategy]]~~). Here, let's work it out for our chosen <math>U</math>. First, we multiply <math>MU</math>:

Next, we find the generators matrix <math>G</math> corresponding to the tuning where these two intervals are unchanged. The formula for this generators matrix is <math>G = U(MU)^{-1}</math> (see [[Generator embedding optimization#Generator embedding|here]] for a full explanation of this formula). Here, let's work it out for our chosen <math>U</math>. First, we multiply <math>MU</math>:

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Reading the columns from <math>G</math>, the first one {{vector|1 0 0}} confirms our period of 2/1, and the second column {{vector|-1 1 0}} gives our generator 3/2. Which is unsurprising. In cents, that's ~~1200¢~~ × ~~log₂~~(3/2) ≈ 701.~~955¢~~. The next unchanged interval will give a more interesting result.

Reading the columns from <math>G</math>, the first one {{vector|1 0 0}} confirms our period of 2/1, and the second column {{vector|-1 1 0}} gives our generator 3/2. Which is unsurprising. In cents, that's {{nowrap|1200 × log2(3/2) ≈ 701.955{{cent}}}}. The next unchanged interval will give a more interesting result.

==== Second diamond extrema ====

So let's do 5/4 now, {{vector|-2 0 1}}. We prepare a matrix out of these two unchanged intervals, 2/1 and 5/4, and call it <math>U</math>:

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(MU)^{-1} \\

\left[ \begin{array} {rrr}

1 & \~~frac12~~ \\

1 & \frac{1}{2} \\

0 & \~~frac14~~ \\

0 & \frac{1}{4} \\

\end{array} \right]

\end{array}

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(MU)^{-1} \\

\left[ \begin{array} {rrr}

1 & \~~frac12~~ \\

1 & \frac{1}{2} \\

0 & \~~frac14~~ \\

0 & \frac{1}{4} \\

\end{array} \right]

\end{array}

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1 & 0 \\

0 & 0 \\

0 & \~~frac14~~ \\

0 & \frac{1}{4} \\

\end{array} \right]

\end{array}

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This tells us our generator is {~~{vector|~~0 0 ~~<math>~~\~~frac14~~</math>}}, or 5^(1/4). In cents, that's ~~1200¢~~ × ~~log₂~~(~~5¹⸍⁴~~) ≈ 696.~~578¢~~.

This tells us our generator is <math>\monzo{0 & 0 & \frac{1}{4}}</math>, or 5{{frac|1|4}}. In cents, that's {{nowrap|1200 × log2(5{{frac|1|4}}) ≈ 696.578{{c}}}}.

==== Third diamond extrema ====

Okay, one more unchanged interval to check: 6/5, which is {{vector|1 1 -1}}. We prepare a matrix out of these two unchanged intervals, 2/1 and 6/5, and call it <math>U</math>:

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(MU)^{-1} \\

\left[ \begin{array} {rrr}

1 & \~~frac23~~ \\

1 & \frac{2}{3} \\

0 & -\~~frac13~~ \\

0 & -\frac{1}{3} \\

\end{array} \right]

\end{array}

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(MU)^{-1} \\

\left[ \begin{array} {rrr}

1 & \~~frac23~~ \\

1 & \frac{2}{3} \\

0 & -\~~frac13~~ \\

0 & -\frac{1}{3} \\

\end{array} \right]

\end{array}

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G \\

\left[ \begin{array} {rrr}

1 & \~~frac13~~ \\

1 & \frac{1}{3} \\

0 & -\~~frac13~~ \\

0 & -\frac{1}{3} \\

0 & \~~frac13~~ \\

0 & \frac{1}{3} \\

\end{array} \right]

\end{array}

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This tells us our generator is ~~{{vector|~~<math>\~~frac13</math> <math>~~-\~~frac13</math> <math>~~\~~frac13~~</math>}}, or expressed another way, (10/3)^(1/3). In cents, that's ~~1200¢~~ × ~~log₂~~((10/3)~~¹⸍³~~) ≈ 694.~~786¢~~.

