Hahn distance: Difference between revisions

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 [[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.

 

<math>\displaystyle

& \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 \\

=& \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)

</math>

 

We may take this formula and apply it to any triple of real numbers {{nowrap|‖(''a'', ''b'', ''c'')‖_hahn {{=}} {{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

 

<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.

 

 

<math>\displaystyle

=& \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

</math>

 

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

 

<math>\displaystyle

\begin{align}

=& \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

</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.

 

{| class="wikitable"

|+ style="font-size: 105%;" | Hahn distance of 5-limit intervals

|-

! 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

|-

| [[81/80]]

| 4

| 2

| 2

| 2

| 2

| [[135/128]]

| 4

| 3

| 2

| 2

| 2

|}

 

[[Category:Math]]

[[Category:Interval complexity measures]]