Other Inharmonic Polyrhythms

Diminished seventh as Inharmonic Polyrhythm
The diminished seventh is made by stacking minor thirds and in twelve equal tuning all the intervals are 300 cents (three equal tempered semitones).

The equal tempered minor third can be expressed using the fourth root of two. Four of them combine to make the octave.

Who is this page for?

Anyone interested in unusual rhythms or microtonal pitches - and the golden ratio, pi etc - composers, mathematicians, or just for fun.

What are they?

You could call these polytempi or polyrhythms or polymetrics. Anyway what you have is a steady pattern of beats with no whole number of beats in the measure.

The pitches played are in the same relationship as the rhythms.

None of these rhythms ever repeat exactly though some get very close to a repeat. E.g. the pi rhythm almost coincides to within much less than a millisecond after 113 "measures".

Π beats to a measure.

Π is the circumference of a disk with diameter 1. Notice how the beats nearly coincide at 22 beats and 7 "measures" - the famous close approximation to Π as 22/7.

At the next good approximation, 355/113. they will seem to exactly coincide. That's because 355/113 is correct to 7 decimal places - so there's much less than a millisecond between them

The musical interval is a major sixth roughly a tenth tone flat (781.7954 cents)

e beats to a measure

e is Euler's number - the base of natural logarithms and a number that occurs frequently in maths. (Not to be confused with Euler's constant).

The musical interval is a fourth roughly a sixth tone sharp (531.234 cents)

Square root of two.

This is famous as the first number to be proved impossible to express as a pure fraction to the bemusement of the Pythagorean school in Ancient Greece. The impossibility proof is due to Hypatus, who, legend says, was thrown overboard at sea by his fellow Pythagoreans as a result of his discovery. See incomensurable magnitudes (wikipedia).

The musical interval is the twelve equal tritone, at 600 cents, which is dissonant. If you play using the white keys of the piano it's the interval from B to F. In twelve equal tuning it is the same as the tuning of the diminised fifth, so is one of the constituent intervals of the diminished seventh.

Generally in Western music it is considered to have an uneasy, scary feel to it. It's nickname since at least the early 18th century is the diabolus in musica ("the Devil in music")

Diminished seventh type chord in seven equal

Two steps of seven equal give you a third which is just 4 cents (a fiftieth of a tone) from the 11/9 neutral third (here 11/9 is the ratio of frequencies of the two notes in the dyad).

So - just as a normal diminished seventh is got by stacking equal tempered minor thirds, you get a kind of neutral equivalent of it by stacking these neutral thirds. It has some similarities with the diminished seventh - completely symmetrical so no favoured "root" note. Though it has seven notes instead of four.

Near seven equal tunings are used in Thailand. Also found in parts of Africa.

More about seven equal here:
7edo

Play all these videos one after another

Practise Tips

It's fun to play along with one of these rhythms, just playing your music with one of the beats, while the others go on in the background of your playing. Probably a good exercise to help develop steady sense of rhythm, and independence.

Use these videos as a resource

You can use any of these videos as a resource for your own website or wikis, or make more of them yourself - see Add videos like these to your own site

Play these rhythms and animations at any tempo with Bounce Metronome

You can use Bounce Metronome Pro to practise these and many more rhythms at any tempo, including changing tempo.