This tells us our generator is <math>\monzo{\frac{1}{3} & -\frac{1}{3} & \frac{1}{3}}</math>, or expressed another way, ({{frac|10|3}}){{frac|1|3}}. In cents, that's {{nowrap|1200 × log2(({{frac|10|3}}){{frac|1|3}}) ≈ 694.786{{c}}}}.

==== Determining the final range ====

We now have our generator sizes that give us pure consonances in the tonality diamond: 701.955¢, 696.578¢, and 694.786¢. The minimum of those is 694.786¢ and the maximum is 701.955¢, so that's our diamond tradeoff range. Anywhere inside that range, we are making at least one of our diamond consonances purer; outside it, we're making them all less pure.

== See also ==

For examples and other information, see [[tuning ranges of regular temperaments]].

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

== References ==

~~== References ==~~

[[Category:Regular temperament theory]]

[[Category:Terms]]

@@ Line 1: / Line 1: @@
-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.
+{{texmap}}
+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&nbsp;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 {{w|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.
+[[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 {{w|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 (2/1, octaves) and the rest of the eigenmonzos any set of {{nowrap|''r'' &minus; 1}} members of the ''q''-odd limit [[tonality diamond]], whenever such a tuning is defined.
 == Original name ==
 In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]], which defined this tuning range, it was known as the "purer" tuning range.<ref>Milne, A. J., Sethares, W. A., and Plamondon, J. (2006). X System. Technical report, Thumtronics, Inc. https://oro.open.ac.uk/21510/1/X_System.pdf</ref> On the wiki and in the regular temperament community, for many years this tuning range was referred to simply as the "nice" tuning range.
 == Vs. diamond monotone ==
 Diamond tradeoff tunings are always guaranteed to occur, but [[diamond monotone]] tunings are not.
 == Examples ==
 === 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.
 The next thing to understand is that tunings in the middle between the pure-4/3 tuning and the pure-6/5 tuning (including the pure-5/4 tuning, because that's in the middle) are in a sense "reasonable compromises" because whenever you're making one interval of the diamond worse, you're always making another one better. You're in the realm of tradeoffs.
-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.
+On the other hand, if you go outside these boundaries&mdash;for example, if you make 4/3 even flatter than pure&mdash;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.
 === Computation ===
@@ Line 42: / Line 40: @@
 ==== First diamond extrema ====
 Let's do it for 4/3 first, {{vector|2 -1 0}}. So, we prepare a matrix out of these two unchanged intervals, 2/1 and 4/3, and call it <math>U</math>:
@@ Line 55: / Line 52: @@
-Next, we find the generators matrix <math>G</math> corresponding to the tuning where these two intervals are unchanged. The formula for this generators matrix is <math>G = U(MU)^{-1}</math> (for a full explanation of this formula, see [[Dave Keenan and Douglas Blumeyer's guide to RTT tuning#Unchanged interval tuning strategy]]). Here, let's work it out for our chosen <math>U</math>. First, we multiply <math>MU</math>:
+Next, we find the generators matrix <math>G</math> corresponding to the tuning where these two intervals are unchanged. The formula for this generators matrix is <math>G = U(MU)^{-1}</math> (see [[Generator embedding optimization#Generator embedding|here]] for a full explanation of this formula). Here, let's work it out for our chosen <math>U</math>. First, we multiply <math>MU</math>:
