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→Algebraic explanation: get rough version out
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=== Algebraic explanation===
=== Algebraic explanation===


(WIP)
This explanation relies on comparing the results of the multivector and matrix approaches to temperament arithmetic, and showing algebraically how the matrix approach can only achieve the same answer as the multivector approach on the condition that it keeps all but one vector between the added matrices the same, that is, not only are the temperaments addable, but their L_dep appears explicitly in the added matrices.
 
To compare results, we eventually get both approaches into a multivector form. With the multivector approach, we wedge the vector set first and then add the resultant multivectors to get a new multivector. With the matrix approach, we treat the vector set as a matrix and add first, then treat the resultant matrix as a vector set and wedge those vectors to get a new multivector. In the
 
The diagrams below are organized into a 2×2 layout. The left part shows the multivector approach, and the right part shows the matrix approach. The top part shows how the results of two approaches match when the L_dep is successfully explicit, and the bottom part shows how the results fail to match when it is not.
 
This first diagram demonstrates this situation for a d=3, g=2 case.
{| class="wikitable"
|+
!
!
!
! colspan="11" |
!
! colspan="11" |
!
|-
!
!
!
| colspan="11" rowspan="1" |multivector approach
!
| colspan="11" rowspan="1" |matrix approach
!
|-
!
!
!
! colspan="11" |
!
! colspan="11" |
!
|-
! rowspan="5" |
| colspan="1" rowspan="5" |explicit L_dep
 
⟨[a b c⟩]
! rowspan="5" |
|a
|b
|c
| rowspan="2" |
|a
|b
|c
| rowspan="2" |
| colspan="3" rowspan="2" |
! rowspan="5" |
|a
|b
|c
| colspan="1" rowspan="2" |+
|a
|b
|c
| colspan="1" rowspan="2" |=
|2a
|2b
|2c
! rowspan="5" |
|-
|d
|e
|f
|g
|h
|i
|d
|e
|f
|g
|h
|i
|d+g
|e+h
|f+i
|-
| colspan="3" rowspan="1" |∧
|
| colspan="3" rowspan="1" |∧
|
| colspan="3" |
| colspan="3" |
|
| colspan="3" |
|
| colspan="3" rowspan="1" |∧
|-
|bf-ce
|af-cd
|ae-bd
| +
|bi-ch
|ai-cg
|ah-bg
|=
|bf-ce + bi-ch
|af-cd + ai-cg
|ae-bd + ah-bg
| colspan="3" rowspan="2" |
| rowspan="2" |
| colspan="3" rowspan="2" |
| rowspan="2" |
|2b(f+i) - 2c(e+h)
|2a(f+i) - 2c(d+g)
|2a(e+h) - 2b(d+g)
|-
| colspan="3" |
|
| colspan="3" |
|
|b(f+i) - c(e+h)
|a(f+i) - c(d+g)
|a(e+h) - b(d+g)
|b(f+i) - c(e+h)
|a(f+i) - c(d+g)
|a(e+h) - b(d+g)
|-
!
!
!
! colspan="11" |
!
! colspan="11" |
!
|-
! rowspan="5" |
| rowspan="5" |hidden L_dep
! rowspan="5" |
|a
|b
|c
| rowspan="2" |
|j
|k
|l
|
| colspan="3" rowspan="2" |
! rowspan="5" |
|a
|b
|c
| colspan="1" rowspan="2" |+
|j
|k
|l
| colspan="1" rowspan="2" |=
|a+j
|b+k
|c+l
! rowspan="5" |
|-
|d
|e
|f
|g
|h
|i
|
|d
|e
|f
|g
|h
|i
|d+g
|e+h
|f+i
|-
| colspan="3" rowspan="1" |∧
|
| colspan="3" rowspan="1" |∧
|
| colspan="3" |
| colspan="3" |
|
| colspan="3" |
|
| colspan="3" rowspan="1" |∧
|-
|bf-ce
|af-cd
|ae-bd
| +
|ki-lh
|ji-lg
|jh-kg
|=
|bf - ce + ki - lh
|af - cd + ji - lg
|ae - bd + jh - kg
| colspan="3" rowspan="2" |
| rowspan="2" |
| colspan="3" rowspan="2" |
| rowspan="2" |
|(b+k)(f+i) - (c+l)(e+h)
|(a+j)(f+i) - (c+l)(d+g)
|(a+j)(e+h) - (b+k)(d+g)
|-
| colspan="3" |
|
| colspan="3" |
|
|bf - ce + ki - lh
|af - cd + ji - lg
|ae - bd + jh - kg
|bf + bi + kf + ki - ce - ch - le - lh
|af + ai + jf + ji - cd - cg - ld - lg
|ae + ah + je + jh - bd - bg - kd - kg
|-
!
!
!
! colspan="11" |
!
! colspan="11" |
!
|}
This second diagram demonstrates this situation for a d=5, g=3 case.
{| class="wikitable"
|+
!
! colspan="2" |
!
! colspan="32" |
!
! colspan="22" |
!
|-
!
! colspan="2" |
!
| colspan="32" rowspan="1" |multivector approach
!
| colspan="22" rowspan="1" |matrix approach
!
|-
!
! colspan="2" |
!
! colspan="32" |
!
! colspan="22" |
!
|-
! rowspan="7" |
| colspan="1" rowspan="7" |explicit L_dep
 
