Temperament addition: Difference between revisions

Basically, in the case of <math>d=2</math>, <math>g_{\text{max}}=1</math> (in non-trivial cases, i.e. not JI or the unison temperament), so any two different ETs or commas you pick are going to be <span style="color: #B6321C;">linearly independent</span> (because the only way they could be <span style="color: #3C8031;">linearly dependent</span> would be to be the same temperament). And yet we know we can still entry-wise add them to new vectors that are [[Douglas_Blumeyer_and_Dave_Keenan%27s_Intro_to_exterior_algebra_for_RTT#Decomposability|decomposable]], because they're already vectors (decomposing means to express a [[Douglas_Blumeyer_and_Dave_Keenan%27s_Intro_to_exterior_algebra_for_RTT#From_vectors_to_multivectors|multivector]] in the form of a list of monovectors, so decomposing a multivector that's already a monovector like this is tantamount to merely putting array braces around it.)

===Geometric explanation===

So for <math>g_{\text{min}}=2</math> and <math>g_{\text{max}}=3</math> we got two different possibilities for <span style="color: #B6321C;"><math>l_{\text{ind}}</math></span>: 1 and 2, and for each of these two possibilities, we found it twice. We can see then that these match up, that is, that the <math>g_{\text{min}}=2</math> case with <span style="color: #B6321C;"><math>l_{\text{ind}}=1</math></span> matches with the <math>g_{\text{max}}=3</math> case with <span style="color: #B6321C;"><math>l_{\text{ind}}=1</math></span>, and the <span style="color: #B6321C;"><math>l_{\text{ind}}=2</math></span> cases match in the same way.

Here's a summary table of our geometric findings so far:

|1

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=== Algebraic explanation===