Diamond tradeoff
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 ratios 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 q-odd-limit diamond tradeoff range of a rank-r p-limit temperament is the convex hull in tuning space of the set of all tunings with r eigenmonzos (unchanged-intervals) chosen as follows: one eigenmonzo (2/1, octaves) 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
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.[1] 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.
Computation
Here we will demonstrate the calculation of the diamond tradeoff tuning range for meantone.
Here is the mapping, [math]M[/math]:
[math]
\left[ \begin{array} {rrr}
1 & 0 & -4 \\
0 & 1 & 4 \\
\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, 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, [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]:
[math]
\left[ \begin{array} {rrr}
1 & 2 \\
0 & -1 \\
0 & 0 \\
\end{array} \right]
[/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 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{array} {c}
M \\
\left[ \begin{array} {rrr}
1 & 0 & -4 \\
0 & 1 & 4 \\
\end{array} \right]
\end{array}
\begin{array} {c}
U \\
\left[ \begin{array} {rrr}
1 & 2 \\
0 & -1 \\
0 & 0 \\
\end{array} \right]
\end{array}
\begin{array} {c} \\ = \end{array}
\begin{array} {c}
MU \\
\left[ \begin{array} {rrr}
1 & 1 \\
0 & -1 \\
\end{array} \right]
\end{array}
[/math]
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{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{array} \right]
\end{array}
[/math]
Then find [math]G[/math] which is [math]U(MU)^{-1}[/math]:
[math]
\begin{array} {c}
U \\
\left[ \begin{array} {rrr}
1 & 2 \\
0 & -1 \\
0 & 0 \\
\end{array} \right]
\end{array}
\begin{array} {c}
(MU)^{-1} \\
\left[ \begin{array} {rrr}
1 & 1 \\
0 & -1 \\
\end{array} \right]
\end{array}
\begin{array} {c} \\ = \end{array}
\begin{array} {c}
G \\
\left[ \begin{array} {rrr}
1 & -1 \\
0 & 1 \\
0 & 0 \\
\end{array} \right]
\end{array}
[/math]
Reading the columns from [math]G[/math], the first one [1 0 0⟩ confirms our period of 2/1, and the second column [-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.
Second diamond extrema
So let's do 5/4 now, [-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]
\left[ \begin{array} {rrr}
1 & -2 \\
0 & 0 \\
0 & 1 \\
\end{array} \right]
[/math]
We multiply [math]MU[/math]:
[math]
\begin{array} {c}
M \\
\left[ \begin{array} {rrr}
1 & 0 & -4 \\
0 & 1 & 4 \\
\end{array} \right]
\end{array}
\begin{array} {c}
U \\
\left[ \begin{array} {rrr}
1 & -2 \\
0 & 0 \\
0 & 1 \\
\end{array} \right]
\end{array}
\begin{array} {c} \\ = \end{array}
\begin{array} {c}
MU \\
\left[ \begin{array} {rrr}
1 & -2 \\
0 & 4 \\
\end{array} \right]
\end{array}
[/math]
We take the inverse:
[math]
\begin{array} {c}
\\
\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]G[/math] which is [math]U(MU)^{-1}[/math]:
[math]
\begin{array} {c}
U \\
\left[ \begin{array} {rrr}
1 & -2 \\
0 & 0 \\
0 & 1 \\
\end{array} \right]
\end{array}
\begin{array} {c}
(MU)^{-1} \\
\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 & \frac{1}{4} \\
\end{array} \right]
\end{array}
[/math]
This tells us our generator is [math]\monzo{0 & 0 & \frac{1}{4}}[/math], or 51⁄4. In cents, that's 1200 × log2(51⁄4) ≈ 696.578 ¢.
Third diamond extrema
Okay, one more unchanged interval to check: 6/5, which is [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]
\left[ \begin{array} {rrr}
1 & 1 \\
0 & 1 \\
0 & -1 \\
\end{array} \right]
[/math]
We multiply [math]MU[/math]:
[math]
\begin{array} {c}
M \\
\left[ \begin{array} {rrr}
1 & 0 & -4 \\
0 & 1 & 4 \\
\end{array} \right]
\end{array}
\begin{array} {c}
U \\
\left[ \begin{array} {rrr}
1 & 1 \\
0 & 1 \\
0 & -1 \\
\end{array} \right]
\end{array}
\begin{array} {c} \\ = \end{array}
\begin{array} {c}
MU \\
\left[ \begin{array} {rrr}
1 & 2 \\
0 & -3 \\
\end{array} \right]
\end{array}
[/math]
We take the inverse:
[math]
\begin{array} {c}
\\
\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]G[/math] which is [math]U(MU)^{-1}[/math]:
[math]
\begin{array} {c}
U \\
\left[ \begin{array} {rrr}
1 & 1 \\
0 & 1 \\
0 & -1 \\
\end{array} \right]
\end{array}
\begin{array} {c}
(MU)^{-1} \\
\left[ \begin{array} {rrr}
1 & \frac{2}{3} \\
0 & -\frac{1}{3} \\
\end{array} \right]
\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 [math]\monzo{\frac{1}{3} & -\frac{1}{3} & \frac{1}{3}}[/math], or expressed another way, (10⁄3)1⁄3. In cents, that's 1200 × log2((10⁄3)1⁄3) ≈ 694.786 ¢.
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.
References
- ↑ 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