S-expression: Difference between revisions

Line 1,117:

Alternatively stated: S(''k'' - 1)/S(''k'' + 1) is superparticular for ''k'' ≠ 0 (mod 4) and is odd-particular when k = 0 (mod 4). This alternative statement highlights an interesting fact that the four harmonics related by tempering S(''k'' - 1)/S(''k'' + 1) are (''k'' - 2):(''k'' - 1):(''k'' + 1):(''k'' + 2) through tempering ((''k''+2)/(''k''-2)) / ((''k''+1)/(''k''-1))2 meaning the ''k''th harmonic is the only one not included and therefore a semiparticular is odd-particular if the excluded "harmonic in the middle" (around which the two on each side are symmetric in terms of placement) is a multiple of 4 and is superparticular otherwise.

== Abstraction ==

=== The maths. ===

Let ''H'' be a [[Wikipedia:Abelian group|commutative group]] with generators hi, ..., hk, ..., hj (such that i ≤ k ≤ j).

These generators are a series indexed by the integers that are analogous to a portion of the harmonic series, but "analogous" is extremely abstract here, because:

The fact that they are indexed by a range of integers is the ''only'' analogy that is guaranteed to hold, but as it turns out, is sufficient for defining analogies of superparticulars and thus S-expressions.

Thus: (the analogue of) a superparticular is of the form hk+1 hk-1 = hk+1 / hk (we'll use multiplicative notation) meaning that (the analogue of) S''k'' is:

<math>

\begin {align}

{\rm S}(k) = \frac{h_k^2}{h_{k-1} h_{k+1}} = (h_k / h_{k-1})/(h_{k+1} / h_k) = h_k h_{k-1}^{-1} (h_{k+1} h_k^{-1})^{-1} = h_k^2 h_{k-1}^{-1} h_{k+1}^{-1}

\end {align}

</math>

Then (the analogues of) S-monzos correspond to the ''exponents'' of the generators, such that:

<math>

\begin {align}

{\rm S}(k) =\ .. h_{k-3}^0 h_{k-2}^0 h_{k-1}^{-1} h_k^2 h_{k+1}^{-1} h_{k+2}^0 h_{k+3}^0 ..\ = [..,\ h_{k-3},\ h_{k-2},\ h_{k-1},\ h_k,\ h_{k+1},\ h_{k+2},\ h_{k+3},\ ..]^{\Large[..,\ 0,\ 0, -1,\ 2, -1,\ 0,\ 0,\ ..]}

\end {align}

</math>

This completes the analogy. What this means is:

Every infinite comma family defined in terms of an S-expression will have an infinite number of analogues, because of the maths of S-monzos continuing to work as expected.

The meanings of these analogues are up to us to interpret, however. This brings us to applications, which we will examine next, with a focus on the musical ones:

=== Applications: ===

Unless otherwise specified, we will let ''k'' be in the positive integers (Z+) and we will let the group operation be the multiplication of rationals, but this abstraction is much more powerful than that.

If you are working with a certain expression for hk, it is suggested to use Sa where "a" is a letter that abbreviates the meaning of what you are using.

Letter suggestions are provided below for expressions suspected to be theoretically useful/interesting in the direction of designing desirable temperaments.

If we use the letter "a" for some analogy Sa''k'', then because of the guarantees of the analogy, we will always be able to speak of a-square-particulars, a-ultraparticulars, a-semiparticulars, a-1/n-square-particulars and a-lopsided-commas, and have it make abstract sense.

To emphasize: this is because all comma families expressed in terms of expressions involving H's group operation applied to elements Sa''k''p for ''k'', ''p'' in Z will have analogues if ''k'' is a valid index for Sa.

Finally, while we usually speak of temperaments, note that S-expressions, as a tool for aiding [[RTT]], have a wide variety of fruitful applications, exactly because [[RTT]] already itself has a wide variety of fruitful applications not explicitly involving tempering, especially in the design of scales where the [[constant structure]] property is desirable, where [[exotemperament]]s can shine as representing a deep, coarse logic.

=== hk = k ===

The trivial example, equal to normal S-expressions and S-monzos, before abstraction.

=== hk = 2k + 1 ===

An analogy of S-expressions and S-monzos for [[EDT]]s which can be used as a corresponding [[RTT]] tool for when we only want to focus on the arithmetic of odds, using [[3/1]] as the new [[equave]].

