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 [[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 (2/1, 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.

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.

== Vs. diamond monotone ==

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.

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.

== ~~Example~~ ==

=== Computation ===

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

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

Here is the mapping, <math>M</math>:

~~Here is the mapping, <math>M</math>:~~

<math>

\begin{~~bmatrix~~}

\left[ \begin{array} {rrr}

1 & 0 & -4 \\

0 & 1 & 4 \\

\end{~~bmatrix~~}

\end{array} \right]

</math>

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.

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]], or unchanged-interval). And we want to know this for the situation where octaves are pure.

==== 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>:

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

<math>

\begin{~~bmatrix~~}

\left[ \begin{array} {rrr}

1 & 2 \\

0 & -1 \\

0 & 0 \\

\end{~~bmatrix~~}

\end{array} \right]

</math>

One property of an unchanged interval of a tuning is that it is [https://mathworld.wolfram.com/Eigenvector.html eigenvector] of the projection matrix<ref>For now, the best explanation of projection matrices seems to be on the [[fractional monzo]]s page.</ref> for the tuning where the eigenvalue <math>λ</math> is 1. If the projection matrix is <math>P</math>, by the definition of eigenvectors, that means <math>P⋅U = λ⋅U</math>, or <math>P⋅U = 1⋅U</math>, or simply <math>P⋅U = U</math>. In other words, the projection matrix maps the interval to itself; it is unchanged by the tuning. Because we know what <math>U</math> is, we could solve for <math>P</math> now. But we don't want <math>P</math>; we want the generators. Fortunately, <math>P</math> is defined in terms of our desired generators, <math>G</math>, and our mapping, <math>M</math>, like this: <math>P = GM</math>. Now, <math>G</math> is the [https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse pseudoinverse] of <math>M</math><ref>we can understand this fact in this context to mean that when you multiply <math>M</math> and <math>G</math> the other way around, <math>MG</math>, you get <math>I</math>, the identity matrix, and this is because <math>G</math> represents the generators in terms of the primes, just like any ordinary prime count vector ([[monzo]]), except often using fractional powers, so we map it with a mapping just like any ordinary interval, and if you look at the columns of <math>I</math> and think of them as generator count vectors, you can see that it represents our targets. In other words, <math>G</math> is the prime count vectors for which each one gets mapped to a different single one of the temperament's generators.</ref>; we often write the pseudoinverse of <math>M</math> as <math>M⁺</math>, so let's write <math>G</math> instead as <math>M⁺</math>. So if <math>P = M⁺⋅M</math>, then we can substitute that in for <math>P</math>, and our equation will now be <math>M⁺⋅M⋅U = U</math>. But we want to solve for our generators, so that's <math>M⁺</math>. So if we right multiply both sides by the inverse of <math>(M⋅U)</math>, we get <math>M⁺ = U(MU)^{-1}</math>; the <math>MU</math> on the left side cancels out. The rest is busywork.

~~We multiply~~ <~~span~~><math>~~M⋅U~~</math></~~span~~>:

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>:

<math>

\begin{~~bmatrix~~}

\begin{array} {c}

M \\

\left[ \begin{array} {rrr}

1 & 0 & -4 \\

0 & 1 & 4 \\

\end{~~bmatrix~~}

\end{array} \right]

\begin{~~bmatrix~~}

\end{array}

\begin{array} {c}

U \\

\left[ \begin{array} {rrr}

1 & 2 \\

0 & -1 \\

0 & 0 \\

\end{~~bmatrix~~}=

\end{array} \right]

\begin{~~bmatrix~~}

\end{array}

\begin{array} {c} \\ = \end{array}

\begin{array} {c}

MU \\

\left[ \begin{array} {rrr}

1 & 1 \\

0 & -1 \\

\end{~~bmatrix~~}

\end{array} \right]

\end{array}

</math>

We take the inverse ~~<math>(M⋅U)^{-1}</math>~~ (which in this case is the same):

We take the inverse (which in this case is the same) (the best way to find a matrix inverse is to use a mathematical calculation tool such as Wolfram Language, but it can be done by hand; for a step-by-step demonstration of how to take the inverse, see [[Defactoring algorithms#Inversion by hand]]):

