Temperament addition: Difference between revisions

Line 1,044:

|}

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 linearly independent (because the only way they could be linearly dependent 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.)

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 linearly independent (because the only way they could be linearly dependent 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===

@@ Line 1,044: / Line 1,044: @@
 |}
-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.)
+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===