@@ Line 152: / Line 149: @@
-Reading the columns from <math>G</math>, the first one {{vector|1 0 0}} confirms our period of 2/1, and the second column {{vector|-1 1 0}} gives our generator 3/2. Which is unsurprising. In cents, that's 1200¢ × log₂(3/2) ≈ 701.955¢. The next unchanged interval will give a more interesting result.
+Reading the columns from <math>G</math>, the first one {{vector|1 0 0}} confirms our period of 2/1, and the second column {{vector|-1 1 0}} gives our generator 3/2. Which is unsurprising. In cents, that's {{nowrap|1200 × log<sub>2</sub>(3/2) ≈ 701.955{{cent}}}}. The next unchanged interval will give a more interesting result.
 ==== Second diamond extrema ====
 So let's do 5/4 now, {{vector|-2 0 1}}. We prepare a matrix out of these two unchanged intervals, 2/1 and 5/4, and call it <math>U</math>:
@@ Line 223: / Line 219: @@
 (MU)^{-1} \\
 \left[ \begin{array} {rrr}
-& \frac12 \\
+& \frac{1}{2} \\
-& \frac14 \\
+& \frac{1}{4} \\
 \end{array} \right]
 \end{array}
@@ Line 248: / Line 244: @@
 (MU)^{-1} \\
 \left[ \begin{array} {rrr}
-& \frac12 \\
+& \frac{1}{2} \\
-& \frac14 \\
+& \frac{1}{4} \\
 \end{array} \right]
 \end{array}
@@ Line 260: / Line 256: @@
 & 0 \\
 & 0 \\
-& \frac14 \\
+& \frac{1}{4} \\
 \end{array} \right]
 \end{array}
@@ Line 267: / Line 263: @@
-This tells us our generator is {{vector|0 0 <math>\frac14</math>}}, or 5^(1/4). In cents, that's 1200¢ × log₂(5¹⸍⁴) ≈ 696.578¢.
+This tells us our generator is <math>\monzo{0 & 0 & \frac{1}{4}}</math>, or 5<sup>{{frac|1|4}}</sup>. In cents, that's {{nowrap|1200 × log<sub>2</sub>(5<sup>{{frac|1|4}}</sup>) ≈ 696.578{{c}}}}.
 ==== Third diamond extrema ====
 Okay, one more unchanged interval to check: 6/5, which is {{vector|1 1 -1}}. We prepare a matrix out of these two unchanged intervals, 2/1 and 6/5, and call it <math>U</math>:
@@ Line 338: / Line 333: @@
 (MU)^{-1} \\
 \left[ \begin{array} {rrr}
-& \frac23 \\
+& \frac{2}{3} \\
-& -\frac13 \\
+& -\frac{1}{3} \\
 \end{array} \right]
 \end{array}
@@ Line 363: / Line 358: @@
 (MU)^{-1} \\
 \left[ \begin{array} {rrr}
-& \frac23 \\
+& \frac{2}{3} \\
-& -\frac13 \\
+& -\frac{1}{3} \\
 \end{array} \right]
 \end{array}
@@ Line 373: / Line 368: @@
 G \\
 \left[ \begin{array} {rrr}
-& \frac13 \\
+& \frac{1}{3} \\
-& -\frac13 \\
+& -\frac{1}{3} \\
-& \frac13 \\
+& \frac{1}{3} \\
 \end{array} \right]
 \end{array}
@@ Line 382: / Line 377: @@
-This tells us our generator is {{vector|<math>\frac13</math> <math>-\frac13</math> <math>\frac13</math>}}, or expressed another way, (10/3)^(1/3). In cents, that's 1200¢ × log₂((10/3)¹⸍³) ≈ 694.786¢.
+This tells us our generator is <math>\monzo{\frac{1}{3} & -\frac{1}{3} & \frac{1}{3}}</math>, or expressed another way, ({{frac|10|3}})<sup>{{frac|1|3}}</sup>. In cents, that's {{nowrap|1200 × log<sub>2</sub>(({{frac|10|3}})<sup>{{frac|1|3}}</sup>) ≈ 694.786{{c}}}}.
 ==== Determining the final range ====
 We now have our generator sizes that give us pure consonances in the tonality diamond: 701.955¢, 696.578¢, and 694.786¢. The minimum of those is 694.786¢ and the maximum is 701.955¢, so that's our diamond tradeoff range. Anywhere inside that range, we are making at least one of our diamond consonances purer; outside it, we're making them all less pure.
 == See also ==
+For examples and other information, see [[tuning ranges of regular temperaments]].
-For examples and other information, see the topic page [[Tuning_ranges_of_regular_temperaments|Tuning ranges of regular temperaments]].
+== References ==
+<references />
-== References ==
 [[Category:Regular temperament theory]]
 [[Category:Terms]]