<math>

\begin{~~bmatrix~~}

\begin{array} {c}

\\

\left[ \begin{array} {rrr}

1 & 1 \\

0 & -1 \\

\end{array} \right]^{-1}

\end{array}

\begin{array} {c} \\ = \end{array}

\begin{array} {c}

(MU)^{-1} \\

\left[ \begin{array} {rrr}

1 & 1 \\

0 & -1 \\

\end{~~bmatrix~~}

\end{array} \right]

\end{array}

</math>

Then find ~~~~<math>M⁺</math~~></span~~> which is ~~~~<math>U⋅(~~M⋅U~~)^{-1}</math~~></span~~>:

Then find <math>G</math> which is <math>U(MU)^{-1}</math>:

<math>

\begin{~~bmatrix~~}

\begin{array} {c}

U \\

\left[ \begin{array} {rrr}

1 & 2 \\

0 & -1 \\

0 & 0 \\

\end{~~bmatrix~~}

\end{array} \right]

\begin{~~bmatrix~~}

\end{array}

\begin{array} {c}

(MU)^{-1} \\

\left[ \begin{array} {rrr}

1 & 1 \\

0 & -1 \\

\end{~~bmatrix~~}=

\end{array} \right]

\begin{~~bmatrix~~}

\end{array}

\begin{array} {c} \\ = \end{array}

\begin{array} {c}

G \\

\left[ \begin{array} {rrr}

1 & -1 \\

0 & 1 \\

0 & 0 \\

\end{~~bmatrix~~}

\end{array} \right]

\end{array}

</math>

Reading the columns from <math>M⁺</math>, the first one confirms our period of 2/1, and the second column 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.

So let's do 5/4 now. We prepare a matrix out of these two unchanged intervals, 2/1 and 5/4, and call it ~~~~<math>U</math~~></span~~>:

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>:

<math>

\begin{~~bmatrix~~}

\left[ \begin{array} {rrr}

1 & -2 \\

0 & 0 \\

0 & 1 \\

\end{~~bmatrix~~}

\end{array} \right]

</math>

We multiply ~~~~<math>~~M⋅U~~</math~~></span~~>:

We multiply <math>MU</math>:

<math>

\begin{~~bmatrix~~}

\begin{array} {c}

M \\

\left[ \begin{array} {rrr}

1 & 0 & -4 \\

0 & 1 & 4 \\

\end{~~bmatrix~~}

\end{array} \right]

\begin{~~bmatrix~~}

\end{array}

\begin{array} {c}

U \\

\left[ \begin{array} {rrr}

1 & -2 \\

0 & 0 \\

0 & 1 \\

\end{~~bmatrix~~}=

\end{array} \right]

\begin{~~bmatrix~~}

\end{array}

\begin{array} {c} \\ = \end{array}

\begin{array} {c}

MU \\

\left[ \begin{array} {rrr}

1 & -2 \\

0 & 4 \\

\end{~~bmatrix~~}

\end{array} \right]

\end{array}

</math>

We take the inverse ~~<math>(M⋅U)^{-1}</math>~~:

We take the inverse:

<math>

\begin{~~bmatrix~~}

1 & \~~frac12~~ \\

\begin{array} {c}

0 & \~~frac14~~ \\

\\

\end{~~bmatrix~~}

\left[ \begin{array} {rrr}

1 & -2 \\

0 & 4 \\

\end{array} \right]^{-1}

\end{array}

\begin{array} {c} \\ = \end{array}

\begin{array} {c}

(MU)^{-1} \\

\left[ \begin{array} {rrr}

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

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

\end{array} \right]

\end{array}

</math>

Then find ~~~~<math>M⁺</math~~></span~~> which is ~~~~<math>U⋅(~~M⋅U~~)^{-1}</math~~></span~~>:

Then find <math>G</math> which is <math>U(MU)^{-1}</math>:

<math>

\begin{~~bmatrix~~}

\begin{array} {c}

U \\

\left[ \begin{array} {rrr}

1 & -2 \\

0 & 0 \\

0 & 1 \\

\end{~~bmatrix~~}

\end{array} \right]

\begin{~~bmatrix~~}

\end{array}

1 & \~~frac12~~ \\

0 & \~~frac14~~ \\

\begin{array} {c}

\end{~~bmatrix~~}=

(MU)^{-1} \\

\begin{~~bmatrix~~}

\left[ \begin{array} {rrr}

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

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

\end{array} \right]