⟨[a b c d e⟩
 
[f g h i j⟩]
|r₁
! rowspan="7" |
| colspan="2" rowspan="1" |a
| colspan="2" rowspan="1" |b
| colspan="2" rowspan="1" |c
| colspan="2" rowspan="1" |d
| colspan="2" rowspan="1" |e
| rowspan="3" |
| colspan="2" rowspan="1" |a
| colspan="2" rowspan="1" |b
| colspan="2" rowspan="1" |c
| colspan="2" rowspan="1" |d
| colspan="2" rowspan="1" |e
| rowspan="3" |
| colspan="10" rowspan="3" |
! rowspan="7" |
|a
|b
|c
|d
|e
| colspan="1" rowspan="3" |+
|a
|b
|c
|d
|e
| colspan="1" rowspan="3" |=
| colspan="2" rowspan="1" |2a
| colspan="2" rowspan="1" |2b
| colspan="2" rowspan="1" |2c
| colspan="2" rowspan="1" |2d
| colspan="2" rowspan="1" |2e
! rowspan="7" |
|-
|r₂
| colspan="2" rowspan="1" |f
| colspan="2" rowspan="1" |g
| colspan="2" rowspan="1" |h
| colspan="2" rowspan="1" |i
| colspan="2" rowspan="1" |j
| colspan="2" rowspan="1" |f
| colspan="2" rowspan="1" |g
| colspan="2" rowspan="1" |h
| colspan="2" rowspan="1" |i
| colspan="2" rowspan="1" |j
|f
|g
|h
|i
|j
|f
|g
|h
|i
|j
| colspan="2" rowspan="1" |2f
| colspan="2" rowspan="1" |2g
| colspan="2" rowspan="1" |2h
| colspan="2" rowspan="1" |2i
| colspan="2" rowspan="1" |2j
|-
|r₃
| colspan="2" rowspan="1" |k
| colspan="2" rowspan="1" |l
| colspan="2" rowspan="1" |m
| colspan="2" rowspan="1" |n
| colspan="2" rowspan="1" |o
| colspan="2" rowspan="1" |p
| colspan="2" rowspan="1" |q
| colspan="2" rowspan="1" |r
| colspan="2" rowspan="1" |s
| colspan="2" rowspan="1" |t
|k
|l
|m
|n
|o
|p
|q
|r
|s
|t
| colspan="2" rowspan="1" |k+p
| colspan="2" rowspan="1" |l+q
| colspan="2" rowspan="1" |m+r
| colspan="2" rowspan="1" |n+s
| colspan="2" rowspan="1" |o+t
|-
|
| colspan="10" rowspan="1" |∧
|
| colspan="10" rowspan="1" |∧
|
| colspan="10" |
| colspan="5" |
|
| colspan="5" |
|
| colspan="10" rowspan="1" |∧
|-
|r₁∧r₂
| rowspan="2" |ag-bf
| rowspan="2" |ah-cf
| rowspan="2" |ai-df
| rowspan="2" |aj-ef
| rowspan="2" |bh-cg
| rowspan="2" |bi-dg
| rowspan="2" |bj-eg
| rowspan="2" |ci-dh
| rowspan="2" |cj-eh
| rowspan="2" |dj-ei
| rowspan="2" |
| rowspan="2" |ag-bf
| rowspan="2" |ah-cf
| rowspan="2" |ai-df
| rowspan="2" |aj-ef
| rowspan="2" |bh-cg
| rowspan="2" |bi-dg
| rowspan="2" |bj-eg
| rowspan="2" |ci-dh
| rowspan="2" |cj-eh
| rowspan="2" |dj-ei
| rowspan="2" |
| colspan="10" rowspan="2" |
| colspan="5" rowspan="3" |
| rowspan="3" |
| colspan="5" rowspan="3" |
| rowspan="3" |
|4ag-4bf
|4ah-4cf
|4ai-4df
|4aj-4ef
|4bh-4cg
|4bi-4dg
|4bj-4eg
|4ci-4dh
|4cj-4eh
|4dj-4ei
|-
|simplify(r₁∧r₂) if necessary
|ag-bf
|ah-cf
|ai-df
|aj-ef
|bh-cg
|bi-dg
|bj-eg
|ci-dh
|cj-eh
|dj-ei
|-
|(r₁∧r₂)∧r₃
|k(bh-cg)
-
 
