Mohajira improv

This is a simple study in four-part harmony in 17-equal temperament. I predominantly used the diatonic scale, but also often used the 9-note "progression" scale which is found in 17-equal as well. I also broke away from scalar structures often, preferring to fill the space in with shifting 13-limit harmony when possible.

More intense theory:
I used the &lt;17 27 40 48 59 63] val for 17-equal, meaning I treated it as though it supported meantone temperament rather than dicot temperament. Progression temperament can be found here: http://x31eq.com/cgi-bin/rt.cgi?ets=8_9&error=34.070&limit=13&invariant=5_3_7_4_6_1_1_2_2_3_3


Etude No. 1 in 17-tone Equal Temperament

I recorded this a while ago, have no idea what it's from

Sunset

This one gets more interesting in the middle around 4:00, still don't remember what this was from

Another improv with Stevie B

This is a really bizarre tuning - pretty much deliberately bizarre - which is "the opposite of Just Intonation." The timbre here is built on the golden ratio, which is often called the "most irrational number": if the first harmonic is called "1", then the second harmonic is 1+phi, the third is 1+2*phi, etc.

Then, it turns out a very good equal temperament that approximates this is 12 equal divisions of the "pseudo octave" of 1+phi ≈ 2.618/1, so I just loaded this up into a sampler, and played around for a minute on a normal keyboard.

I'm still not really sure what to make of the result -- my initial playing around with it gave some really strange results, but it's also kind of cool! I think it has potential, anyway.

Antiharmonics - 12 Equal Divisions of 2.618

Here's another little demo playing around with the "anti-harmonic" idea, but instead of using an equal temperament, I just went with "antiharmonics 12-24." This can be thought of as an extension of Zhea Erose's ideas on "primodality" to these antiharmonics.

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This is a really bizarre tuning - pretty much deliberately bizarre - which is "the opposite of Just Intonation." The timbre here is built on the golden ratio, which is often called the "most irrational number": if the first harmonic is called "1", then the second harmonic is 1+phi, the third is 1+2*phi, etc.

This tuning is derived from harmonics 12-24 from this antiharmonic series.

Antiharmonics - Primodal - 12 - 24

Here's another little demo playing around with the "anti-harmonic" idea, but instead of using an equal temperament, I took the "pseudo-octave" of 1+phi and linearly divided it into 12 linear divisions in Hz. This is similar to the "primodal" approach from Zhea Erose but technically these aren't actually harmonics in the series.

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This is a really bizarre tuning - pretty much deliberately bizarre - which is "the opposite of Just Intonation." The timbre here is built on the golden ratio, which is often called the "most irrational number": if the first harmonic is called "1", then the second harmonic is 1+phi, the third is 1+2*phi, etc.

This tuning is derived from harmonics 12-24 from this antiharmonic series.

Antiharmonics - Arithmetic - 12-AD-(1+phi)

This is a tuning-corrected version of my performance of the "House of the Rising Sun," which I did for Lumatone. The video of that performance is here: https://www.youtube.com/watch?v=IlZv13YZzSM

That rendition of the piece is *almost* in 31-EDO, except for one note. Due to a quirk of the way I'd set up Kontakt, the G half flat is about 12 cents flat, so that during the 4:6:7:8:9 chord, the 7/4 over the A is closer to 955 cents rather than the 967 cents it would be in 31-EDO - so closer to the 7/4 of 24-EDO than 31-EDO. This version corrects that one note so the entire thing is in 31-EDO, making that chord a little sweeter than it was. Other than that, the entire performance is the same.

House of the Rising Sun (Pure 31 Version)

This is a chord progression in Hanson temperament, tuned to 34-equal. In this system, rather than 4 minor thirds landing you at the octave, 6 minor thirds lands you at an octave + a fifth. The minor thirds in this system are even closer to the harmonic series than the familiar ones in 12-equal, which contributes to the "bright" feel of this tuning. Don't think too much about the name "Hanson."

Comma pump in Hanson/Kleismic temperament, 34-equal

This is a chord progression in Blackwood temperament, tuned to 15-equal. In this system, the circle of fifths is truncated at 5 steps, meaning that for C-G-D-A-E-B all of the fifths are a little bit sharp so that the B on top ends up being another C again. This means, by definition, that the E on top is more like an F.

Since this chain by itself doesn't really give us a usable major or minor third, we have to add them manually and in a way that's not accessible by the base circle of fifths that proceeds from the root. This chord progression exploits this property.

Comma pump in Blackwood temperament, 15-equal

These musical examples are tuned to what is known as "mavila" temperament. It is named after the Mavila village of the Chopi people in southern Mozambique, where this tuning system was first discovered by Kraig Grady.

