User:Arseniiv/Timbres: Difference between revisions

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If we find φ − 1 too small a difference, we can add a letter ''C'' with value [''C''] = φ, and apply to ''G'' once rules ''A'' ⟶ ''BA'', ''B'' ⟶ ''A'', and then, again once, ''A'' ⟶ ''C'', ''B'' ⟶ ''A'', and end up with ''ACCACACCACCACACCACACC''… This word doesn’t give us a set which is mapped into itself when multiplied by φ, but is close to that in a sense.

One can also apply a stochastic ruleset ''A'' ⟶ (''AB'' or ''BA''), ''B'' ⟶ ''A'' to ''G'', except the first ''A'' which goes to ''AB'' to be sure the result starts with ''A''. We end up with a set of timbres which is, I believe, every possible one satisfying (1)…(3) for scaling by φ.

One can also apply a stochastic ruleset ''A'' ⟶ (''AB'' or ''BA''), ''B'' ⟶ ''A'' to ''G'', except the first ''A'' which goes to ''AB'' to be sure the result starts with ''A''. <s>We end up with a set of timbres which is, I believe, every possible one satisfying (1)…(3) for scaling by φ.</s>

We ''don’t'' end with a set of timbres which satisfy (2), but we will if we use this ruleset in a special way while generating the limit (not after). Namely, as is already typical, we replace the first ''A'' with ''AB'', then each iteration we’ll get a word which has a previous word as its prefix; and we need to pick a rule for each ''A'' in that prefix which was used beforehand. So we always apply ''A'' ⟶ ''AB'' to the first ''A'' in the word, we always apply a rule to the second ''A'' which we applied after we got a word where the second ''A'' appeared, and so on, and we only have freedom to pick which alternate rules to apply in the newly-generated suffix. That still allows for a countable set of variations of the limit word. Now, all these ''do'' satisfy (2) though it might be not as obvious as with the fully deterministic process.

===== Preserving the initial A =====

We can preserve the initial A in the string and also rarefy the timbre at the last step by two neat tricks. First, make our iterated ruleset be ''Å'' ⟶ ''ÅB'', ''A'' ⟶ (''AB'' or ''BA''), ''B'' ⟶ ''A''. ''Å'' is a pun for ''A''₀ and is preserved and marks the start, but all other ''A''s that get generated are normal. Then after generation we apply ''another'' “rarefying” ruleset ''Å'' ⟶ ''ÅB'', ''A'' ⟶ ''C'', ''B'' ⟶ ''A'' just once. No matter were rules stochastic or not, this approach gets us starting with ÅB which means partials 1, φ.

==== Silver timbres ====

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Our new word is ''ACACAACACAACACACAACACAA''…

We can apply ''A'' ⟶ (''AC'' or ''CA'') stochastically too (ensuring the first ''A'' is fixed, as above).

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 If we find φ − 1 too small a difference, we can add a letter ''C'' with value [''C''] = φ, and apply to ''G'' once rules ''A'' ⟶ ''BA'', ''B'' ⟶ ''A'', and then, again once, ''A'' ⟶ ''C'', ''B'' ⟶ ''A'', and end up with ''ACCACACCACCACACCACACC''… This word doesn’t give us a set which is mapped into itself when multiplied by φ, but is close to that in a sense.
-One can also apply a stochastic ruleset ''A'' ⟶ (''AB'' or ''BA''), ''B'' ⟶ ''A'' to ''G'', except the first ''A'' which goes to ''AB'' to be sure the result starts with ''A''. We end up with a set of timbres which is, I believe, every possible one satisfying (1)…(3) for scaling by φ.
+One can also apply a stochastic ruleset ''A'' ⟶ (''AB'' or ''BA''), ''B'' ⟶ ''A'' to ''G'', except the first ''A'' which goes to ''AB'' to be sure the result starts with ''A''. <s>We end up with a set of timbres which is, I believe, every possible one satisfying (1)…(3) for scaling by φ.</s>
+We ''don’t'' end with a set of timbres which satisfy (2), but we will if we use this ruleset in a special way while generating the limit (not after). Namely, as is already typical, we replace the first ''A'' with ''AB'', then each iteration we’ll get a word which has a previous word as its prefix; and we need to pick a rule for each ''A'' in that prefix which was used beforehand. So we always apply ''A'' ⟶ ''AB'' to the first ''A'' in the word, we always apply a rule to the second ''A'' which we applied after we got a word where the second ''A'' appeared, and so on, and we only have freedom to pick which alternate rules to apply in the newly-generated suffix. That still allows for a countable set of variations of the limit word. Now, all these ''do'' satisfy (2) though it might be not as obvious as with the fully deterministic process.
+===== Preserving the initial A =====
+We can preserve the initial A in the string and also rarefy the timbre at the last step by two neat tricks. First, make our iterated ruleset be ''Å'' ⟶ ''ÅB'', ''A'' ⟶ (''AB'' or ''BA''), ''B'' ⟶ ''A''. ''Å'' is a pun for ''A''₀ and is preserved and marks the start, but all other ''A''s that get generated are normal. Then after generation we apply ''another'' “rarefying” ruleset ''Å'' ⟶ ''ÅB'', ''A'' ⟶ ''C'', ''B'' ⟶ ''A'' just once. No matter were rules stochastic or not, this approach gets us starting with ÅB which means partials 1, φ.
 ==== Silver timbres ====
@@ Line 222: / Line 228: @@
 Our new word is ''ACACAACACAACACACAACACAA''…
+We can apply ''A'' ⟶ (''AC'' or ''CA'') stochastically too (ensuring the first ''A'' is fixed, as above).