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{{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.


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<math>\displaystyle  
<math>\displaystyle  
\begin{align}
\begin{aligned}
& \lVert 3^a \cdot 5^b \cdot 7^c \rVert_\text {hahn} \\
& \lVert 3^a \cdot 5^b \cdot 7^c \rVert_\text {hahn} \\
=& \left(\lvert a \rvert + \lvert b \rvert + \lvert c \rvert + \lvert a + b + c \rvert\right)/2 \\
=& \left(\lvert a \rvert + \lvert b \rvert + \lvert c \rvert + \lvert a + b + c \rvert\right)/2 \\
=& \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)
=& \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{aligned}
</math>
</math>


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<math>\displaystyle  
<math>\displaystyle  
\begin{align}
\begin{aligned}
& \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} \\
=& \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
=& \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{aligned}
</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


<math>\displaystyle  
<math>\displaystyle  
\begin{align}
\begin{aligned}
& \left\lVert \lvert x_1\ x_2\ x_3\ x_4\ x_5\ x_6 \rangle \right\rVert \\
& \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
=& \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}
\end{aligned}
</math>
</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.
== 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]]
Retrieved from "https://en.xen.wiki/w/Hahn_distance"