l(ah-cf)
 
+
 
m(ag-bf)
|k(bi-dg)
-
 
l(ai-df)
 
+
 
n(ag-bf)
|k(bj-eg)
-
 
l(aj-ef)
 
+
 
o(ag-bf)
|k(ci-dh)
-
 
m(ai-df)
 
+
 
n(ah-cf)
|k(cj-eh)
-
 
m(aj-ef)
 
+
 
o(ah-cf)
|k(dj-ei)
-
 
n(aj-ef)
 
+
 
o(ai-df)
|l(ci-dh)
-
 
m(bi-dg)
 
+
 
n(bh-cg)
|l(cj-eh)
-
 
m(bj-eg)
 
+
 
o(bh-cg)
|l(dj-ei)
-
 
n(bj-eg)
 
+
 
o(bi-dg)
|m(dj-ei)
-
 
n(cj-eh)
 
+
 
o(ci-dh)
| +
|p(bh-cg)-q(ah-cf)+r(ag-bf)
|p(bi-dg)-q(ai-df)+s(ag-bf)
|p(bj-eg)-q(aj-ef)+t(ag-bf)
|p(ci-dh)-r(ai-df)+s(ah-cf)
|p(cj-eh)-r(aj-ef)+t(ah-cf)
|p(dj-ei)-s(aj-ef)+t(ai-df)
|q(ci-dh)-r(bi-dg)+s(bh-cg)
|q(cj-eh)-r(bj-eg)+t(bh-cg)
|q(dj-ei)-s(bj-eg)+t(bi-dg)
|r(dj-ei)-s(cj-eh)+t(ci-dh)
|=
|(k+p)(bh-cg)-(l+q)(ah-cf)+(m+r)(ag-bf)
|(k+p)(bi-dg)-(l+q)(ai-df)+(n+s)(ag-bf)
|(k+p)(bj-eg)-(l+q)(aj-ef)+(o+t)(ag-bf)
|(k+p)(ci-dh)-(m+r)(ai-df)+(n+s)(ah-cf)
|(k+p)(cj-eh)-(m+r)(aj-ef)+(o+t)(ah-cf)
|(k+p)(dj-ei)-(n+s)(aj-ef)+(o+t)(ai-df)
|(l+q)(ci-dh)-(m+r)(bi-dg)+(n+s)(bh-cg)
|(l+q)(cj-eh)-(m+r)(bj-eg)+(o+t)(bh-cg)
|(l+q)(dj-ei)-(n+s)(bj-eg)+(o+t)(bi-dg)
|(m+r)(dj-ei)-(n+s)(cj-eh)+(o+t)(ci-dh)
|(k+p)(bh-cg)-(l+q)(ah-cf)+(m+r)(ag-bf)
|(k+p)(bi-dg)-(l+q)(ai-df)+(n+s)(ag-bf)
|(k+p)(bj-eg)-(l+q)(aj-ef)+(o+t)(ag-bf)
|(k+p)(ci-dh)-(m+r)(ai-df)+(n+s)(ah-cf)
|(k+p)(cj-eh)-(m+r)(aj-ef)+(o+t)(ah-cf)
|(k+p)(dj-ei)-(n+s)(aj-ef)+(o+t)(ai-df)
|(l+q)(ci-dh)-(m+r)(bi-dg)+(n+s)(bh-cg)
|(l+q)(cj-eh)-(m+r)(bj-eg)+(o+t)(bh-cg)
|(l+q)(dj-ei)-(n+s)(bj-eg)+(o+t)(bi-dg)
|(m+r)(dj-ei)-(n+s)(cj-eh)+(o+t)(ci-dh)
|-
!
!
!
!
! colspan="32" |
!
! colspan="22" |
!
|-
! rowspan="7" |
| rowspan="7" |hidden L_dep
|r₁
! rowspan="7" |
| colspan="2" rowspan="1" |a
| colspan="2" rowspan="1" |b
| colspan="2" rowspan="1" |c
| colspan="2" rowspan="1" |d
| colspan="2" rowspan="1" |e
| rowspan="3" |
| colspan="2" rowspan="1" |a
| colspan="2" rowspan="1" |b
| colspan="2" rowspan="1" |c
| colspan="2" rowspan="1" |d
| colspan="2" rowspan="1" |e
| rowspan="3" |
| colspan="10" rowspan="3" |
! rowspan="7" |
|a
|b
|c
|d
|e
