Douglas Blumeyer's RTT How-To: Difference between revisions

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Let’s review what we’ve seen so far. 5-limit JI is 3-dimensional. When we have a rank-3 temperament of 5-limit JI, 0 commas are tempered out. When we have a rank-2 temperament of 5-limit JI, 1 comma is tempered out. When we have a rank-1 temperament of 5-limit JI, 2 commas are tempered out.<ref>Probably, a rank-0 temperament of 5-limit JI would temper 3 commas out. All I can think a rank-0 temperament could be is a single pitch, or in other words, everything is tempered out. So perhaps in some theoretical sense, a comma basis in 5-limit made out of 3 vectors, thus a square 3×3 matrix, as long as none of the lines are parallel, should minimally represent every interval in the space.</ref>

There’s a straightforward formula here: <math>d - n = r</math>, where <math>d</math> is dimensionality, <math>n</math> is nullity, and <math>r</math> is rank. We’ve seen every one of those words so far except nullity. Nullity simply means the count of commas tempered out ''(see Figure 4c)''.

There’s a straightforward formula here: <math>d - n = r</math>, where <math>d</math> is dimensionality, <math>n</math> is nullity, and <math>r</math> is rank. We’ve seen every one of those words so far except '''nullity'''. [[Nullity]] simply means the count of commas tempered out ''(see Figure 4c)''.

So far, everything we’ve done has been in terms of 5-limit, which has dimensionality of 3. Before we generalize our knowledge upwards, into the 7-limit, let’s take a look at how things one step downwards, in the simpler direction, in the 3-limit, which is only 2-dimensional.

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[[File:Temperaments by rnd.png|400px|thumb|left|'''Figure 4d.''' Some temperaments by dimension, rank, and nullity]]

=== the 7-limit ===

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 Let’s review what we’ve seen so far. 5-limit JI is 3-dimensional. When we have a rank-3 temperament of 5-limit JI, 0 commas are tempered out. When we have a rank-2 temperament of 5-limit JI, 1 comma is tempered out. When we have a rank-1 temperament of 5-limit JI, 2 commas are tempered out.<ref>Probably, a rank-0 temperament of 5-limit JI would temper 3 commas out. All I can think a rank-0 temperament could be is a single pitch, or in other words, everything is tempered out. So perhaps in some theoretical sense, a comma basis in 5-limit made out of 3 vectors, thus a square 3×3 matrix, as long as none of the lines are parallel, should minimally represent every interval in the space.</ref>
-There’s a straightforward formula here: <span><math>d - n = r</math></span>, where <span><math>d</math></span> is dimensionality, <span><math>n</math></span> is nullity, and <span><math>r</math></span> is rank. We’ve seen every one of those words so far except nullity. Nullity simply means the count of commas tempered out ''(see Figure 4c)''.
+There’s a straightforward formula here: <span><math>d - n = r</math></span>, where <span><math>d</math></span> is dimensionality, <span><math>n</math></span> is nullity, and <span><math>r</math></span> is rank. We’ve seen every one of those words so far except '''nullity'''. [[Nullity]] simply means the count of commas tempered out ''(see Figure 4c)''.
 So far, everything we’ve done has been in terms of 5-limit, which has dimensionality of 3. Before we generalize our knowledge upwards, into the 7-limit, let’s take a look at how things one step downwards, in the simpler direction, in the 3-limit, which is only 2-dimensional.
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 [[File:Temperaments by rnd.png|400px|thumb|left|'''Figure 4d.''' Some temperaments by dimension, rank, and nullity]]
 === the 7-limit ===