By virtue of the structure of this unique tuning, every existing piece of common practice music has a "shadow" equivalent rendition in mavila. That is, when Bach wrote Fur Elise, he actually wrote two songs - the one that you know, and the anti-diatonic equivalent in mavila temperament. It's only that the anti-diatonic versions have never been heard before.

Mavila is a tuning system whereby four stacked perfect fifths, rather than getting you to a major third, gets you to a minor third - meaning that the fifths are flat. Conversely, four stacked perfect fourths gets you to a major third, rather than a minor third. This has some very strange implications for music.

The mavila diatonic scale is similar to the normal diatonic scale - except interval classes are flipped. Wherever there was a major third, you'll find a minor third, and vice versa. Half steps become whole steps and whole steps become half steps (closer to neutral second range, however). When you sharpen the leading tone in minor, you end up sharpening it down instead, meaning you flatten it. Also, minor is now major - you end up with three parallel natural/harmonic/melodic major scales, and only one minor scale. Instead of a diminished triad in the major scale, there is now an augmented triad.

In these examples, the chosen mavila temperament is 23-equal, which was chosen to provide greater accuracy in approximating the harmonic series. 16-equal is another good choice, with 9-equal also being good enough for government work.

The Mavila Experiments - Mozart - Rondo Alla Turca - 23-EDO

The Mavila Experiments - Beethoven - Fur Elise - 23-EDO

The Mavila Experiments - Bach - Invention #1 - 23-EDO

==16-EDO VERSION==

These musical examples are tuned to what is known as "mavila" temperament. It is named after the Mavila village of the Chopi people in southern Mozambique, where this tuning system was first discovered by Kraig Grady.

Because of the structure of this unique tuning, it is true that every existing piece of common practice music has a "shadow" version in mavila temperament. That is, when Beethoven wrote Fur Elise, he actually wrote two compositions - the one that you know, and the anti-diatonic equivalent in mavila temperament. It's only that the anti-diatonic versions have never been heard before.

Mavila is a tuning system whereby four stacked perfect fifths, rather than getting you to a major third, gets you to a minor third - meaning that the fifths are flat. Conversely, four stacked perfect fourths gets you to a major third, rather than a minor third. This has some very strange implications for music. The mavila diatonic scale is similar to the normal diatonic scale - except interval classes are flipped. Wherever there was a major third, you'll find a minor third, and vice versa. Half steps become whole steps and whole steps become half steps (closer to neutral second range, however). When you sharpen the leading tone in minor, you end up sharpening it down instead, meaning you flatten it. Also, minor is now major - you end up with three parallel natural/harmonic/melodic major scales, and only one minor scale. Instead of a diminished triad in the major scale, there is now an augmented triad.

In these examples, the chosen mavila temperament is 16-equal, which was chosen because it strikes a good middle ground between approximating the harmonic series and making the interval categories as distinct as possible. In contrast, 23-equal improves the fit of this temperament to the harmonic series, whereas 25-equal makes the intervals more distinct from one another.

A special thanks goes out to Graham Breed; his custom Lilypond code and several weeks worth of gchat discussion are what made all of this possible.


The Mavila Experiments - Bach - Invention #8 - 16-EDO

The Mavila Experiments - Bach - Invention #1 - 16-EDO

The Mavila Experiments - Beethoven - Fur Elise - 16-EDO

The Mavila Experiments - Mozart - Rondo Alla Turca - 16-EDO

The Mavila Experiments - Beethoven - Moonlight Sonata - 16-EDO

The Mavila Experiments - Bach - Invention #13 - 16-EDO

==CRAZY 9-EDO VERSION - FOR USE ONLY IF YOU'RE TRAPPED ON A DESERT ISLAND WITH ONLY ONE INSTRUMENT IN 9-EDO==

These musical examples are tuned to what is known as "mavila" temperament. It is named after the Mavila village of the Chopi people in southern Mozambique, where this tuning system was first discovered by Kraig Grady.

Because of the structure of this unique tuning, it is true that every existing piece of common practice music has a "shadow" version in mavila temperament. That is, when Beethoven wrote Fur Elise, he actually wrote two compositions - the one that you know, and the anti-diatonic equivalent in mavila temperament. It's only that the anti-diatonic versions have never been heard before.

Mavila is a tuning system whereby four stacked perfect fifths, rather than getting you to a major third, gets you to a minor third - meaning that the fifths are flat. Conversely, four stacked perfect fourths gets you to a major third, rather than a minor third. This has some very strange implications for music. The mavila diatonic scale is similar to the normal diatonic scale - except interval classes are flipped. Wherever there was a major third, you'll find a minor third, and vice versa. Half steps become whole steps and whole steps become half steps (closer to neutral second range, however). When you sharpen the leading tone in minor, you end up sharpening it down instead, meaning you flatten it. Also, minor is now major - you end up with three parallel natural/harmonic/melodic major scales, and only one minor scale. Instead of a diminished triad in the major scale, there is now an augmented triad.