| colspan="1" rowspan="3" |+
|a
|b
|c
|d
|e
| colspan="1" rowspan="3" |=
| colspan="2" rowspan="1" |2a
| colspan="2" rowspan="1" |2b
| colspan="2" rowspan="1" |2c
| colspan="2" rowspan="1" |2d
| colspan="2" rowspan="1" |2e
! rowspan="7" |
|-
|r₂
| colspan="2" rowspan="1" |f
| colspan="2" rowspan="1" |g
| colspan="2" rowspan="1" |h
| colspan="2" rowspan="1" |i
| colspan="2" rowspan="1" |j
| colspan="2" rowspan="1" |u
| colspan="2" rowspan="1" |v
| colspan="2" rowspan="1" |w
| colspan="2" rowspan="1" |x
| colspan="2" rowspan="1" |y
|f
|g
|h
|i
|j
|u
|v
|w
|x
|y
| colspan="2" rowspan="1" |f+u
| colspan="2" rowspan="1" |g+v
| colspan="2" rowspan="1" |w+h
| colspan="2" rowspan="1" |i+x
| colspan="2" rowspan="1" |j+y
|-
|r₃
| colspan="2" rowspan="1" |k
| colspan="2" rowspan="1" |l
| colspan="2" rowspan="1" |m
| colspan="2" rowspan="1" |n
| colspan="2" rowspan="1" |o
| colspan="2" rowspan="1" |p
| colspan="2" rowspan="1" |q
| colspan="2" rowspan="1" |r
| colspan="2" rowspan="1" |s
| colspan="2" rowspan="1" |t
|k
|l
|m
|n
|o
|p
|q
|r
|s
|t
| colspan="2" rowspan="1" |k+p
| colspan="2" rowspan="1" |l+q
| colspan="2" rowspan="1" |m+r
| colspan="2" rowspan="1" |n+s
| colspan="2" rowspan="1" |o+t
|-
|
| colspan="10" rowspan="1" |∧
|
| colspan="10" rowspan="1" |∧
|
| colspan="10" |
| colspan="5" |
|
| colspan="5" |
|
| colspan="10" rowspan="1" |∧
|-
|r₁∧r₂
| rowspan="2" |ag-bf
| rowspan="2" |ah-cf
| rowspan="2" |ai-df
| rowspan="2" |aj-ef
| rowspan="2" |bh-cg
| rowspan="2" |bi-dg
| rowspan="2" |bj-eg
| rowspan="2" |ci-dh
| rowspan="2" |cj-eh
| rowspan="2" |dj-ei
| rowspan="2" |
| rowspan="2" |av-bu
| rowspan="2" |aw-cu
| rowspan="2" |ax-du
| rowspan="2" |ay-eu
| rowspan="2" |bw-cv
| rowspan="2" |bx-dv
| rowspan="2" |by-ev
| rowspan="2" |cx-dw
| rowspan="2" |cy-ew
| rowspan="2" |dy-ex
| rowspan="2" |
| colspan="10" rowspan="2" |
| colspan="5" rowspan="3" |
| rowspan="3" |
| colspan="5" rowspan="3" |
| rowspan="3" |
|2a(g+v) - 2b(f+u)
|2a(w+h)
 
