Generator embedding optimization: Difference between revisions

Line 7,868:

* in the case of a retuning map, both are ''prime'' tuning maps, while in this case both are ''generator'' tuning maps.

We can make use of the Greek letter delta and its association with differences. So let's use <math>𝜹_i</math> as a substitute for <math>𝒈_i - 𝒈_{0}</math>. We may call it a '''delta of generator tuning maps'''. The delta <math>𝜹_i</math> takes the index <math>i</math> of whichever tied tuning is the one the anchor tuning is subtracted from. (More payoff for our zero-indexing of those; our deltas here, like our blend map entries, will therefore be one-indexed as per normal.)

We can make use of the Greek letter delta and its association with differences. So let's use <math>𝜹_i</math> as a substitute for <math>𝒈_i - 𝒈_{0}</math>. We may call it a ''delta of generator tuning maps''. The delta <math>𝜹_i</math> takes the index <math>i</math> of whichever tied tuning is the one the anchor tuning is subtracted from. (More payoff for our zero-indexing of those; our deltas here, like our blend map entries, will therefore be one-indexed as per normal.)

Substituting back into our formula, then, we find:

Line 7,904:

But we can do a bit better. Let's find a variable that could refer to this whole matrix, whose rows are each generator tuning map deltas. The natural thing seems to be to use the capital version of the Greek letter delta, which is <math>\textit{Δ}</math>. However, this letter is so strongly associated with use as an operator, for representing the difference in values of the thing just to its right, that probably this isn't the best idea. How about instead we just use the Latin letter <math>D</math>, for "delta".

But we can do a bit better. Let's find a variable that could refer to this whole matrix, whose rows are each generator tuning map deltas. The natural thing seems to be to use the capital version of the Greek letter delta, which is <math>\textit{Δ}</math>. However, this letter is so strongly associated with use as an operator, for representing the difference in values of the thing just to its right, that probably this isn't the best idea. How about instead we just use the Latin letter <math>D</math>, for "delta". This is our '''(generator tuning map) deltas matrix'''

This lets us simplify the formula down to this:

Line 7,912:

𝒈 = 𝒈_{0} + 𝒃D

</math>

===How to identify tunings===

@@ Line 7,868: / Line 7,868: @@
 * in the case of a retuning map, both are ''prime'' tuning maps, while in this case both are ''generator'' tuning maps.
-We can make use of the Greek letter delta and its association with differences. So let's use <math>𝜹_i</math> as a substitute for <math>𝒈_i - 𝒈_{0}</math>. We may call it a '''delta of generator tuning maps'''. The delta <math>𝜹_i</math> takes the index <math>i</math> of whichever tied tuning is the one the anchor tuning is subtracted from. (More payoff for our zero-indexing of those; our deltas here, like our blend map entries, will therefore be one-indexed as per normal.)
+We can make use of the Greek letter delta and its association with differences. So let's use <math>𝜹_i</math> as a substitute for <math>𝒈_i - 𝒈_{0}</math>. We may call it a ''delta of generator tuning maps''. The delta <math>𝜹_i</math> takes the index <math>i</math> of whichever tied tuning is the one the anchor tuning is subtracted from. (More payoff for our zero-indexing of those; our deltas here, like our blend map entries, will therefore be one-indexed as per normal.)
 Substituting back into our formula, then, we find:
@@ Line 7,904: / Line 7,904: @@
-But we can do a bit better. Let's find a variable that could refer to this whole matrix, whose rows are each generator tuning map deltas. The natural thing seems to be to use the capital version of the Greek letter delta, which is <math>\textit{Δ}</math>. However, this letter is so strongly associated with use as an operator, for representing the difference in values of the thing just to its right, that probably this isn't the best idea. How about instead we just use the Latin letter <math>D</math>, for "delta".
+But we can do a bit better. Let's find a variable that could refer to this whole matrix, whose rows are each generator tuning map deltas. The natural thing seems to be to use the capital version of the Greek letter delta, which is <math>\textit{Δ}</math>. However, this letter is so strongly associated with use as an operator, for representing the difference in values of the thing just to its right, that probably this isn't the best idea. How about instead we just use the Latin letter <math>D</math>, for "delta". This is our '''(generator tuning map) deltas matrix'''
 This lets us simplify the formula down to this:
@@ Line 7,912: / Line 7,912: @@
 𝒈 = 𝒈_{0} + 𝒃D
 </math>
 ===How to identify tunings===