A Complete Pitch Geometry

In previous posts, I explained how close 53ET is to being a 5-limit system, and how to create intuitive diagrams that do away with obscure lists of pitch ratios and vague explainations. Pitch geometry is a way of diagramming the actual harmonic series as viewed on a Just Fourths tuned string instrument so that things line up (or mis-align) based on the harmonic series. This gives us a practical way to view it without getting overly technical.

And best of all, it's baked directly into a real instrument that can be played. You can see simple ratios up to 13 with it. In particular, you can see everything from 2/1, 3/2, 4/3, 5/4, ... , 16/15. Which happens to be every "superparticular" ratio (meaning (N+1)/N) up to the commonly used Just Major Semitone. It's the Just Intonation version of the chromatic note. Click the picture to see the fine details!

I would like to mention this book by Cameron Powers:

http://www.amazon.com/Lost-Secrets-Perfect-Harmony-Indigenous/dp/1933983183

It is about Middle Eastern scales in particular, but it's also very much about the sort of future that touchscreen instruments are going to bring with them; Just Intonation becoming normal practice once again, assisted by the digital age.

Geo Synthesizer (aka Pythagoras, Geo Synth) is not just a fun toy for playing with virtuosity on iPhone and iPad. It was designed from the beginning to feed virtuosity in every way that it can. From allowing for very fast chromatic play, to learning the deep magic of the harmonic series (Just Intonation based microtonality), or just playing fretless; it is designed to teach you things you might not already know. This way you can learn the satisfaction of constantly improving your musicianship, rather than the short-lived satisfaction of an instrument that's merely easy to use.

Octaves, or 2-limit (2,1/2)
This is a trivial system of notes. It is everything you can create out of positive and negative powers of 2, meaning: 2^-1, 2^0, 2^1,... (ie: 1/2, 1, 2,...). It is obvious that octaves are one of the more basic equivalences that anybody can tune by ear alone.

Pythagorean Tuning, or 3-limit (3/2, 4/3 - blue lines)

Tuning a piano is not as straightforward as it seems, because the instrument is a real-world resonating body. In the real-world, the 12 tone system is a numerological fiction that roughly approximates the behavior of the harmonic series. Like a pair of drummers that play different time signatures but have the same number of beats per minute, a pair of waves that progress at a rate in simple ratios will regularly meet at the exact same point in the shortest time possible.

A simple way to tune a piano would be to start from one note that we will use as the reference note that is "in tune", and listen to the fifth overtone in the string. We then tune another string to that pitch, or an octave lower than it to stay in the same octave range. By the time we have done this to 5 strings, we have a Pythagorean Tuned pentatonic scale. If we do this for 7 strings, then we have a Pythagorean Tuned diatonic scale. It is obvious that these seven notes will not be equally spaced when we play them in order, because there will be whole tones and half tones. But the unevenness goes even deeper than this, because if we extend out to 12 strings, then we have a Pythagorean tuned chromatic scale. This scale will sound beautiful as long as you don't do anything that assumes that the 12 tones wrap around at the ends. This 12 tone scale overshoots the octave by about 1/9 of a Just whole tone.

The Blue lines that go up and down by fourths and fifths are the 3-limit. So this is every ratio that you can make with powers of primes up to the number 3. For example: 1/3, 2/3, 3/2, 9/8, etc. Because of how the 2-limit (powers of octaves!) and the 3-limit (powers of fifths, and therefore fourths as well) line up, it very nearly creates a 53 note per octave equal tempered scale. It isn't exact, but it is close enough that you can forget about the numbers not lining up at the extremities, as it's a perfect Pythagorean scale in practice. It is the first usefully close approximation to a Just Circle Of Fifths.

Just Intonation, or 5-Limit (5/4, 6/5 - green lines)
Usually, when Just Intonation is spoken of, 5-limit is what is meant. This means inclusion of perfect thirds, major and minor. It is a very fortunate coincidence that the 53 note per octave scale, happens to line up almost exactly. Almost every world-music system is using some variant of this scale. Flexible intonation instruments like Sitar, Violin, Voice, etc will invariably use this system or some subset of it.

Higher Limits, 7-limit (yellow), 11-limit (turquoise), 13-limit (purple)

These lines were recently added so that I could locate limits that get used in special contexts. Arabic Maqam is quoted as using a few different variations on what a "quartertone" is. It is typically notated as the dead center between a 12ET major third (or described as a 3/4 tone). But this is not what actually happens. Those intervals literally taken are awful for chording. The just intoned variations that can be found by ear and practice are what are used.

The scale families known as "Bayati" have quartertones in their bottom tetrachord. This would mean, D, E quarterflat, F, G. If what is meant is the note that is very close to the chromatic quartertone, then it means the ratio 11/10. This ratio, as well as 12/11 is unreachable in 53ET, which is otherwise excellent for this type of music. If what is meant is the note that fits almost exactly into 53ET, then it means 13/12.

Count The Lines

In the picture above, I can go over the various pitch ratios counter-clockwise.

• 3/2 - This is the Just fifth. It is a blue line
• 4/3 - This is a Just fourth. It is blue because it's the dual of 3/2.
  
• 5/4 - This is the Major Third. It is green. (to the left of F# on bottom row in the picture)
  
• 6/5 - This is the Minor Third. It is green.
• 7/6 - Yellow.
  
• 8/7 - Yellow.
  
• 9/8 - The Just Whole Tone. Grey. This is a scale building block
• 10/9 - The Minor Just Whole Tone. Gray. It arises often.
• 11/10 - Turquoise. A high quartertone. It falls almost exactly between two 53ET frets.
  
• 12/11 - Turquoise. The quartertone that matches a chromatic Quartertone closely. It falls almost exactly between two 53ET frets.
  
• 13/12 - Purple. A low quartertone that almost falls in the 53ET scale.
• 14/13 - Purple. A low quartertone. It falls almost exactly between two 53ET frets. Note: because of this pattern of quartertones, some people advocate splitting the way that 12ET is split to 24 to get quartertones. If you use 106ET, then you can specify these quartertones pretty exactly.
  
• 15/14 - Gray.
• 16/15 - The Just Major Semitone. A building block of scales.
• ...
• ... (they get smaller and smaller)
  
• ...
• 81/80 - It's not explicitly shown here, but it is very special. It almost coincides with 1 53ET fret, which is the distance between the vertical lines. It is the difference between two whole tones and a major third. Ie: (9/8 * 9/8) / (4/5) = 81/64 * 4/5 =81/80. The 53ET fret is also almost the amount by which 6 whole tones overshoot an octave. Many tuning systems want two whole tones to equal a major third, so tempering is done to make this so.
  

The good thing about the way it is drawn out onto the instrument, you don't have to think too hard about pitch ratios or anything like that. You just line up lines. The colors give an indication what family the pitch ratio is in. This style of playing doesn't use frets to keep you in tune, but uses instruments with a lot of resonance, so that the resonance is what keeps you in tune; which is getting help from the laws of physics rather than hard constraints.

Amazingly, it's not hard to play pretty accurately on the phone. You can certainly be more accurate than 12ET is. You can turn up the delay feedback to the top, and set the delay rate to 0.01seconds to make the audio resonate with a drone pitch (which I can't fine tune, but it's fretless, so you can adjust!). What is important is knowing exactly what you are doing with pitches, and by exactly how much you are falling short.

Read my older post on 5-limit for a deeper explanation of Pitch Geometry for 53ET and Just Intonation. It's almost identical to the user interface for the instrument, but is much more clear.

http://rrr00bb.blogspot.com/2011/08/53et-5-limit-in-geo-explained.html