Hahn distance: Difference between revisions
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\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> = | 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{ | <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. | ||
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\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> | ||