Hahn distance: Difference between revisions

Line 1:

In ~~[[wikipedia:~~ Graph (mathematics)|graph theory]], the ~~[[wikipedia:~~ Distance (graph theory)|distance] between two vertices a and b is defined as the minimum number of edges in a path connecting them, or in other words the minimum length of a connecting path; if there is no path connection them, the distance is regarded as infinite. Given a set of just ~~intervals~~, or more usually, of classes of octave-equivalent intervals, we can define a corresponding graph whose vertices are the intervals and which contain an edge between two intervals if the ratio between them is a consonance. Normally the unison is not counted as a consonance, and we therefore obtain in this way a graph with no loops which is very useful in various ways, such as in the study of scales.

In {{w|Graph (mathematics)|graph theory}}, the {{w|Distance (graph theory)|distance}} between two vertices a and b is defined as the minimum number of edges in a path connecting them, or in other words the minimum length of a connecting path; if there is no path connection them, the distance is regarded as infinite. Given a set of [[just interval]]s, or more usually, of [[pitch class|classes of octave-equivalent intervals]], we can define a corresponding graph whose vertices are the intervals and which contain an edge between two intervals if the ratio between them is a [[consonance]]. Normally the [[unison]] is not counted as a consonance, and we therefore obtain in this way a graph with no loops which is very useful in various ways, such as in the study of scales.

If we apply the above construction to the set of p-limit interval classes, using as consonances the q-odd-limit consonances, excluding the unison and ~~octaves~~, where q is an odd number q ≥ p which less than the next prime after p, the resulting graph could be called the Hahn graph, and distance on it is q-limit Hahn distance between two octave classes.

If we apply the above construction to the set of [[harmonic limit|''p''-limit]] interval classes, using as consonances the [[odd limit|''q''-odd-limit]] consonances, excluding the unison and [[octave]]s, where ''q'' is an odd number ''q'' ≥ ''p'' which less than the next prime after ''p'', the resulting graph could be called the Hahn graph, and distance on it is ''q''-limit Hahn distance between two octave classes.

Up to the 7-limit, Hahn distance has a very nice formula give by

<math>\displaystyle

\begin{align}

& \lVert 3^a \cdot 5^b \cdot 7^c \rVert_\text {hahn} \\

=& (\lvert a \rvert + \lvert b \rvert + \lvert c \rvert + \lvert a + b + c \rvert)/2 \\

=& \max(\lvert a \rvert, \lvert b \rvert, \lvert c \rvert, \lvert a + b \rvert, \lvert b + c \rvert, \lvert c + a \rvert, \lvert a + b + c \rvert)

\end{align}

</math>

~~<math>= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)</math>~~

We may take this formula and apply it to any triple of real numbers ‖(a, b, c)‖<sub>hahn</sub> = (|a| + |b| + |c| + |a + b + c|)/2.

We may take this formula and apply it to any triple of real numbers ||(a, b, c)~~||_hahn~~ = (|a|+|b|+|c|+|a+b+c|)/2.

If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice; it also defines a seminorm on 7-limit [[Monzos_and_Interval_Space|interval space]]. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by

<math>\displaystyle \left\lVert (a, b, c) \right\rVert_\text {sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}</math>

and discussed [[~~The_Seven_Limit_Symmetrical_Lattices|here~~]]. We can regard Hahn distance as an alternative to symmetrical Euclidean distance which is more closely tied to the consonance graph of the lattice.

and discussed in [[The Seven Limit Symmetrical Lattices]]. We can regard Hahn distance as an alternative to symmetrical Euclidean distance which is more closely tied to the consonance graph of the lattice.