A suggestion is to use the notation So''k'', if this is not unambiguous, with the letter "o" standing for "odd". Thus:

<math>

\begin {align}

{\rm So}(k) = h_k^2 h_{k-1}^{-1} h_{k+1}^{-1} = \frac{ (2k+1)^2 }{ (2k-1)(2k+3) } = \frac{ 4k^2 + 4k + 1 }{ 4k^2 - 4k - 3 }

\end {align}

</math>

=== hk = (k + 1)/k ===

An analogy of S-expressions and S-monzos aiming at deeply faithful modelling of the harmonic series through modelling distances between superparticulars accurately.

A suggestion is to use the notation Ss''k'', if this is not unambiguous, with the letter "s" standing for "superparticular" or "super" generally.

We will see that this implies Ss''k'' is the difference between two adjacent S''k'' (an ultraparticular), implying Ss''k'' * Ss(''k'' - 1) is a semiparticular. Thus:

<math>

\begin {align}

{\rm Ss}(k) = h_k^2 h_{k-1}^{-1} h_{k+1}^{-1} = \large \frac{ \frac{(k+1)}{k} \cdot \frac{(k+1)}{k} }{ \frac{k}{k-1}\cdot\frac{k+2}{k+1} } \normalsize = \frac{ {\rm S}(k+1) }{ {\rm S}(k) }

\implies {\rm Ss}(k){\rm Ss}(k-1) = \frac{ {\rm S}(k+1) }{ {\rm S}(k) } \cdot \frac{ {\rm S}(k) }{ {\rm S}(k-1) } = \frac{ {\rm S}(k+1) }{ {\rm S}(k-1) }

\end {align}

</math>

This implies that s-ultraparticulars and s-semiparticulars are now about ratios between

[[Category:Elementary math]]

[[Category:superparticular ratios]]

[[Category:Terms]]

