User:Frostburn/SonicWeave - Revision history

Frostburn: Expand on rational frequencies and define relative to absolute conversion.

2024-01-05T13:58:01Z

Expand on rational frequencies and define relative to absolute conversion.

← Older revision		Revision as of 13:58, 5 January 2024
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	== Rational frequencies as vectors ==		== Rational frequencies as vectors ==
	When dealing with musical data we mostly care about frequencies measured in Hertz. To represent frequencies other than 1 Hz we need more basis vectors and by the [[Wikipedia:Fundamental theorem of arithmetic\|fundamental theorem of arithmetic]] we can choose the exponents of prime numbers to get a vector representation of any rational multiple of 1 Hz.		When dealing with musical data we mostly care about frequencies measured in Hertz. To represent frequencies other than 1 Hz we need more basis vectors and by the [[Wikipedia:Fundamental theorem of arithmetic\|fundamental theorem of arithmetic]] we can choose the exponents of prime numbers to get a vector representation of any rational multiple of 1 Hz.

			Using the basis of (the exponents of) the second, prime 2 and prime 3 we can plot 1 Hz at (-1, 0, 0), 2 Hz at (-1, 1, 0), 3 Hz at (-1, 0, 1) and 4/3 Hz at (-1, 2, -1). [TODO: image]

			To make sense of vector addition we say that objects in this space represent the frequency you get at the intersection of the line from origin and the Hertz plane e.g. the sum of 9 Hz and 4 Hz is 36 Hz² and represent the frequency 6 Hz i.e. the geometric average of 9 Hz and 4 Hz. This is one part of what the '''ablog''' (short for ''absolute logarithmic'') function in SonicWeave does. [TODO: image]

			This works for everything except for objects with 0 as the exponent of the second. These objects form the relative echelon in Sonic Weave where the conversion from linear factors to logarithmic pitch is handled by the '''logarithmic''' function. In this document we notate it using a right-facing arrow:

			<math>\overrightarrow{2^x 3^y 5^z \ldots} \mapsto x e_2 + y e_3 + z e_5 + \ldots, x, y, z, \ldots \in \mathbb{Q}</math>

			''Relative logarithmic'' vector addition is the same as multiplication of the represented ''linear'' values.

			<math>\overrightarrow{a} + \overrightarrow{b} = \overrightarrow{a b}</math>

			=== Conversion from relative to absolute ===
			The second part of '''ablog''' requires a unison frequency declaration like <code>1/1 = 440 Hz</code>. In this document we denote <math>e_0 = \overrightarrow{1 Hz}</math> and the declared unison frequency as <math>e_0'</math> e.g <math>e_0' = \overrightarrow{440 Hz}</math>

			''Relative logarithmic'' values are converted to ''absolute logarithmic'' frequencies simply by adding <math>e_0'</math>. e.g. '''ablog(logarithmic(81/55))''' is

			<math>
			\overrightarrow{\frac{81}{55}} + e_0' = (4 e_3 - e_5 - e_{11}) + (3 e_2 + e_5 + e_{11} + e_0) = 3 e_2 + 4 e_3 + e_0 = \overrightarrow{648 Hz}
			</math>

Frostburn: Start over and establish that exponents of SI units can be used as basis.

2024-01-05T12:58:28Z

Start over and establish that exponents of SI units can be used as basis.

Show changes

Frostburn: Expand type hierarchy but leave TODOs for obscure types.

2023-12-10T14:04:14Z

Expand type hierarchy but leave TODOs for obscure types.

@@ Line 60: / Line 60: @@
 \mathrm{xfjs} &\supset \{\mathrm{n3}, \mathrm{m4.5}\} \\
 \mathrm{pitch} &= \mathrm{ji} \cup \mathrm{nedo} \cup \mathrm{nedji} \cup \mathrm{cents} \cup \mathrm{monzo} \cup \mathrm{xfjs}
 \end{align}
 </math>
@@ Line 71: / Line 80: @@
 Similarly, the free pitch type is unlikely to be relevant in day-to-day use of SW3.

Frostburn: Sketch up to real logarithmic types.

