User:Arseniiv/Timbres - Revision history

Arseniiv: /* Preserving the initial A */ small fixes

2023-12-28T10:27:51Z

Preserving the initial A: small fixes

← Older revision		Revision as of 10:27, 28 December 2023
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	===== Preserving the initial A =====		===== 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, 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.		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 ====		==== Silver timbres ====

Arseniiv: /* Golden timbres */ ådding rule hacks with Å

2023-12-28T10:00:53Z

Golden timbres: ådding rule hacks with Å

← Older revision		Revision as of 10:00, 28 December 2023
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	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.		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, 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 ====		==== Silver timbres ====

Arseniiv: /* Golden timbres */ my mistake

2022-11-14T11:15:12Z

Golden timbres: my mistake

← Older revision		Revision as of 11:15, 14 November 2022
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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.		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.

	==== Silver timbres ====		==== Silver timbres ====

Arseniiv: /* Silver timbres */ added stochastic variation here too

2022-11-13T23:14:57Z

Silver timbres: added stochastic variation here too

← Older revision		Revision as of 23:14, 13 November 2022
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	Our new word is ''ACACAACACAACACACAACACAA''…		Our new word is ''ACACAACACAACACACAACACAA''…

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

Arseniiv: /* A general algorithm to make timbres like these */ elaborations

2022-11-13T23:11:13Z

A general algorithm to make timbres like these: elaborations

← Older revision		Revision as of 23:11, 13 November 2022
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	# its differences ''s''<sub>''n'' + 1</sub> − ''s''<sub>''n''</sub> are finitely many.		# its differences ''s''<sub>''n'' + 1</sub> − ''s''<sub>''n''</sub> are finitely many.

	~~Let’s pick~~ a finite alphabet {''A'', …} and for each letter ''L'' define its ''value'' [''L''], a positive real number so that:		We’ll work with those differences. Pick a finite alphabet {''A'', …} and for each letter ''L'' define its ''value'' [''L''], a positive real number so that:
	* [''A''] = 1;		* [''A''] = 1;
	* for each letter ''L'', there is a word ''L''<sub>1</sub>⋯''L''<sub>''n''</sub> such that its total value [''L''<sub>1</sub>] + … + [''L''<sub>''n''</sub>] = α ⋅ [''L''].		* for each letter ''L'', there is a word ''L''<sub>1</sub>⋯''L''<sub>''n''</sub> such that its total value [''L''<sub>1</sub>] + … + [''L''<sub>''n''</sub>] = α ⋅ [''L''].

	~~Then let’s~~ make a context-free [[Wikipedia:L-system]] over this alphabet. Its start word will be ''A'' and each of its production rules ''L'' ⟶ ''w'' should “scale by α”—satisfy [''w''] = α [''L'']. We also want to observe this L-system has a “left limit” in a sense that for each integer ''N'', there is an iteration of the system sharing a prefix of length ''N'' with every further iteration. Having that, we can construct a limiting word of the system as the unique word which shares prefixes of unbounded non-decreasing lengths with each iteration of the system.		So there is a way to scale each difference α times and break it into unscaled ones, and there is a way to start (''A''). We’re going to build a sequence of differences iteratively.

			Let’s make a context-free [[Wikipedia:L-system]] over this alphabet. Its start word will be ''A'' and each of its production rules ''L'' ⟶ ''w'' should “scale by α”—satisfy [''w''] = α [''L''] (above, we guaranteed there are such ''w'' for each letter ''L''). We also want to observe this L-system has a “left limit” in the sense that for each integer ''N'', there is an iteration of the system sharing a prefix of length ''N'' with every further iteration. Having that, we can construct a limiting word of the system as the unique word which shares prefixes of unbounded non-decreasing lengths with each iteration of the system.