In the 13-limit the formula for Hahn distance can be given as

<math>~~|| |~~x_1\ x_2\ x_3\ x_4\ x_5\ x_6~~> ||_~~{hahn} = ~~</math>~~

<math>\displaystyle

\begin{align}

~~<math>~~(|y|+|x_3|+|x_4|+|x_5|+|x_6|+|y+x_3+x_4+x_5+x_6|)/2</math>

& \left\lVert \lvert x_1\ x_2\ x_3\ x_4\ x_5\ x_6 \rangle \right\rVert_\text{hahn} \\

=& (\lvert y \rvert + \lvert x_3 \rvert + \lvert x_4 \rvert + \lvert x_5 \rvert + \lvert x_6 \rvert + \lvert y + x_3 + x_4 + x_5 + x_6 \rvert)/2

\end{align}

</math>

where y = signum(x2)ceil(|x2/2|); here "signum" is +1 or -1 depending on the sign of x2 and "ceil" is the ceiling function. Hahn distance for the 9 or 11 limit can also be found from this formula.

It should be noted that this formula defines a ~~[http://en.wikipedia.org/wiki/Metric_space~~ metric space distance function] but not a norm, and hence does not define a normed vector space, making the 9, 11 or 13 limit pitch classes into a lattice. We can modify it to

It should be noted that this formula defines a {{w|Metric space|metric space distance function}} but not a norm, and hence does not define a normed vector space, making the 9-, 11- or 13-limit pitch classes into a lattice. We can modify it to

~~<math>|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> || = </math>~~

<math>\displaystyle

\begin{align}

& \left\lVert \lvert x_1\ x_2\ x_3\ x_4\ x_5\ x_6 \rangle \right\rVert \\

=& \lvert x_2/2 \rvert + \lvert x_3 \rvert + \lvert x_4 \rvert + \lvert x_5 \rvert + \lvert x_6 \rvert + \lvert x_2/2 + x_3 + x_4 + x_5 + x_6 \rvert

\end{align}

</math>

This makes the 9.5.7.11.13 sublattice symmetrical, corresponded to even distance values from the origin, with the full lattice corresponding to all positive integer distances.