2023-12-10T11:49:21Z

Sketch up to real logarithmic types.

← Older revision		Revision as of 11:49, 10 December 2023
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	Theres a linear domain where 3/2 + 3/2 means 3 (as a ratio of two frequencies) and a logarithmic domain where 3\2 + 3\2 means 8 (as a ratio of two frequencies).		Theres a linear domain where 3/2 + 3/2 means 3 (as a ratio of two frequencies) and a logarithmic domain where 3\2 + 3\2 means 8 (as a ratio of two frequencies).
	== Tiers ==		== Tiers ==
	Types are organized into tiers consisting of integers, rationals, radicals (i.e. rationals raised to rational powers) and reals.		Types are organized into tiers consisting of booleans, integers, rationals, radicals (i.e. rationals raised to rational powers) and reals.

	<math>		<math>
	\mathbb{Z} \subset \mathbb{Q} \subset \mathrm{radical} \subset \mathbb{R}		\mathbb{B} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathrm{radical} \subset \mathbb{R}
	</math>		</math>
	== Type system ==		== Type system ==
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	<math>		<math>
	\begin{align}		\begin{align}
	\mathrm{boolean} &= \{0, 1\} \\		\mathrm{boolean} &= \mathbb{B} = \{0, 1\} \\
	\mathrm{integer} &= \mathbb{Z} \supset \{1, 2, 3, 4\} \\		\mathrm{integer} &= \mathbb{Z} \supset \{1, 2, 3, 4\} \\
	\mathrm{fraction} &\supset \{3/2, 5/3\} \\		\mathrm{fraction} &\supset \{3/2, 5/3\} \\
	\mathrm{decimal} &\supset \{(1.2), (1.4)\} \\		\mathrm{decimal} &\supset \{(1.2), (1.4), (1,5)\} \\
	\mathrm{rational} &= \mathbb{Q} = \mathrm{integer} \cup \mathrm{fraction} \cup \mathrm{decimal}		\mathrm{rational} &= \mathbb{Q} = \mathrm{integer} \cup \mathrm{fraction} \cup \mathrm{decimal}
	\end{align}		\end{align}
	</math>		</math>

			Note: Decimals require surrounding parenthesis when using a decimal dot but may be entered plain when using a decimal comma.

	=== Radical linear types ===		=== Radical linear types ===
	<math>		<math>
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	\begin{align}		\begin{align}
	\mathrm{integer!} &\supset \{69!, 420!\} \\		\mathrm{integer!} &\supset \{69!, 420!\} \\
	\mathrm{decimal!} &\supset \{3.14159!, 2.718!\} \\		\mathrm{decimal!} &\supset \{(3.14159!), (2.718!)\} \\
	\mathrm{real} &= \mathbb{R} = \mathrm{radical} \cup \mathrm{integer!} \cup \mathrm{decimal!}		\mathrm{real} &= \mathbb{R} = \mathrm{radical} \cup \mathrm{integer!} \cup \mathrm{decimal!}
	\end{align}		\end{align}
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	Note: Real linear types are mostly an artifact of the catch-all property extended time monzos. Not recommended for everyday use.		Note: Real linear types are mostly an artifact of the catch-all property extended time monzos. Not recommended for everyday use.
			=== Rational logarithmic types ===
			<math>
			\begin{align}
			\mathrm{fjs} &\supset \{ \mathrm{P5}, \mathrm{M3}^5 \} \\
			\mathrm{ji} &= \mathrm{fjs}
			\end{align}
			</math>
			=== Radical logarithmic types ===
			<math>
			\begin{align}
			\mathrm{nedo} &\supset \{ 5\backslash 7, 7\backslash 12 \} \\
			\mathrm{nedji} &\supset \{ 7\backslash 13<3>, 1\backslash 3<5/3> \} \\
			\mathrm{cents} &\supset \{.5, 1.955, 100., c \} \\
			\mathrm{monzo} &\supset \{[-4, 4, -1>, [1/2, 1/3> \} \\
			\mathrm{xfjs} &\supset \{\mathrm{n3}, \mathrm{m4.5}\} \\
			\mathrm{pitch} &= \mathrm{ji} \cup \mathrm{nedo} \cup \mathrm{nedji} \cup \mathrm{cents} \cup \mathrm{monzo} \cup \mathrm{xfjs}
			\end{align}
			</math>
			=== Real logarithmic types ===
			<math>
			\begin{align}
			\mathrm{cents!} &\supset \{.777!, 1901.955!, 69.!, c!\} \\
			\mathrm{freePitch} &= \mathrm{pitch} \cup \mathrm{cents!}
			\end{align}
			</math>

