general methods for linear temperament notation

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Dave Keenan
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Re: general methods for linear temperament notation

Post by Dave Keenan »

ᴄᴍʟᴏᴇɢᴄᴍʟᴜɪɴ wrote: Wed Apr 21, 2021 9:36 am
Dave Keenan wrote: Thu Mar 18, 2021 6:40 am I notice that he not only uses the same notation |a b c ...> for prime exponents and generator counts, he doesn't use plus signs and he even calls the latter "tempered monzos" — yikes — helping making my case that Gene's monzo/val terminology is unhelpful and just raises a barrier to entry.
As you know I've been pushing myself to level up my understanding of RTT recently. I had an insight that helps explain to me Mike's decision here. If you forgive the problem of needlessly renaming vector to monzo, then "tempered monzo" is reasonable (as is using the same notation), because actually prime exponent vectors and generator count vectors are the same thing when you consider that JI can be represented as a mapping matrix:

\( \left[ \begin{array} {rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right] \)
A lovely use of latex. That's a good insight in general. But of course they are only the same because you've chosen to make the generators be the primes. If instead you'd used my "musicians choice" generators, in this case 1:2, 2:3 and 4:5, the matrix would be

\( \left[ \begin{array} {rrr}
1 & 1 & 2 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right] \)


and its generator count vectors would not then be prime exponent vectors. I think the term monzo is equivalent to prime exponent vector, not just any old vector. And a "tempered prime exponent vector" is something else entirely, if it is anything at all.
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cmloegcmluin
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Re: general methods for linear temperament notation

Post by cmloegcmluin »

Sure. The prime exponent vectors would only match generator count vectors for a single form of the mapping matrix, the one I gave above, which I do not have a name for; it is simply the form that Mike uses in his lectures on the wiki.

I copy-pasted the latex from you, silly. :P
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