Hahn distance - Revision history

Xenllium at 22:40, 27 January 2026

2026-01-27T22:40:24Z

← Older revision		Revision as of 22:40, 27 January 2026
Line 42:		Line 42:

	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.		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.

			== Examples ==
			{\| class="wikitable"
			\|+ style="font-size: 105%;" \| Hahn distance of 5-limit intervals
			\|-
			! Ratio
			! 5-odd-limit
			! 9-odd-limit
			! 15-odd-limit
			! 25-odd-limit
			! 27-odd-limit
			\|-
			\| [[6/5]]
			\| 1
			\| 1
			\| 1
			\| 1
			\| 1
			\|-
			\| [[10/9]]
			\| 2
			\| 1
			\| 1
			\| 1
			\| 1
			\|-
			\| [[16/15]]
			\| 2
			\| 2
			\| 1
			\| 1
			\| 1
			\|-
			\| [[25/24]]
			\| 2
			\| 2
			\| 2
			\| 1
			\| 1
			\|-
			\| [[27/25]]
			\| 3
			\| 2
			\| 2
			\| 2
			\| 1
			\|-
			\| [[45/32]]
			\| 3
			\| 2
			\| 2
			\| 2
			\| 2
			\|-
			\| [[75/64]]
			\| 3
			\| 3
			\| 2
			\| 2
			\| 2
			\|-
			\| [[81/80]]
			\| 4
			\| 2
			\| 2
			\| 2
			\| 2
			\|-
			\| [[135/128]]
			\| 4
			\| 3
			\| 2
			\| 2
			\| 2
			\|}

	[[Category:Math]]		[[Category:Math]]

Sintel: -legacy

2025-03-29T18:01:03Z

-legacy

← Older revision		Revision as of 18:01, 29 March 2025
Line 1:		Line 1:
	~~{{Legacy}}~~
	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.		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.

ArrowHead294 at 13:14, 19 March 2025

2025-03-19T13:14:13Z

← Older revision		Revision as of 13:14, 19 March 2025
Line 31:		Line 31:
	</math>		</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.		where y = signum(x2){{ceil\|{{abs\|x2/2}}}}; here "signum" is +1 or −1 depending on the sign of x2 and {{ceil\|''x''}} 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 {{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		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

ArrowHead294 at 17:17, 18 February 2025

2025-02-18T17:17:14Z

← Older revision		Revision as of 17:17, 18 February 2025
Line 9:		Line 9:
	\begin{align}		\begin{align}
	& \lVert 3^a \cdot 5^b \cdot 7^c \rVert_\text {hahn} \\		& \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 \\		=& \left(\lvert a \rvert + \lvert b \rvert + \lvert c \rvert + \lvert a + b + c \rvert\right)/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)		=& \max\left(\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\right)
	\end{align}		\end{align}
	</math>		</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 {{nowrap\|‖(''a'', ''b'', ''c'')‖<sub>hahn</sub> {{=}} {{sfrac\|{{!}}''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		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>		<math>\displaystyle \left\lVert (a, b, c) \right\rVert_\text {sym} = \sqrt{a^2 + b^2 + c^2 + ab + bc + ca}</math>

	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.		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.
Line 27:		Line 27:
	\begin{align}		\begin{align}
	& \left\lVert \lvert x_1\ x_2\ x_3\ x_4\ x_5\ x_6 \rangle \right\rVert_\text{hahn} \\		& \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		=& \left(\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\right)/2
	\end{align}		\end{align}
	</math>		</math>

Lériendil at 16:46, 17 February 2025

2025-02-17T16:46:02Z

← Older revision		Revision as of 16:46, 17 February 2025
Line 1:		Line 1:
	{{Legacy~~\|Hahn_distance~~}}		{{Legacy}}
	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.		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.

Lériendil: deploying new cat

2025-02-17T16:18:48Z

deploying new cat

← Older revision		Revision as of 16:18, 17 February 2025
Line 1:		Line 1:
			{{Legacy\|Hahn_distance}}
	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.		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.

Lériendil: Changed protection level for "Hahn distance" ([Edit=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)))

2025-02-17T16:03:53Z

Changed protection level for "Hahn distance" ([Edit=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)) [Move=Allow only administrators] (expires 17:03, 17 February 2025 (UTC)))

← Older revision	Revision as of 16:03, 17 February 2025
(No difference)

FloraC: Improve wiki markup; style; linking

2023-11-26T14:02:11Z

Improve wiki markup; style; linking

← Older revision		Revision as of 14:02, 26 November 2023
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		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		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		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.		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.		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.

FloraC: Recategorize

2023-11-26T11:13:55Z

Recategorize

← Older revision		Revision as of 11:13, 26 November 2023
Line 34:		Line 34:

	[[Category:Math]]		[[Category:Math]]
	[[Category:Interval complexity ~~measure~~]]		[[Category:Interval complexity measures]]

	{{Todo\| add examples \| cleanup }}		{{Todo\| add examples \| cleanup }}

FloraC: internalize wikipedia links; recategorize

2023-08-29T13:56:55Z

internalize wikipedia links; recategorize

← Older revision		Revision as of 13:56, 29 August 2023
Line 1:		Line 1:
	In [~~http~~:~~//en.wikipedia.org/wiki/Graph_~~(mathematics) graph theory], the [~~http~~:~~//en.wikipedia.org/wiki/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 [[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.

	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 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.
Line 32:		Line 32:

	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.		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.
	~~[[Category:distance]]~~
	[[Category:~~math~~]]		[[Category:Math]]
	[[Category:measure]]		[[Category:Interval complexity measure]]
	~~[[Category:todo:add_examples]]~~
			{{Todo\| add examples \| cleanup }}