Intonation Information

If you've got a PC, have a look at my software that should help you to hear this stuff.

Now, here are some pages that help with the practicalities of making music in alternative tunings:

5-limit scales
A complete list of scales that fulfil certain criteria about 5-limit harmony.
7-limit scales
Some 7-limit scales fulfilling similar criteria. Not a complete list, because there's so much variety in 7-limit harmony. Includes my "blues scales" which are well worth a listen.
Scales on a decimal lattice and decimal notation
I devised my decimal notation and lattice as a way of representing the revolutionary Miracle tuning rediscovered by Paul Erlich and Dave Keenan. This page shows some examples of existing scales adapted to this new notation that's ideally suited for 11-limit music.
Temperamental information
A page each on meantone, miracle, schismic and diaschismic temperament.
Tripod Notation
Another crazy notation system, for magic temperament or 9-limit harmony that tempers out 225:224. About 740 kiB.
7 plus 3 scales
These are really cool, and I'll try and get some MIDI examples to demonstrate how cool, but probably won't.
My schismic fourth tuning
This is a way of getting a conventional keyboard to play precise, non-meantone scales. Have a look if you always thought extended mappings are too difficult.
My modified instruments
Write-ups of a refretted guitar and a MIDI keyboard with a symmetrical layout.
Intervals in 31 note equal temperament
A table showing the ratio space interpretation of intervals now in the extended 11-limit.

And here are some pages on more abstract tuning theory:

The Regular Mapping Paradigm
A high-level account of the change in thinking some of these theories have contributed to.
Tuning lattices
Background information on triangular and tetrahetral lattices. These are useful in defined scales, and are used in a couple of the pages above. Also, an introduction to 7-limit harmony
Matrices for beginners
An introduction to my tuning matrices. These are useful for mathematical analysis of scales. Well, maybe not that useful, but great fun nonetheless. Very much out of date now.
Notation and Keyboarding
Stuff about Bosanquet keyboards and the like.
Weighted Prime Based Errors and Complexities
Some boring details about assigning numbers to temperaments. In PDF form so I can use equations. 505 kiB. After I declared this finished Kees van Prooijen got in touch to say that his proposal for what we call the Kees metric or expressibility is on his website. There are some other interesting things there so have a look.
Temperament Class Finding for Mathematicians
Another PDF condensing the problem of finding temperament classes using formulae from the previous paper. No references to music to make it as short as possible. Hopefully in a language mathematicians can understand. One side of A5 or about 40 kiB. Also, Temperament Class Finding as a Lattice Problem.
Complete Rank 2 Temperament Searches
A PDF showing a way of finding rank 2 temperaments within a given error and complexity that's provably complete. Also as a single column PDF in a bigger font that may be better to read on screen but takes up a lot more paper. Both around 180 kiB.
ET Error Cutoffs for ETs
Proving that a certain function that's always supposed to increase does always increase.
RMS-Based Error and Complexity Measures Involving Composite Intervals
About how the weighted prime errors might actually mean something. Also as a single column PDF like before. Both around 250kiB.
Exploring Parametric Badness
Mathematical details and examples of a way of measuring how bad a temperament class is. About 202kiB.
Anomalous Saturated Suspensions
Some chords not covered by Partch's o/utonalities.
Linear temperaments
A general procedure for defining linear temperaments in terms of tuning matrices.
Catalog of linear temperaments
Modest list of temperaments of note. Also a really big list.
Temperament finding program
Lists temperaments for particular limits, following ideas in the matrix links above. A more advanced program finds temperaments from unison vectors.
Equal temperaments
And some stuff about ETs and matrices.
Maximal evenness proofs
Proofs of some fairly obvious properties about maximally even scales
My MOS applet
This is an applet for doing calculations relating to MOS scales. There's not point going there if you can't run Java.
Unconstrained waffle
A supplement to my introduction to alternative tunings


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