@@ Line 1,117: / Line 1,117: @@
 Alternatively stated: S(''k'' - 1)/S(''k'' + 1) is superparticular for ''k'' ≠ 0 (mod 4) and is odd-particular when k = 0 (mod 4). This alternative statement highlights an interesting fact that the four harmonics related by tempering S(''k'' - 1)/S(''k'' + 1) are (''k'' - 2):(''k'' - 1):(''k'' + 1):(''k'' + 2) through tempering ((''k''+2)/(''k''-2)) / ((''k''+1)/(''k''-1))<sup>2</sup> meaning the ''k''th harmonic is the only one not included and therefore a semiparticular is odd-particular if the excluded "harmonic in the middle" (around which the two on each side are symmetric in terms of placement) is a multiple of 4 and is superparticular otherwise.
+== Abstraction ==
+=== The maths. ===
+Let ''H'' be a [[Wikipedia:Abelian group|commutative group]] with generators h<sub>i</sub>, ..., h<sub>k</sub>, ..., h<sub>j</sub> (such that i ≤ k ≤ j).
+These generators are a series indexed by the integers that are analogous to a portion of the harmonic series, but "analogous" is extremely abstract here, because:
+The fact that they are indexed by a range of integers is the ''only'' analogy that is guaranteed to hold, but as it turns out, is sufficient for defining analogies of superparticulars and thus S-expressions.
+Thus: (the analogue of) a superparticular is of the form h<sub>k+1</sub> h<sub>k</sub><sup>-1</sup> = h<sub>k+1</sub> / h<sub>k</sub> (we'll use multiplicative notation) meaning that (the analogue of) S''k'' is:
+<math>
+\begin {align}
+{\rm S}(k) = \frac{h_k^2}{h_{k-1} h_{k+1}} = (h_k / h_{k-1})/(h_{k+1} / h_k) = h_k h_{k-1}^{-1} (h_{k+1} h_k^{-1})^{-1} = h_k^2 h_{k-1}^{-1} h_{k+1}^{-1}
+\end {align}
+</math>
+Then (the analogues of) S-monzos correspond to the ''exponents'' of the generators, such that:
+<math>
+\begin {align}
+{\rm S}(k) =\ .. h_{k-3}^0 h_{k-2}^0 h_{k-1}^{-1} h_k^2 h_{k+1}^{-1} h_{k+2}^0 h_{k+3}^0 ..\ = [..,\ h_{k-3},\ h_{k-2},\ h_{k-1},\ h_k,\ h_{k+1},\ h_{k+2},\ h_{k+3},\ ..]^{\Large[..,\ 0,\ 0, -1,\ 2, -1,\ 0,\ 0,\ ..]}
+\end {align}
+</math>
+This completes the analogy. What this means is:
+Every infinite comma family defined in terms of an S-expression will have an infinite number of analogues, because of the maths of S-monzos continuing to work as expected.
+The meanings of these analogues are up to us to interpret, however. This brings us to applications, which we will examine next, with a focus on the musical ones:
+=== Applications: ===
+Unless otherwise specified, we will let ''k'' be in the positive integers (Z<sub>+</sub>) and we will let the group operation be the multiplication of rationals, but this abstraction is much more powerful than that.
+If you are working with a certain expression for h<sub>k</sub>, it is suggested to use Sa where "a" is a letter that abbreviates the meaning of what you are using.
+Letter suggestions are provided below for expressions suspected to be theoretically useful/interesting in the direction of designing desirable temperaments.
+If we use the letter "a" for some analogy Sa''k'', then because of the guarantees of the analogy, we will always be able to speak of a-square-particulars, a-ultraparticulars, a-semiparticulars, a-1/n-square-particulars and a-lopsided-commas, and have it make abstract sense.
+To emphasize: this is because all comma families expressed in terms of expressions involving H's group operation applied to elements Sa''k''<sup>p</sup> for ''k'', ''p'' in Z will have analogues if ''k'' is a valid index for Sa.
+Finally, while we usually speak of temperaments, note that S-expressions, as a tool for aiding [[RTT]], have a wide variety of fruitful applications, exactly because [[RTT]] already itself has a wide variety of fruitful applications not explicitly involving tempering, especially in the design of scales where the [[constant structure]] property is desirable, where [[exotemperament]]s can shine as representing a deep, coarse logic.
+=== h<sub>k</sub> = k ===
+The trivial example, equal to normal S-expressions and S-monzos, before abstraction.
+=== h<sub>k</sub> = 2k + 1 ===
+An analogy of S-expressions and S-monzos for [[EDT]]s which can be used as a corresponding [[RTT]] tool for when we only want to focus on the arithmetic of odds, using [[3/1]] as the new [[equave]].
+A suggestion is to use the notation So''k'', if this is not unambiguous, with the letter "o" standing for "odd". Thus:
+<math>
+\begin {align}
+{\rm So}(k) = h_k^2 h_{k-1}^{-1} h_{k+1}^{-1} = \frac{ (2k+1)^2 }{ (2k-1)(2k+3) } = \frac{ 4k^2 + 4k + 1 }{ 4k^2 - 4k - 3 }
+\end {align}
+</math>
+=== h<sub>k</sub> = (k + 1)/k ===
+An analogy of S-expressions and S-monzos aiming at deeply faithful modelling of the harmonic series through modelling distances between superparticulars accurately.
+A suggestion is to use the notation Ss''k'', if this is not unambiguous, with the letter "s" standing for "superparticular" or "super" generally.
+We will see that this implies Ss''k'' is the difference between two adjacent S''k'' (an ultraparticular), implying Ss''k'' * Ss(''k'' - 1) is a semiparticular. Thus:
+<math>
+\begin {align}
+{\rm Ss}(k) = h_k^2 h_{k-1}^{-1} h_{k+1}^{-1} = \large \frac{ \frac{(k+1)}{k} \cdot  \frac{(k+1)}{k} }{ \frac{k}{k-1}\cdot\frac{k+2}{k+1} } \normalsize = \frac{ {\rm S}(k+1) }{ {\rm S}(k) }
+\implies {\rm Ss}(k){\rm Ss}(k-1) = \frac{ {\rm S}(k+1) }{ {\rm S}(k) } \cdot \frac{ {\rm S}(k) }{ {\rm S}(k-1) } = \frac{ {\rm S}(k+1) }{ {\rm S}(k-1) }
+\end {align}
+</math>
+This implies that s-ultraparticulars and s-semiparticulars are now about ratios between
 [[Category:Elementary math]]
 [[Category:superparticular ratios]]
 [[Category:Terms]]