	This limit ''ℓ'' is what we’re after. Take <math>s_n = \sum_{i=0}^n [\ell_n]</math>, that is, differences of ''s'' are the values of ℓ’s letters. By construction we see conditions (2) and (3) satisfied, and if (1) is not, we just divide all of ''s'' by ''s''<sub>0</sub>.		This limit ''ℓ'' is what we’re after. Take <math>s_n = \sum_{i=0}^n [\ell_n]</math>, that is, differences of ''s'' are the values of ℓ’s letters. By construction we see conditions (2) and (3) satisfied, and if (1) is not, we just divide all of ''s'' by ''s''<sub>0</sub>.

Arseniiv: /* A general algorithm to make timbres like these */ new section

2022-11-13T22:59:06Z

A general algorithm to make timbres like these: new section

← Older revision		Revision as of 22:59, 13 November 2022
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	This tweak is an idea from XA Discord. When I originally tried the same idea but with 5, 10, 15, 20 tuned a step lower, the timbre sounded pretty disastrous by itself, disregarding any harmonic possibilities, but I didn’t think to try to salvage that.		This tweak is an idea from XA Discord. When I originally tried the same idea but with 5, 10, 15, 20 tuned a step lower, the timbre sounded pretty disastrous by itself, disregarding any harmonic possibilities, but I didn’t think to try to salvage that.

			== A general algorithm to make timbres like these ==

			Now let’s make some constrants fixed and look at a solution that came to me 2022-11-13 by night after meddling with golden and silver timbres again.

			We’d like to construct an increasing sequence ''s'' of real numbers such that:
			# ''s''<sub>0</sub> = 1;
			# a set of its values is mapped into itself under multiplication by a given α ≥ 1;
			# its differences ''s''<sub>''n'' + 1</sub> − ''s''<sub>''n''</sub> are finitely many.

			Let’s pick a finite alphabet {''A'', …} and for each letter ''L'' define its ''value'' [''L''], a positive real number so that:
			* [''A''] = 1;
			* for each letter ''L'', there is a word ''L''<sub>1</sub>⋯''L''<sub>''n''</sub> such that its total value [''L''<sub>1</sub>] + … + [''L''<sub>''n''</sub>] = α ⋅ [''L''].

			Then let’s make a context-free [[Wikipedia:L-system]] over this alphabet. Its start word will be ''A'' and each of its production rules ''L'' ⟶ ''w'' should “scale by α”—satisfy [''w''] = α [''L'']. We also want to observe this L-system has a “left limit” in a sense that for each integer ''N'', there is an iteration of the system sharing a prefix of length ''N'' with every further iteration. Having that, we can construct a limiting word of the system as the unique word which shares prefixes of unbounded non-decreasing lengths with each iteration of the system.

			This limit ''ℓ'' is what we’re after. Take <math>s_n = \sum_{i=0}^n [\ell_n]</math>, that is, differences of ''s'' are the values of ℓ’s letters. By construction we see conditions (2) and (3) satisfied, and if (1) is not, we just divide all of ''s'' by ''s''<sub>0</sub>.

			A perhaps better way to guarantee (1) is to pick production rule(s) for ''A'' so that its successor(s) always starts with ''A'' itself. Moreover, if there’s only one production rule for each letter, and ''A''’s successor is more than ''A'' itself, then we have a limit automatically: ''A'' is a prefix of its immediate derivation, which is a larger prefix of ''its'' immediate derivation, and so on.

			In simple words, we constructed a way for each difference we’re allowed to use, when stretched by α, to be subdivided using just those same differences. This makes quite a regular structure. I conjecture for a deterministic L-system it’s always the case that <math>s_n = C n + \mathcal O(1)</math>.

			=== Generalizations ===

			One can use different production rulesets (for example one on even iterations and another on odd iterations) given that the limit exists and they all do scale by α. One can also apply rulesets ''to'' the limit word (possibly another ω times (and more) which is probably pointless). If you find differences of ''s'' too small, you can apply another rules to the limit word picked in such a way that letters with small values disappear. One also can apply rules only in some places. The last two kinds of tweaks may invalidate conditions (1) and (2), though there ''is'' a sense to speak about partial satisfaction of (2), in a statistical sense, which can then remain.

