Diamond tradeoff: Difference between revisions

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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> (see [[~~Dave Keenan & Douglas Blumeyer's guide to RTT/All-interval tuning schemes~~|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>:

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 {{nowrap|1200 × log<sub>2</sub>(3/2) ≈ 701.955{{c}}}}. The next unchanged interval will give a more interesting result.

==== Second diamond extrema ====

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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> (see [[Dave Keenan & Douglas Blumeyer's guide to RTT/All-interval tuning schemes|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>:
+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 {{nowrap|1200 &times; log<sub>2</sub>(3/2) &#8776; 701.955{{c}}}}. The next unchanged interval will give a more interesting result.
 ==== Second diamond extrema ====