**Unusual musical
scales**

Click here to go to the page for the Tumbling Dekany (a polytope music generator).

In my paper *Harmonic errors in
equal tempered musical scales*, I have investigated equal divisions of
the octave other than 12 (equal tempered) and have rated their usefulness based
on the quality of the available harmonies and the number of divisions required
(the fewer the better). This paper may also serve as an introduction to microtonalism.

Here's a start at explaining *An algorithm
for dynamically choosing the best 12 notes from an extended meantone*.

Prompted by a challenge issued by Paul Erlich, and
inspired by Margo Schulter's meantone spectrum chart,
I produced these visualisations of the

*Harmonic errors in
single-chain-of-fifths tunings*

*Harmonic errors in
double-chain-of-fifths tunings*

That is, the errors in the intervals of those octave-based tunings whose notes
are generated by chains of uniform fifths, whether the chain be closed (equal tempered)
or open (e.g. meantone).

I also discovered this strange 9-limit temperament and the harmonic properties of this 11-note chain-of-minor-thirds scale.

First, here's a coloured 9-limit lattice for the ordinary diatonic scale so you get the idea.

Now here are some coloured lattices for Paul Erlich's decatonic scales:

Symmetrical, 7-limit

Symmetrical, 9-limit

Pentachordal,
7-limit

Pentachordal,
9-limit

9-limit chord chart
for the above.

Here are the results of my computer search for generators of 7-limit MOS scales. I have since added triple chains 1/3 octave apart to the search and found no new generators.

Here are some Excel spreadsheets with charts for predicting the dissonance
of intervals.

SetharesDissonance.zip 636kB

HarmonicComplexity.zip 155kB

A chart classifying all the tetrachords available in 72-ET (a Ms Word document).

See my note on the naming of musical intervals

and One way to name the intervals of any EDO from 5 to 72 (with 22edo as an example).

Here's a method for optimally distributing any comma.

Here's a dekany graph with 3-fold symmetry.

Here's a variation on the above dekany graph to reduce clutter.

An Excel spreadsheet for calculating the size of MOS scales (those with Myhill's property).

An Excel spreadsheet for calculating ratios near to an interval given in cents.

Keyboard mapping spreadsheet. It gives an ideal generalised-keyboard mapping for any linear temperament.

*The Noble Mediant: Complex ratios and
metastable musical intervals* coauthored with
Margo Schulter.

*The Conundrum of the Bent Tuning Forks* on combination-tone
chords.

*A Middle Path: Between Just Intonation and the
Equal Temperaments* by Paul Erlich.