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"Pajara Temperament", a Tuning System that Supports Standard Western Tuning ("12-tone equal temperament")

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Howdy composers! I just uploaded a video talking about "Pajara Temperament", a temperament and scale system generated by a "circle of minor seconds" (instead of a "circle of fifths") that repeats at the half-octave (instead of the octave). It's got its own unique scale structures (like the "symmetrical decatonic scale"), which can be played standard western tuning, which makes it easy to experiment with without actually going "microtonal" right away.


Please feel free to ask any questions you might have and to let me know what you think =)

Here's the script for the video:

The scales we’ve used historically in western music (the pentatonic, diatonic, and chromatic scales) can be derived from continuous chains of perfect fifths. Unbroken chains of a single interval naturally arrive at melodically consistent scales at certain lengths, called “Moment of Symmetry” scales, and it’s probably no coincidence that, when that chain is made of fifths, or, more accurately, good approximations of the frequency ratio 3:2, MOS scales occur at lengths of 5, 7, and 12. As I’ve shown in previous tuning theory videos, that derivation also allows us to systematically adjust the tuning of those scales by changing the size of all instances of that generating interval. (For example, to better approximate the frequency ratio 5:4, we can slightly flat our fifths away from pure so that after stacking four fifths we land on a significantly flatter major third.)

We can, however, stack intervals other than the fifth on top of each other and repeat them within larger “period” intervals (that may or may not be the octave) to derive other melodically consistent scales that can have their tuning changed regularly by adjusting the size of each instance of that single generating interval.

One such system is called “Pajara Temperament”. Pajara’s period is the half-octave, and its generator is an approximation of the just diatonic semitone, with the frequency ratio 16:15. Pajara may be of special note to western musicians since standard western tuning, aka 12 equal divisions of the octave, supports Pajara temperament and its MOS scales. The MOS scales that Pajara generates are very different from our familiar western scales, however.

One such a scale is the symmetrical decatonic scale (which is really just a 5-note MOS scale that repeats at the half octave). Similarly to how we can illustrate western scales as an array of fifths and octaves, we can illustrate Pajara as an array of half octaves and semitones, though our western note names lose their meaning in this context, and what’s important instead is each note’s location in the 2D array:

A#-B -C -Db-D

E -F -F#-G -Ab

A#-B -C -Db-D

The 5:4 frequency ratio (which is the bottom half of a major triad) is arrived at by going up one period and down two generators, and the 6:5 frequency ratio (which is the top half of a major triad) is arrived at by going up 3 generators.

Given those facts, we can show that this MOS scale gives us access to 2 major chords, 2 minor chords, and 1 augmented chord in each half octave. Here are those chords in ascending order in standard western tuning up two half octaves:

And here’s a short example piece using this scale tuned to standard western tuning:

This piece can be rendered in any Pajara tuning, however, and retain it’s general melodic structure. It’s harmonic purity can also be improved upon by using a generator sharper than 100 cents. At approximately 109.1 cents, we end up with 22-tone equal temperament, which sounds like this:

In between 12 and 22, a generator of approximately 105.9 cents generates 34 equal divisions of the octave, and is pretty close to optimal in terms of harmonic purity in Pajara temperament:

We can tune the generator even sharper than the one that generates 22-edo to arrive at tunings where the large and small steps of the scale approach each other in size, leading to interesting melodic characteristics, at the cost of some harmonic accuracy. For instance, a generator of approximately 115.4 cents results in 52 equal divisions of the octave, and sounds like this:

With a generator of 120 cents, the sizes of the large and small steps finally converge, resulting in 10 tone equal temperament with only a single “neutral” triad. Interestingly, after having written the piece in a more accurate Pajara tuning, I can’t help but still hear the original major and minor tonalities:

And when rendered with a more forgiving timbre, I think it has an especially nice quality to it:

Any piece of music written using this scale in western tuning can then be tuned to other Pajara tunings without losing its melodic structure, the same way any piece of music written in the pentatonic, diatonic, and chromatic scales can be tuned to other Syntonic tunings (aka Meantone tunings) without losing its melodic structure.

I recommend giving this scale a shot, seeing what you come up with, and then, if you’re feeling adventurous, rendering it in other Pajara tunings!

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