Diamond tradeoff: Difference between revisions

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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 ~~invariant~~ interval (sometimes called an [[eigenmonzo]]). And we want to know this for the situation where octaves are pure.

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.

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

<math>

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</math>

Reading the columns from <math>T</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 ~~invariant~~ interval will give a more interesting result.

Reading the columns from <math>T</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 ~~invariants~~, 2/1 and 5/4, and call it <math>U</math>:

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>:

<math>

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

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

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>:

<math>

@@ Line 38: / Line 38: @@
 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 invariant interval (sometimes called an [[eigenmonzo]]). And we want to know this for the situation where octaves are pure.
+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.
-Let's do it for 4/3 first. So, we prepare a matrix out of these two invariants, 2/1 and 4/3, and call it <span><math>U</math></span>:
+Let's do it for 4/3 first. So, we prepare a matrix out of these two unchanged, 2/1 and 4/3, and call it <span><math>U</math></span>:
 <math>
@@ Line 96: / Line 96: @@
 </math>
-Reading the columns from <span><math>T</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 invariant interval will give a more interesting result.
+Reading the columns from <span><math>T</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 invariants, 2/1 and 5/4, and call it <span><math>U</math></span>:
+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>:
 <math>
@@ Line 156: / Line 156: @@
 This tells us our generator is 5^(1/4). In cents, that's 1200¢ × log₂(5¹⸍⁴) ≈ 696.578¢.
-Okay, one more invariant interval to check: 6/5. We prepare a matrix out of these two invariants, 2/1 and 6/5, and call it <span><math>U</math></span>:
+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>:
 <math>