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Hahn distance: Difference between revisions

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Fix <math> environments for MathJax v4 (align->aligned, etc.)
 
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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>


Line 24: Line 24:


<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>


Line 35: Line 35:


<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"