Mike Battaglia
battaglia01@mathstodon.xyz
<p>Music by day, digital signal processing and scientific computing by... day. MS Data Science at UTAustin, currently doing a second Master's in Computer Science. Jazz pianist (<a href="https://music.apple.com/us/album/lover-of-nature/945538481" target="_blank" rel="nofollow noopener" translate="no"><span class="invisible">https://</span><span class="ellipsis">music.apple.com/us/album/lover</span><span class="invisible">-of-nature/945538481</span></a>) and Lumatonist(<a href="https://www.youtube.com/watch?v=uIYg8b2p8JY" target="_blank" rel="nofollow noopener" translate="no"><span class="invisible">https://www.</span><span class="">youtube.com/watch?v=uIYg8b2p8JY</span><span class="invisible"></span></a>). Featured artist at Lumatone Keyboard.</p>
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The main idea I haven&#39;t seen you talk about is to then look at the group \(\text{Hom}(G, \Bbb Z)\) of homomorphisms from the JI group to the integers. If we think of \(G\) as being embedded in a real vector space \(\Bbb R^n\), then this group \(G*\) can be thought of as being embedded as a lattice in the dual space. This is called the lattice or group of &quot;vals&quot;, short for &quot;valuations&quot; (this name is directly from Gene, FWIW). Each val represents a way o...
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There is a LOT more and I don&#39;t have time to write it all now, but just for interest: - We can identify regular temperaments as subspaces of \(\Bbb R^n\) intersecting the lattice \(G\) - We can also identify them as subspaces \(\Bbb R^{n*}\) intersecting the lattice of vals \(G*\); the corresponding vals &quot;support&quot; the temperament by tempering out the comma in question - We can stack vals vertically to form &quot;mapping matrices&quot; and put them in various no...