			Similarly, the free pitch type is unlikely to be relevant in day-to-day use of SW3.

Frostburn: /* Real linear types */ Include missing lower tier.

2023-12-10T11:29:34Z

Real linear types: Include missing lower tier.

← Older revision		Revision as of 11:29, 10 December 2023
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	\mathrm{integer!} &\supset \{69!, 420!\} \\		\mathrm{integer!} &\supset \{69!, 420!\} \\
	\mathrm{decimal!} &\supset \{3.14159!, 2.718!\} \\		\mathrm{decimal!} &\supset \{3.14159!, 2.718!\} \\
	\mathrm{real} &= \mathbb{R} = \mathrm{integer!} \cup \mathrm{decimal!}		\mathrm{real} &= \mathbb{R} = \mathrm{radical} \cup \mathrm{integer!} \cup \mathrm{decimal!}
	\end{align}		\end{align}
	</math>		</math>

	Note: Real linear types are mostly an artifact of the catch-all property extended time monzos. Not recommended for everyday use.		Note: Real linear types are mostly an artifact of the catch-all property extended time monzos. Not recommended for everyday use.

Frostburn: Sketch SonicWeave type system up to real linear types.

2023-12-10T11:28:24Z

Sketch SonicWeave type system up to real linear types.

New page

This is my second attempt at creating a a Domain Specific Language (DSL) called ''SonicWeave'' for manipulating frequencies, ratios and pitches in Scale Workshop 3.

Still very much a work in progress. Expect things to shift around as I design a parseable grammar.
== Values ==
Values consist of strings, functions and extended time monzos which combine a rational time exponent, rational prime exponents, a multiplicative rational residual and a catch-all real cents offset.
== Domains ==
Theres a linear domain where 3/2 + 3/2 means 3 (as a ratio of two frequencies) and a logarithmic domain where 3\2 + 3\2 means 8 (as a ratio of two frequencies).
== Tiers ==
Types are organized into tiers consisting of integers, rationals, radicals (i.e. rationals raised to rational powers) and reals.

<math>
\mathbb{Z} \subset \mathbb{Q} \subset \mathrm{radical} \subset \mathbb{R}
</math>
== Type system ==
Values are dressed into types to implement domain-specific operator overloading and type-aware function overloading.
=== Rational linear types ===
<math>
\begin{align}
\mathrm{boolean} &= \{0, 1\} \\
\mathrm{integer} &= \mathbb{Z} \supset \{1, 2, 3, 4\} \\
\mathrm{fraction} &\supset \{3/2, 5/3\} \\
\mathrm{decimal} &\supset \{(1.2), (1.4)\} \\
\mathrm{rational} &= \mathbb{Q} = \mathrm{integer} \cup \mathrm{fraction} \cup \mathrm{decimal}
\end{align}
</math>
=== Radical linear types ===
<math>
\begin{align}
\mathrm{radicalExpression} &\supset \{\sqrt{3}, 2^\frac{5}{3}\} \\
\mathrm{radical} &= \mathrm{rational} \cup \mathrm{radicalExpression}
\end{align}
</math>
=== Real linear types ===
<math>
\begin{align}
\mathrm{integer!} &\supset \{69!, 420!\} \\
\mathrm{decimal!} &\supset \{3.14159!, 2.718!\} \\
\mathrm{real} &= \mathbb{R} = \mathrm{integer!} \cup \mathrm{decimal!}
\end{align}
</math>

Note: Real linear types are mostly an artifact of the catch-all property extended time monzos. Not recommended for everyday use.