			=== Examples ===

			==== Harmonic timbre ====

			Don’t add any letters, use any rule of form ''A'' ⟶ ''A''<sup>''α''</sup> where ''α'' > 1 is an integer.

			The limit is ''AAAAAAAAA''… and so ''s'' is 1, 2, 3, 4, …

			==== Golden timbres ====

			The scale factor α is the golden ratio φ.

			Add a letter ''B'' with [''B''] = φ − 1, and the rules are ''A'' ⟶ ''AB'' and ''B'' ⟶ ''A''. (Note that φ [''A''] = φ = 1 + φ − 1 = [''AB''], and that φ [''B''] = φ<sup>2</sup> − φ = 1 = [''A''].)

			The limit is ''G'' ≡ ''ABAABABAABAABABAABABA''…, and so ''s'' is 1, φ, φ + 1, φ + 2, 2 φ + 1, 2 φ + 2, 3 φ + 1, 3 φ + 2, 3 φ + 3, 4 φ + 2, …

			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.

			==== Silver timbres ====

			Our α is √2 + 1.

			Add a letter ''B'' with value [''B''] = √2 − 1, and the rules are ''A'' ⟶ ''AAB'', ''B'' ⟶ ''A''.

			The limit of that is ''AABAABAAABAABAAAB''… but [''B''] ≈ 0.4 might seem to small, so…

			Apply to it once ''A'' ⟶ ''AC'', ''B'' ⟶ ''A'' where [''C''] = √2. ''Now'' we get a nice and proper timbre because we had space to consolidate differences ''and'' not shatter (2)—because ''AC'' is a reformulation of an older ''AAB'' consequent.

			Our new word is ''ACACAACACAACACACAACACAA''…

Arseniiv: /* An inharmonic timbre tuned to 17edo */ layout in pre

2021-10-14T19:26:17Z

An inharmonic timbre tuned to 17edo: layout in pre

← Older revision		Revision as of 19:26, 14 October 2021
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	Partials are, in [[17edo\|\17]] steps,		Partials are, in [[17edo\|\17]] steps,
	<pre># 1 2 3 4 5 6 7 8		<pre># 1 2 3 4 5 6 7 8
	0, 17, 27, 34, 40, 44, 48, 51,		0, 17, 27, 34, 40, 44, 48, 51,
	# 9 10 11 12 13 14 15 16		# 9 10 11 12 13 14 15 16
	54, 57, 59, 61, 63, 65, 67, 68,		54, 57, 59, 61, 63, 65, 67, 68,
	#17 18 19 20 21 22 23 24 25		# 17 18 19 20 21 22 23 24 25
	69, 71, 72, 74, 75, 76, 77, 78, 79,</pre>		69, 71, 72, 74, 75, 76, 77, 78, 79,</pre>
	where 5, 10, 15, 20, but not 25, are 1\17 higher than “the best” (as 5 is represented so-so).		where 5, 10, 15, 20, but not 25, are 1\17 higher than “the best” (as 5 is represented so-so).

	This tweak is an idea from XA Discord. When I originally tried the same idea but with 5, 10, 15, 20 tuned a step lower, the timbre sounded pretty disastrous by itself, disregarding any harmonic possibilities, but I didn’t think to try to salvage that.		This tweak is an idea from XA Discord. When I originally tried the same idea but with 5, 10, 15, 20 tuned a step lower, the timbre sounded pretty disastrous by itself, disregarding any harmonic possibilities, but I didn’t think to try to salvage that.