\end{array}

\begin{array} {c} \\ = \end{array}

\begin{array} {c}

G \\

\left[ \begin{array} {rrr}

1 & 0 \\

0 & 0 \\

0 & \~~frac14~~ \\

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

\end{~~bmatrix~~}

\end{array} \right]

\end{array}

</math>

~~This tells us our generator is 5^(1/4). In cents, that's 1200¢ × log₂(5¹⸍⁴) ≈ 696.578¢.~~

Okay, one more unchanged interval to check: 6/5. We prepare a matrix out of these two unchanged intervals, 2/1 and 6/5, and call it ~~~~<math>U</math~~></span~~>:

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>:

<math>

\begin{~~bmatrix~~}

\left[ \begin{array} {rrr}

1 & 1 \\

0 & 1 \\

0 & -1 \\

\end{~~bmatrix~~}

\end{array} \right]

</math>

We multiply ~~~~<math>~~M⋅U~~</math~~></span~~>:

We multiply <math>MU</math>:

<math>

\begin{~~bmatrix~~}

\begin{array} {c}

M \\

\left[ \begin{array} {rrr}

1 & 0 & -4 \\

0 & 1 & 4 \\

\end{~~bmatrix~~}

\end{array} \right]

\begin{~~bmatrix~~}

\end{array}

\begin{array} {c}

U \\

\left[ \begin{array} {rrr}

1 & 1 \\

0 & 1 \\

0 & -1 \\

\end{~~bmatrix~~}=

\end{array} \right]

\begin{~~bmatrix~~}

\end{array}

\begin{array} {c} \\ = \end{array}

\begin{array} {c}

MU \\

\left[ \begin{array} {rrr}

1 & 2 \\

0 & -3 \\

\end{~~bmatrix~~}

\end{array} \right]

\end{array}

</math>

We take the inverse ~~<math>(M⋅U)^{-1}</math>~~:

We take the inverse:

<math>

\begin{~~bmatrix~~}

1 & \~~frac23~~ \\

\begin{array} {c}

0 & -\~~frac13~~ \\

\\

\end{~~bmatrix~~}

\left[ \begin{array} {rrr}

1 & 2 \\

0 & -3 \\

\end{array} \right]^{-1}

\end{array}

\begin{array} {c} \\ = \end{array}

\begin{array} {c}

(MU)^{-1} \\

\left[ \begin{array} {rrr}

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

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

\end{array} \right]

\end{array}

</math>

Then find ~~~~<math>M⁺</math~~></span~~> which is ~~~~<math>U⋅(~~M⋅U~~)^{-1}</math~~></span~~>:

Then find <math>G</math> which is <math>U(MU)^{-1}</math>:

<math>

\begin{~~bmatrix~~}

\begin{array} {c}

U \\

\left[ \begin{array} {rrr}

1 & 1 \\

0 & 1 \\

0 & -1 \\

\end{~~bmatrix~~}

\end{array} \right]

\begin{~~bmatrix~~}

\end{array}

1 & \~~frac23~~ \\

0 & -\~~frac13~~ \\

\begin{array} {c}

\end{~~bmatrix~~}=

(MU)^{-1} \\

\begin{~~bmatrix~~}

\left[ \begin{array} {rrr}

1 & \~~frac13~~ \\

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

0 & -\~~frac13~~ \\

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

0 & \~~frac13~~ \\

\end{array} \right]

\end{~~bmatrix~~}

\end{array}

\begin{array} {c} \\ = \end{array}

\begin{array} {c}

G \\

\left[ \begin{array} {rrr}

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

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

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

\end{array} \right]

\end{array}

</math>

~~This tells us our generator is (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 ==

[[Category:Regular temperament theory]]