In these examples, the chosen mavila temperament is 9-equal, which was chosen because there are only 9 notes to the octave. It's an odd tuning, in that although it has comparatively awful harmonic properties, and its melodic properties are a mixed bag, it does "work" in a sense: it's a rare novelty that allows you to set up tonality with only 9 notes to the octave. If you plan on getting stuck on a desert island sometime soon, and you only have 9 pieces of wood with which to construct a 9-tone wooden xylophone, it would be worthwhile to know how to use this tuning.

A special thanks goes out to Graham Breed; his custom Lilypond code and several weeks worth of gchat discussion are what made all of this possible.

The Mavila Experiments - Bach - Invention #1 - 9-EDO

The Mavila Experiments - Bach - Invention #8 - 9-EDO

In this set of listening examples, common practice works have been retuned to exotic tunings in which the relative structure of the diatonic scale is always preserved, so that it always takes the form of LLsLLLs (for "Large step" and "small step," respectively). The perfect fifth, which generates the scale, is allowed to vary from as flat as 7-EDO, where the diatonic scale is a perfectly evened-out "neutral" diatonic scale, to as sharp as 5-EDO, where the half step becomes so small that it vanishes, leaving a perfectly evened-out pentatonic scale.

The chosen spectrum of tunings is such that diatonic interval categories such as the "major third" may, at times, approximate different intervals from the harmonic series than they normally would in 12 equal temperament, just as long as the LLsLLLs scale pattern is consistently present in the tuning. These experiments aim to explore the properties of the "categorical perception" effect, and to see to what extent it exists and interrelates with the missing fundamental phenomenon and other psychoacoustic effects.

The "artwork" for this compilation describes the tunings used, and the properties of each one; click "Download Artwork" to view it.

In this particular set, the chosen composition is Bach's Fugue in C Major (BWV952).

Bach - Fugue in C Major (BWV952) - 12-EDO

Bach - Fugue in C Major (BWV952) - 31-EDO

Bach - Fugue in C Major (BWV952) - 19-EDO

Bach - Fugue in C Major (BWV952) - 26-EDO

Bach - Fugue in C Major (BWV952) - 75-EDO (Generator 43\75)

Bach - Fugue in C Major (BWV952) - 7-EDO

Bach - Fugue in C Major (BWV952) - 17-EDO

Bach - Fugue in C Major (BWV952) - 22-EDO

Bach - Fugue in C Major (BWV952) - 27-EDO

Bach - Fugue in C Major (BWV952) - 37-EDO

Bach - Fugue in C Major (BWV952) - 72-EDO (Generator 43\72)

Bach - Fugue in C Major (BWV952) - 5-EDO

Orgone Temperament, discovered by Andrew Heathwaite, has a number of interesting properties: it is a relatively accurate temperament, low in complexity, and most decent tunings of its 7-note MOS are generally low in categorical entropy. Additionally, its 7-note MOS has a step pattern of LLsLsLs, which is readily picked up by the ear as resembling the "half steps" and "whole steps" of the diatonic scale.

This can be regarded either as a bug or a feature - either as an unwanted effect of our "western conditioning," or as an entirely new system with features that resemble, but are not identical to, the diatonic scale. This simple one minute study explores the latter perspective and its implications on "tonality." The basic "major" triad is set to 7:9:11 and the "minor" triad set to its inverse.

The chosen tuning for this piece is 11-EDO.

Tonal Study In Orgone Temperament, 11-EDO

This is a comma pump in porcupine which goes up by three 6/5's and then down two 4/3's back to 1/1. I designed it as an example of something that I thought sounded perfectly "natural," but which doesn't work at all in 12-EDO. I'll use this in a piece somewhere down the road.

The tuning here is 22-EDO.

A Porcupine Comma Pump - 22-EDO version

This is a comma pump in porcupine which goes up by three 6/5's and then down two 4/3's back to 1/1. I designed it as an example of something that I thought sounded perfectly "natural," but which doesn't work at all in 12-EDO. I'll use this in a piece somewhere down the road.

The tuning here is 59-EDO.

A Porcupine Comma Pump - 59-EDO version

This is a comma pump in porcupine which goes up by three 6/5's and then down two 4/3's back to 1/1. I designed it as an example of something that I thought sounded perfectly "natural," but which doesn't work at all in 12-EDO. I'll use this in a piece somewhere down the road.

The tuning here is 23-EDO, which does a much poorer job in approximating the harmonic series when compared to the other tunings listed here, so the chords don't seem to blend as well. That notwithstanding, it still has a certain interesting color to it, which I somewhat enjoy.

A Porcupine Comma Pump - 23-EDO version

This is mostly goofy experimental music theory stuff to see how different microtonal scales sound - no real musical compositions here (yet)

Mike Battaglia

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