Arseniiv: /* News 2021-10 */ a timbre for 17edo

2021-10-14T19:23:43Z

News 2021-10: a timbre for 17edo

← Older revision		Revision as of 19:23, 14 October 2021
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	More specifically, all partials 3<sup>''n''</sup> ''m'' are replaced with (8/3)<sup>''n''</sup> ''m''. This is to make fifths work in some way like fourths. Listen [https://www.dropbox.com/sh/yyasgbykj2ps01c/AAAWKadGOIYqylMnhZkCa7cza?dl=0 here]. For me it sounds surprisingly consonant, considering retuning 5 to 24/5 sounds just evil. Though I can’t say I see what harmonic reifications this timbre brings.		More specifically, all partials 3<sup>''n''</sup> ''m'' are replaced with (8/3)<sup>''n''</sup> ''m''. This is to make fifths work in some way like fourths. Listen [https://www.dropbox.com/sh/yyasgbykj2ps01c/AAAWKadGOIYqylMnhZkCa7cza?dl=0 here]. For me it sounds surprisingly consonant, considering retuning 5 to 24/5 sounds just evil. Though I can’t say I see what harmonic reifications this timbre brings.

			=== An inharmonic timbre tuned to 17edo ===

			[https://www.dropbox.com/sh/m3vcxljnbg82gtq/AAB4L82RLbUvGAvZDlwti5-pa?dl=0 Here].

			Partials are, in [[17edo\|\17]] steps,
			<pre># 1 2 3 4 5 6 7 8
			0, 17, 27, 34, 40, 44, 48, 51,
			# 9 10 11 12 13 14 15 16
			54, 57, 59, 61, 63, 65, 67, 68,
			#17 18 19 20 21 22 23 24 25
			69, 71, 72, 74, 75, 76, 77, 78, 79,</pre>
			where 5, 10, 15, 20, but not 25, are 1\17 higher than “the best” (as 5 is represented so-so).

			This tweak is an idea from XA Discord. When I originally tried the same idea but with 5, 10, 15, 20 tuned a step lower, the timbre sounded pretty disastrous by itself, disregarding any harmonic possibilities, but I didn’t think to try to salvage that.

Arseniiv: /* News 2021-10 */ 3 retuned to 8/3

2021-10-14T17:32:18Z

News 2021-10: 3 retuned to 8/3

← Older revision		Revision as of 17:32, 14 October 2021
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	This time the lowest note of an interval or chord is always the same pitch. This should be more useful for comparisons.		This time the lowest note of an interval or chord is always the same pitch. This should be more useful for comparisons.

			=== A harmonic timbre with 3 retuned to 8/3 ===

			More specifically, all partials 3<sup>''n''</sup> ''m'' are replaced with (8/3)<sup>''n''</sup> ''m''. This is to make fifths work in some way like fourths. Listen [https://www.dropbox.com/sh/yyasgbykj2ps01c/AAAWKadGOIYqylMnhZkCa7cza?dl=0 here]. For me it sounds surprisingly consonant, considering retuning 5 to 24/5 sounds just evil. Though I can’t say I see what harmonic reifications this timbre brings.

Arseniiv: /* Harmonic timbres with missing partials */ many interval and chord sounds

2021-10-10T23:25:29Z

Harmonic timbres with missing partials: many interval and chord sounds

← Older revision		Revision as of 23:25, 10 October 2021
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	7:4 1.75000 968.82591¢		7:4 1.75000 968.82591¢
	2:1 2.00000 1200.00000¢</pre>		2:1 2.00000 1200.00000¢</pre>

			=== Harmonic timbres with missing partials ===

			[https://www.dropbox.com/sh/997zpwsqrx3sg5n/AADvqLayIpN5xPz-IGk8KDnga?dl=0 Here is a folder of samples of various intervals and triads] for testing how deleting multiples of 3, 5 or both makes septimal and undecimal thirds/sixths differ positively (or not) from pental ones; also [[7/5]] (for timbres without multiples of 3, specifically).

			This time the lowest note of an interval or chord is always the same pitch. This should be more useful for comparisons.