@@ Line 1: / Line 1: @@
-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).
+{{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 (2/1, 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.
-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.
 == Vs. diamond monotone ==
 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.
 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.
-== Example ==
+=== Computation ===
+Here we will demonstrate the calculation of the diamond tradeoff tuning range for meantone.
-Here we will demonstrate the calculation of the diamond tradeoff tuning range for meantone.
+Here is the mapping, <math>M</math>:
-Here is the mapping, <span><math>M</math></span>:
 <math>
-\begin{bmatrix}
+\left[ \begin{array} {rrr}
 & 0 & -4 \\
 & 1 & 4 \\
-\end{bmatrix}
+\end{array} \right]
 </math>
-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.
+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]], or unchanged-interval). And we want to know this for the situation where octaves are pure.
+==== 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>:
-Let's do it for 4/3 first. So, we prepare a matrix out of these two unchanged intervals, 2/1 and 4/3, and call it <span><math>U</math></span>:
 <math>
-\begin{bmatrix}
+\left[ \begin{array} {rrr}
 & 2 \\
 & -1 \\
 & 0 \\
-\end{bmatrix}
+\end{array} \right]
 </math>
-One property of an unchanged interval of a tuning is that it is [https://mathworld.wolfram.com/Eigenvector.html eigenvector] of the projection matrix<ref>For now, the best explanation of projection matrices seems to be on the [[fractional monzo]]s page.</ref> for the tuning where the eigenvalue <span><math>λ</math></span> is 1. If the projection matrix is <span><math>P</math></span>, by the definition of eigenvectors, that means <span><math>P⋅U = λ⋅U</math></span>, or <span><math>P⋅U = 1⋅U</math></span>, or simply <span><math>P⋅U = U</math></span>. In other words, the projection matrix maps the interval to itself; it is unchanged by the tuning. Because we know what <span><math>U</math></span> is, we could solve for <span><math>P</math></span> now. But we don't want <span><math>P</math></span>; we want the generators. Fortunately, <span><math>P</math></span> is defined in terms of our desired generators, <span><math>G</math></span>, and our mapping, <span><math>M</math></span>, like this: <span><math>P = GM</math></span>. Now, <span><math>G</math></span> is the [https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse pseudoinverse] of <span><math>M</math></span><ref>we can understand this fact in this context to mean that when you multiply <span><math>M</math></span> and <span><math>G</math></span> the other way around, <span><math>MG</math></span>, you get <span><math>I</math></span>, the identity matrix, and this is because <span><math>G</math></span> represents the generators in terms of the primes, just like any ordinary prime count vector ([[monzo]]), except often using fractional powers, so we map it with a mapping just like any ordinary interval, and if you look at the columns of <span><math>I</math></span> and think of them as generator count vectors, you can see that it represents our targets. In other words, <span><math>G</math></span> is the prime count vectors for which each one gets mapped to a different single one of the temperament's generators.