-
 
2c(f+u)
|2a(i+x)
 
-
 
2d(f+u)
|2a(j+y)
 
-
 
2e(f+u)
|2b(w+h)
 
-
 
2c(g+v)
|2b(i+x)
 
-
 
2d(g+v)
|2b(j+y)
 
-
 
2e(g+v)
|2c(i+x)
 
-
 
2d(w+h)
|2c(j+y)
 
-
 
2e(w+h)
|2d(j+y)
 
-
 
2e(i+x)
|-
|simplify(r₁∧r₂) if necessary
|a(g+v)-b(f+u)
|a(w+h)- c(f+u)
|a(i+x)-d(f+u)
|a(j+y)-e(f+u)
|b(w+h)-c(g+v)
|b(i+x)-d(g+v)
|b(j+y)-e(g+v)
|c(i+x)-d(w+h)
|c(j+y)-e(w+h)
|d(j+y)-e(i+x)
|-
|(r₁∧r₂)∧r₃
|k(bh-cg)
-
 
l(ah-cf)
 
+
 
m(ag-bf)
|k(bi-dg)
-
 
l(ai-df)
 
+
 
n(ag-bf)
|k(bj-eg)
-
 
l(aj-ef)
 
+
 
o(ag-bf)
|k(ci-dh)
-
 
m(ai-df)
 
+
 
n(ah-cf)
|k(cj-eh)
-
 
m(aj-ef)
 
+
 
o(ah-cf)
|k(dj-ei)
-
 
n(aj-ef)
 
+
 
o(ai-df)
|l(ci-dh)
-
 
m(bi-dg)
 
+
 
n(bh-cg)
|l(cj-eh)
-
 
m(bj-eg)
 
+
 
o(bh-cg)
|l(dj-ei)
-
 
n(bj-eg)
 
+
 
o(bi-dg)
|m(dj-ei)
-
 
n(cj-eh)
 
+
 
o(ci-dh)
|<nowiki>+</nowiki>
|p(bw-cv)
-
 
q(aw-cu)
 
+
 
r(av-bu)
|p(bx-dv)
-
 
q(ax-du)
 
+
 
s(av-bu)
|p(by-ev)
 
-
 
q(ay-eu)
 
+
 
t(av-bu)
|p(cx-dw)
-
 
r(ax-du)
 
+
 
s(aw-cu)
|p(cy-ew)
-
 
r(ay-eu)
 
+
 
t(aw-cu)
|p(dy-ex)
 
-
 
s(ay-eu)
 
+
 
t(ax-du)
|q(cx-dw) -
 
r(bx-dv)
 
+
 
s(bw-cv)
|q(cy-ew)
 
-
 
r(by-ev) +
 
t(bw-cv)
|q(dy-ex) -
 
s(by-ev)
 
+
 
t(bw-cv)
|r(dy-ex) -
 
s(cy-ew)
 
