Optimizing MOS scales for DR

This is an expert page. It either assumes experience with xen theory or involves fairly technical procedures.

Optimizing a MOS for one DR chord

TODO: reword

A MOS can exactly tune only one DR chord with two non-free deltas. To find such a tuning, we derive and solve an algebraic equation for the generator.

Example

We start by choosing the MOS scale and equave, and the DR chord.

For example with 5L 2s ⟨2/1⟩, the usual diatonic scale, and we want to approximate 4:5:6, the just major chord, with a delta-rational MOS chord.

Identify the mappings of each of the deltas. The deltas are 5/4, 6/5, 7/6. For a Meantone mapping, these are g⁴/4 -1, g-g⁴/4. This is because in meantone, 1/1, 3/2, 5/4 are 1, g, g⁴/4 respectively, so the deltas are identified by subtracting the each term with the one before it.

In this case, we want the difference between our deltas to become 1, so the delta signature will be +1+1.

To achieve this, we take the difference between the first two deltas and set it to zero, so (g⁴/4 -1) - (g-g⁴/4) = 0. Put in integer terms, it's g⁴ - 2g - 2 = 0. Solving for g, the only root that makes sense is g≈1.49453, which in cents is 695.630c. And thus, with this generator, we will have a DR ~4:5:6 meantone chord!

Note that the equation to solve depends on what chord you want to tune as equal-beating. For example, assuming pure octaves, Meantone admits an equation for tuning the 3:4:5 as equal-beating: g⁴ + 2g − 8 = 0 The latter equation has solution g = 1.4960 = 697.3¢.

If instead we chose a Schismic mapping, the deltas would be g⁸/8 -1, 2/g - g⁸/8, which gives a generator of 498.308c for 4:5:6.

Optimizing a MOS for multiple DR chords

Since a MOS can't nail multiple DR chords exactly, we solve a least-squares error problem; we choose a DR error measure and we minimize the sum of squares of DR errors.