@@ Line 1: / Line 1: @@
-In [[wikipedia: Graph (mathematics)|graph theory]], the [[wikipedia: Distance (graph theory)|distance] between two vertices a and b is defined as the minimum number of edges in a path connecting them, or in other words the minimum length of a connecting path; if there is no path connection them, the distance is regarded as infinite. Given a set of just intervals, or more usually, of classes of octave-equivalent intervals, we can define a corresponding graph whose vertices are the intervals and which contain an edge between two intervals if the ratio between them is a consonance. Normally the unison is not counted as a consonance, and we therefore obtain in this way a graph with no loops which is very useful in various ways, such as in the study of scales.
+In {{w|Graph (mathematics)|graph theory}}, the {{w|Distance (graph theory)|distance}} between two vertices a and b is defined as the minimum number of edges in a path connecting them, or in other words the minimum length of a connecting path; if there is no path connection them, the distance is regarded as infinite. Given a set of [[just interval]]s, or more usually, of [[pitch class|classes of octave-equivalent intervals]], we can define a corresponding graph whose vertices are the intervals and which contain an edge between two intervals if the ratio between them is a [[consonance]]. Normally the [[unison]] is not counted as a consonance, and we therefore obtain in this way a graph with no loops which is very useful in various ways, such as in the study of scales.
-If we apply the above construction to the set of p-limit interval classes, using as consonances the q-odd-limit consonances, excluding the unison and octaves, where q is an odd number q ≥ p which less than the next prime after p, the resulting graph could be called the Hahn graph, and distance on it is q-limit Hahn distance between two octave classes.
+If we apply the above construction to the set of [[harmonic limit|''p''-limit]] interval classes, using as consonances the [[odd limit|''q''-odd-limit]] consonances, excluding the unison and [[octave]]s, where ''q'' is an odd number ''q'' ≥ ''p'' which less than the next prime after ''p'', the resulting graph could be called the Hahn graph, and distance on it is ''q''-limit Hahn distance between two octave classes.
 Up to the 7-limit, Hahn distance has a very nice formula give by
-<math>||3^a 5^b 7^c||_{hahn} = (|a| + |b| + |c| + |a+b+c|)/2</math>
+<math>\displaystyle
+\begin{align}
+& \lVert 3^a \cdot 5^b \cdot 7^c \rVert_\text {hahn} \\
+=& (\lvert a \rvert + \lvert b \rvert + \lvert c \rvert + \lvert a + b + c \rvert)/2 \\
+=& \max(\lvert a \rvert, \lvert b \rvert, \lvert c \rvert, \lvert a + b \rvert, \lvert b + c \rvert, \lvert c + a \rvert, \lvert a + b + c \rvert)
+\end{align}
+</math>
-<math>= max(|a|, |b|, |c|, |a+b|, |b+c|, |c+a|, |a+b+c|)</math>
+We may take this formula and apply it to any triple of real numbers ‖(a, b, c)‖<sub>hahn</sub> = (|a| + |b| + |c| + |a + b + c|)/2.
-We may take this formula and apply it to any triple of real numbers ||(a, b, c)||_hahn = (|a|+|b|+|c|+|a+b+c|)/2.
 If we do that, Hahn distance becomes a norm defining a normed vector space, which we might call Hahn space, and 5 or 7 limit classes of intervals become a lattice; it also defines a seminorm on 7-limit [[Monzos_and_Interval_Space|interval space]]. While Hahn space is not Euclidean, the distance measure it gives is not too different from the symmetrical Euclidean distance given by
-<math>||(a, b, c)||_{sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}</math>
+<math>\displaystyle \left\lVert (a, b, c) \right\rVert_\text {sym} = \sqrt{(a^2 + b^2 + c^2 + ab + bc + ca)}</math>
-and discussed [[The_Seven_Limit_Symmetrical_Lattices|here]]. We can regard Hahn distance as an alternative to symmetrical Euclidean distance which is more closely tied to the consonance graph of the lattice.
+and discussed in [[The Seven Limit Symmetrical Lattices]]. We can regard Hahn distance as an alternative to symmetrical Euclidean distance which is more closely tied to the consonance graph of the lattice.
 In the 13-limit the formula for Hahn distance can be given as
-<math>|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> ||_{hahn} = </math>
+<math>\displaystyle
+\begin{align}
-<math>(|y|+|x_3|+|x_4|+|x_5|+|x_6|+|y+x_3+x_4+x_5+x_6|)/2</math>
+& \left\lVert \lvert x_1\ x_2\ x_3\ x_4\ x_5\ x_6 \rangle \right\rVert_\text{hahn} \\
+=& (\lvert y \rvert + \lvert x_3 \rvert + \lvert x_4 \rvert + \lvert x_5 \rvert + \lvert x_6 \rvert + \lvert y + x_3 + x_4 + x_5 + x_6 \rvert)/2
+\end{align}
+</math>
 where y = signum(x2)ceil(|x2/2|); here "signum" is +1 or -1 depending on the sign of x2 and "ceil" is the ceiling function. Hahn distance for the 9 or 11 limit can also be found from this formula.
-It should be noted that this formula defines a [http://en.wikipedia.org/wiki/Metric_space metric space distance function] but not a norm, and hence does not define a normed vector space, making the 9, 11 or 13 limit pitch classes into a lattice. We can modify it to
+It should be noted that this formula defines a {{w|Metric space|metric space distance function}} but not a norm, and hence does not define a normed vector space, making the 9-, 11- or 13-limit pitch classes into a lattice. We can modify it to
-<math>|| |x_1\ x_2\ x_3\ x_4\ x_5\ x_6> || = </math>
-<math>|x_2/2|+|x_3|+|x_4|+|x_5|+|x_6|+|x_2/2+x_3+x_4+x_5+x_6|</math>
+<math>\displaystyle
+\begin{align}
+& \left\lVert \lvert x_1\ x_2\ x_3\ x_4\ x_5\ x_6 \rangle \right\rVert \\
+=& \lvert x_2/2 \rvert + \lvert x_3 \rvert + \lvert x_4 \rvert + \lvert x_5 \rvert + \lvert x_6 \rvert + \lvert x_2/2 + x_3 + x_4 + x_5 + x_6 \rvert
+\end{align}
+</math>
 This makes the 9.5.7.11.13 sublattice symmetrical, corresponded to even distance values from the origin, with the full lattice corresponding to all positive integer distances.