+
 
t(cx-dw)
|=
|k(bh-cg)
-
 
l(ah-cf)
 
+
 
m(ag-bf)
 
+
 
p(bw-cv)
 
-
 
q(aw-cu)
 
+
 
r(av-bu)
|k(bi-dg)
-
 
l(ai-df)
 
+
 
n(ag-bf)
 
+
 
p(bx-dv)
 
-
 
q(ax-du)
 
+
 
s(av-bu)
|k(bj-eg)
-
 
l(aj-ef)
 
+
 
o(ag-bf)
 
+
 
p(by-ev)
 
-
 
q(ay-eu)
 
+
 
t(av-bu)
|k(ci-dh)
-
 
m(ai-df)
 
+
 
n(ah-cf)
 
+
 
p(cx-dw)
 
-
 
r(ax-du)
 
+
 
s(aw-cu)
|k(cj-eh)
-
 
m(aj-ef)
 
+
 
o(ah-cf)
 
+
 
p(cy-ew)
 
-
 
r(ay-eu)
 
+
 
t(aw-cu)
|k(dj-ei)
-
 
n(aj-ef)
 
+
 
o(ai-df)
 
+
 
p(dy-ex)
 
-
 
s(ay-eu)
 
+
 
t(ax-du)
|l(ci-dh)
-
 
m(bi-dg)
 
+
 
n(bh-cg)
 
+
 
q(cx-dw)
 
-
 
r(bx-dv)
 
+
 
s(bw-cv)
|l(cj-eh)
-
 
m(bj-eg)
 
+
 
o(bh-cg)
 
+
 
q(cy-ew)
 
-
 
r(by-ev)
 
+
 
t(bw-cv)
|l(dj-ei)
-
 
n(bj-eg)
 
+
 
o(bi-dg)
 
+
 
q(dy-ex)
 
-
 
s(by-ev)
 
+
 
t(bw-cv)
|m(dj-ei)
-
 
n(cj-eh)
 
+
 
o(ci-dh)
 
+
 
r(dy-ex)
 
-
 
s(cy-ew)
 
+
 
t(cx-dw)
|(k+p)(b(w+h)-c(g+v)) - (l+q)(a(w+h)- c(f+u)) + (m+r)(a(g+v)-b(f+u))
|(k+p)(b(i+x)-d(g+v)) - (l+q)(a(i+x)-d(f+u)) + (n+s)(a(g+v)-b(f+u))
|(k+p)(b(j+y)-e(g+v)) - (l+q)(a(j+y)-e(f+u)) + (o+t)(a(g+v)-b(f+u))
|(k+p)(c(i+x)-d(w+h)) - (m+r)(a(i+x)-d(f+u)) + (n+s)(a(w+h)- c(f+u))
|(k+p)(c(j+y)-e(w+h)) - (m+r)(a(j+y)-e(f+u)) + (o+t)(a(w+h)- c(f+u))
|(k+p)(d(j+y)-e(i+x)) - (n+s)(a(j+y)-e(f+u)) + (o+t)(a(i+x)-d(f+u))
|(l+q)(c(i+x)-d(w+h)) - (m+r)(b(i+x)-d(g+v)) + (n+s)(b(w+h)-c(g+v))
|(l+q)(c(j+y)-e(w+h)) - (m+r)(b(j+y)-e(g+v)) + (o+t)(b(w+h)-c(g+v))
|(l+q)(d(j+y)-e(i+x)) - (n+s)(b(j+y)-e(g+v)) + (o+t)(b(i+x)-d(g+v))
|(m+r)(d(j+y)-e(i+x)) - (n+s)(c(j+y)-e(w+h)) + (o+t)(c(i+x)-d(w+h))
|-
!
! colspan="2" |
!
! colspan="32" |
!
! colspan="22" |
!
|}
These two examples are by no means a proof, but meditation on the patterns in the variables is at least fairly convincing. 


===Sintel's proof of the <span style="color: #B6321C;">linear-independence</span> conjecture===
===Sintel's proof of the <span style="color: #B6321C;">linear-independence</span> conjecture===
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