</ref>; we often write the pseudoinverse of <span><math>M</math></span> as <span><math>M⁺</math></span>, so let's write <span><math>G</math></span> instead as <span><math>M⁺</math></span>. So if <span><math>P = M⁺⋅M</math></span>, then we can substitute that in for <span><math>P</math></span>, and our equation will now be <span><math>M⁺⋅M⋅U = U</math></span>. But we want to solve for our generators, so that's <span><math>M⁺</math></span>. So if we right multiply both sides by the inverse of <span><math>(M⋅U)</math></span>, we get <span><math>M⁺ = U(MU)^{-1}</math></span>; the <span><math>MU</math></span> on the left side cancels out. The rest is busywork.
-We multiply <span><math>M⋅U</math></span>:
+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>:
 <math>
-\begin{bmatrix}
+\begin{array} {c}
+M \\
+\left[ \begin{array} {rrr}
 & 0 & -4 \\
 & 1 & 4 \\
-\end{bmatrix}
+\end{array} \right]
-\begin{bmatrix}
+\end{array}
+\begin{array} {c}
+U \\
+\left[ \begin{array} {rrr}
 & 2 \\
 & -1 \\
 & 0 \\
-\end{bmatrix}=
+\end{array} \right]
-\begin{bmatrix}
+\end{array}
+\begin{array} {c} \\ = \end{array}
+\begin{array} {c}
+MU \\
+\left[ \begin{array} {rrr}
 & 1 \\
 & -1 \\
-\end{bmatrix}
+\end{array} \right]
+\end{array}
 </math>
-We take the inverse <span><math>(M⋅U)^{-1}</math></span> (which in this case is the same):
+We take the inverse (which in this case is the same) (the best way to find a matrix inverse is to use a mathematical calculation tool such as Wolfram Language, but it can be done by hand; for a step-by-step demonstration of how to take the inverse, see [[Defactoring algorithms#Inversion by hand]]):
 <math>
-\begin{bmatrix}
+\begin{array} {c}
+ \\
+\left[ \begin{array} {rrr}
+& 1 \\
+& -1 \\
+\end{array} \right]^{-1}
+\end{array}
+\begin{array} {c} \\ = \end{array}
+\begin{array} {c}
+(MU)^{-1} \\
+\left[ \begin{array} {rrr}
 & 1 \\
 & -1 \\
-\end{bmatrix}
+\end{array} \right]
+\end{array}
 </math>
-Then find <span><math>M⁺</math></span> which is <span><math>U⋅(M⋅U)^{-1}</math></span>:
+Then find <math>G</math> which is <math>U(MU)^{-1}</math>:
 <math>
-\begin{bmatrix}
+\begin{array} {c}
+U \\
+\left[ \begin{array} {rrr}
 & 2 \\
 & -1 \\
 & 0 \\
-\end{bmatrix}
+\end{array} \right]
-\begin{bmatrix}
+\end{array}
+\begin{array} {c}
+(MU)^{-1} \\
+\left[ \begin{array} {rrr}
 & 1 \\
 & -1 \\
-\end{bmatrix}=
+\end{array} \right]
-\begin{bmatrix}
+\end{array}
+\begin{array} {c} \\ = \end{array}
+\begin{array} {c}
+G \\
+\left[ \begin{array} {rrr}
 & -1 \\
 & 1 \\
 & 0 \\
-\end{bmatrix}
+\end{array} \right]
+\end{array}
 </math>
-Reading the columns from <span><math>M⁺</math></span>, the first one confirms our period of 2/1, and the second column 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.
-So let's do 5/4 now. We prepare a matrix out of these two unchanged intervals, 2/1 and 5/4, and call it <span><math>U</math></span>:
+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>:
 <math>
-\begin{bmatrix}
+\left[ \begin{array} {rrr}
 & -2 \\
 & 0 \\
 & 1 \\
-\end{bmatrix}
+\end{array} \right]
 </math>
-We multiply <span><math>M⋅U</math></span>:
+We multiply <math>MU</math>:
 <math>
-\begin{bmatrix}
+\begin{array} {c}
+M \\
+\left[ \begin{array} {rrr}
 & 0 & -4 \\
 & 1 & 4 \\
-\end{bmatrix}
+\end{array} \right]
-\begin{bmatrix}
+\end{array}
+\begin{array} {c}
+U \\
+\left[ \begin{array} {rrr}
 & -2 \\
 & 0 \\
 & 1 \\
-\end{bmatrix}=
+\end{array} \right]
-\begin{bmatrix}
+\end{array}
+\begin{array} {c} \\ = \end{array}
+\begin{array} {c}
+MU \\
+\left[ \begin{array} {rrr}
 & -2 \\
 & 4 \\
-\end{bmatrix}
+\end{array} \right]
