270edo

A tuning system is a set of pitches that enable the making of music. A tuning system can be used in designing a musical instrument, or as the foundation of a system of notation, or as a theoretical construct for composition and analysis. There are countless possible tuning systems, and even the actual systems in use are very many. One simple class of tuning systems consists of those that divide octaves into some number of equal steps. The conventional tuning system that has spread from Europe around the world, 12edo, divides octaves into twelve equal steps. This system is so dominant, governing the design of instruments, tuners, digital audio workstations, etc., that exploring alternatives is not so easy! But there is a vast world available to explore, with just a bit of effort.

What makes a tuning system good? The availability of consonant intervals is a major criterion. What should count as a consonant interval, that's a challenging issue. A classical starting point is that a consonant interval is one that is very close to a simple frequency ratio. How close? How simple? But at least we are getting close to a concrete idea.

The chart above shows, for a quite wide spectrum of simple ratios, how closely they can be approximated by the available intervals of 12edo. The numbers on the top and left margins define the numerator and denominator of a simple ratio. The green square containing 0.02 in the upper left shows the error when 12edo is approximating the simple ratio 3:1. That ratio would be 19.02 steps of 12edo, but of course the closest actual interval in the tuning system is 19 steps, so the error is 0.02. The error for 5:1 is 0.14, which is not so terribly accurate. I have used bright green to flag the very accurate intervals of 12edo and pale green to flag the moderately accurate intervals. The squares in grey are redundant, because the numerator and denominator are not mutually prime.

The simplest intervals are the ones toward the upper left. The more and brighter green cells that a tuning has, the better the tuning, especially when the green cells are toward the upper left.

One of the simplest alternative tunings is 24edo, where each (half) step of 12edo is divided in half, to form quarter tones. These charts measure errors in terms of the size of the step of the tuning. Since the step size of 24edo is half the size of the steps of 12edo, many of the errors double in size. But some are much smaller. The error for 11:1 is only 0.03. 11:1 falls just about half way between steps of 12edo, so when a new note is added in the middle, it comes very close to 11:1. But overall, comparing 24edo to 12edo, it is not clear which is the winner. 24edo adds a lot of notes for not so much gain.

270edo divides octaves into 270 equal steps instead of the conventional 12, so obviously it is adding an awful lot of extra notes! But its chart shows, it really hits a lot of simple ratios quite accurately. Of course its steps are so small that it can't be too far off! But these errors are in terms of the tiny steps of 270edo. For example, 3:1 is 427.94 steps of 270edo. The available interval of 428 steps is just 0.06 of a step sharp.

Here are three snapshots of the evolution of a composition from my thermodynamic composition system:

• temp=865.154772702373; cost=17487362.4275858;
• temp=447.765805311866; cost=8883781.06482988;
• temp=10; cost=3690000; (oops, this is from memory... I didn't record the exact numbers!)

Even at the final stage, the piece has not settled into monotony:

For this piece I counted as consonant a wide range of simple intervals, such as 14:13, 13:12, 12:11, etc. These ratios can be combined to form commas like 676:675, 1001:1000, and 2401:2400 that 270edo tempers out. Traversals of these commas can get stuck, as topological defects, in the piece. Exactly what pattern of such traversals remains in this final stage... that would be a bit complicated to figure out!

Emergent Order

I use a thermodynamic simulation in my algorithmic composition software. Randomized decisions are at the heart of this simulation. A piece of music might have ten thousand note events, a voice and a point in time at which a pitch is to be sounded. The pitch to be sounded at each note event is randomly chosen. Each possible pitch that could be played, selected from the options provided by the tuning system, is assigned a probability. A pitch that fits well into the musical context will be assigned a high probability; a pitch that would sound very out of place is assigned a low probability. The musical context is defined by the pitches that have been chosen at related note events:

• vertical relationships: pitches sounding at the same time in other voices
• horizontal relationships: pitches immediately before or after in the same voices
• thematic relationships: pitches sounding at more distant times that are in the same corresponding place in a different expression of a musical phrase or motif or theme.

The choice of the pitch to sound at one note event depends on the pitches chosen to sound at other note events. There is a problem of circularity here! How can the first choices be made? Won't the result depend heavily on the order in which choices are made?

