index

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.

A motley collection of resources relating to the Blackjack scale and the Miracle temperament in general..

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