Now, here are some pages that help with the practicalities of
making music in alternative tunings:
- Evolutionary Harmony with a Schismatic-Fourth Keyboard Mapping
- This is a way of getting a conventional keyboard to play
precise, non-meantone scales.
It's explained from the point of view of
mixing strange new harmony into a familiar context.
- Parapyth on a Single Keyboard
- This is a variation of the schismatic-fourth keyboard mapping
for a planar temperament with no mapping for the prime 5.
- 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,
- Tripod Notation
Another crazy notation system, for
or 9-limit harmony that tempers out 225:224.
About 740 kiB.
- Tricycle Notation
- A variation with a simpler appearance that ends up
working with miracle temperament.
- 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 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
- 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Very much out of date now.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
Both around 250kiB.
- Exploring Parametric Badness
- Mathematical details and examples of a way of measuring
how bad a temperament class is.
- 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
- 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