+\end{array}
 </math>
-We take the inverse <span><math>(M⋅U)^{-1}</math></span>:
+We take the inverse:
 <math>
-\begin{bmatrix}
-& \frac12 \\
+\begin{array} {c}
-& \frac14 \\
+ \\
-\end{bmatrix}
+\left[ \begin{array} {rrr}
+& -2 \\
+& 4 \\
+\end{array} \right]^{-1}
+\end{array}
+\begin{array} {c} \\ = \end{array}
+\begin{array} {c}
+(MU)^{-1} \\
+\left[ \begin{array} {rrr}
+& \frac{1}{2} \\
+& \frac{1}{4} \\
+\end{array} \right]
+\end{array}
 </math>
-Then find <span><math>M⁺</math></span> which is <span><math>U⋅(M⋅U)^{-1}</math></span>:
+Then find <math>G</math> which is <math>U(MU)^{-1}</math>:
 <math>
-\begin{bmatrix}
+\begin{array} {c}
+U \\
+\left[ \begin{array} {rrr}
 & -2 \\
 & 0 \\
 & 1 \\
-\end{bmatrix}
+\end{array} \right]
-\begin{bmatrix}
+\end{array}
-& \frac12 \\
-& \frac14 \\
+\begin{array} {c}
-\end{bmatrix}=
+(MU)^{-1} \\
-\begin{bmatrix}
+\left[ \begin{array} {rrr}
+& \frac{1}{2} \\
+& \frac{1}{4} \\
+\end{array} \right]
+\end{array}
+\begin{array} {c} \\ = \end{array}
+\begin{array} {c}
+G \\
+\left[ \begin{array} {rrr}
 & 0 \\
 & 0 \\
-& \frac14 \\
+& \frac{1}{4} \\
-\end{bmatrix}
+\end{array} \right]
+\end{array}
 </math>
-This tells us our generator is 5^(1/4). In cents, that's 1200¢ × log₂(5¹⸍⁴) ≈ 696.578¢.
-Okay, one more unchanged interval to check: 6/5. We prepare a matrix out of these two unchanged intervals, 2/1 and 6/5, and call it <span><math>U</math></span>:
+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>:
 <math>
-\begin{bmatrix}
+\left[ \begin{array} {rrr}
 & 1 \\
 & 1 \\
 & -1 \\
-\end{bmatrix}
+\end{array} \right]
 </math>
-We multiply <span><math>M⋅U</math></span>:
+We multiply <math>MU</math>:
 <math>
-\begin{bmatrix}
+\begin{array} {c}
+M \\
+\left[ \begin{array} {rrr}
 & 0 & -4 \\
 & 1 & 4 \\
-\end{bmatrix}
+\end{array} \right]
-\begin{bmatrix}
+\end{array}
+\begin{array} {c}
+U \\
+\left[ \begin{array} {rrr}
 & 1 \\
 & 1 \\
 & -1 \\
-\end{bmatrix}=
+\end{array} \right]
-\begin{bmatrix}
+\end{array}
+\begin{array} {c} \\ = \end{array}
+\begin{array} {c}
+MU \\
+\left[ \begin{array} {rrr}
 & 2 \\
 & -3 \\
-\end{bmatrix}
+\end{array} \right]
+\end{array}
 </math>
-We take the inverse <span><math>(M⋅U)^{-1}</math></span>:
+We take the inverse:
 <math>
-\begin{bmatrix}
-& \frac23 \\
+\begin{array} {c}
-& -\frac13 \\
+ \\
-\end{bmatrix}
+\left[ \begin{array} {rrr}
+& 2 \\
+& -3 \\
+\end{array} \right]^{-1}
+\end{array}
+\begin{array} {c} \\ = \end{array}
+\begin{array} {c}
+(MU)^{-1} \\
+\left[ \begin{array} {rrr}
+& \frac{2}{3} \\
+& -\frac{1}{3} \\
+\end{array} \right]
+\end{array}
 </math>
-Then find <span><math>M⁺</math></span> which is <span><math>U⋅(M⋅U)^{-1}</math></span>:
+Then find <math>G</math> which is <math>U(MU)^{-1}</math>:
 <math>
-\begin{bmatrix}
+\begin{array} {c}
+U \\
+\left[ \begin{array} {rrr}
 & 1 \\
 & 1 \\
 & -1 \\
-\end{bmatrix}
+\end{array} \right]
-\begin{bmatrix}
+\end{array}
-& \frac23 \\
-& -\frac13 \\
+\begin{array} {c}
-\end{bmatrix}=
+(MU)^{-1} \\
-\begin{bmatrix}
+\left[ \begin{array} {rrr}
-& \frac13 \\
+& \frac{2}{3} \\
-& -\frac13 \\
+& -\frac{1}{3} \\
-& \frac13 \\
+\end{array} \right]
-\end{bmatrix}
+\end{array}
+\begin{array} {c} \\ = \end{array}
+\begin{array} {c}
+G \\
+\left[ \begin{array} {rrr}
+& \frac{1}{3} \\
+& -\frac{1}{3} \\
+& \frac{1}{3} \\
+\end{array} \right]
+\end{array}
 </math>
-This tells us our generator is (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 />
 [[Category:Regular temperament theory]]