I use a variety of technniques for the initial choices, but the main way around the problem of circularity is that choices are made again and again. After initial pitches have been chosen for all ten thousand note events, those choices are revisited. Some single note event is randomly selected from among the ten thousand. The probabilities are computed for this note event, based on the pitches currently assigned to related note events. A fresh random pitch choice is made, using these probabilities, and the chosen pitch is assigned to this note event. This process is repeated again and again, many millions of times. So the pitch to be assigned to a single note event will be chosen again and again, thousands of times. Between each choice, though, the pitches assigned to the related note events will also have changed, so the probabilities will be different each time the pitch is chosen.

The calculation of the probabilities is based on a cost function. A low cost is assigned to pitches that fit well with pitches at related note events. A system temperature parameter is used in deriving probabilities from the costs. When possible pitches have cost differences that are small relative to the temperature, they will be assigned similar probabilities. When cost differences are large relative to the temperature, then the probabilities will be very different.

The overall process of pitch selection usually involves starting the system at a high temperature, assigning and reassigning pitches to note events again and again, then slowly lowering the temperature, again reassigning pitches many times at each temperature. The pitch choice made at one note event will affect the choices to be made at related note events, and then those choices will affect yet other choices, and this propagation of choices will let the whole system organize itself.

A total cost for the system can be computed, as simply the sum of the costs for all the note events in the system. At high temperatures, high cost pitches have a higher probability, so the total system cost will be high. As the temperature is lowered, the total system cost goes down. A curious feature of thermodynamic systems like this is that the decrease in cost with temperature is often not smooth. Phase transitions occur, where long range order arises and the system cost suddenly decreases. The graph above, with temperature on the horizontal axis and cost on the vertical axis, shows a sudden drop in cost around temperature 230.

A tonal center would be a typical kind of long range order in a musical system. Looking at a particular note event, if the pitches at the related note events are quite unrelated harmonically, then there will be no strong bias in the probabilities for assigning a new pitch at this event. But once the pitches at the related note events are all harmonically close to some tonal center, then there will be a strong bias to assign a harmonically related pitch at this event, too.

The slow decrease in system temperature allows long range order to emerge spontaneously. I took eighteen snapshots of the evolution of the system in a run of this software. The first snapshots are at a very high temperature, so the pitches are quite disordered. The last snapshots are at a very low temperature, after long range order has emerged and established itself. At these extremes of very little order or very strong order, the pieces are rather boring. The most musically interesting pieces are in the middle, at the boundary between order and disorder.

1. temp=4983.67001252014; cost=10544386.267572;
2. temp=3178.35877439225; cost=8365076.82243706;
3. temp=1768.29608835035; cost=6691731.54375484;
4. temp=907.870229277442; cost=5341434.10767405;
5. temp=437.105167762909; cost=4254305.00978789;
6. temp=253.151000149135; cost=3362954.84572504;
7. temp=226.226281845489; cost=2575694.51859118;
8. temp=213.857636486513; cost=1992138.33672853;
9. temp=194.206977218147; cost=1593476.40311909;
10. temp=168.06408964933; cost=1268744.01912949;
11. temp=144.276865461673; cost=1010195.00479415;
12. temp=119.940307483187; cost=804750.28245681;
13. temp=103.794718800345; cost=639506.147892938;
14. temp=88.3911327966952; cost=494414.291970105;
15. temp=77.7312146561222; cost=382966.425501626;
16. temp=69.4638522645718; cost=300979.198963986;
17. temp=60.1130776123832; cost=239863.346055088
18. temp=48.7834419649326; cost=190764.398975199

Even the final piece here has not settled into utter monotony. When a tuning system tempers out simple commas, the system can get stuck in some pattern of comma traversal. That seems to have happened here. The tuning system I used here is 34edo. Studying the scores a bit, I think a traversal of the diaschisma is what got caught.

Well Temperament

Temperament is tuning with compromises. There are many criteria, that can't all be satisfied perfectly:

• including many consontant intervals;
• not including too many notes per octave;
• symmetry, where the intervals available from one note are also available from other notes;
• simple paths from a note don't lead to a note that is slightly off the starting point, giving rise to drifting or sounding out of tune.

Well temperament is a class of tunings that allow all twelve key signatures without any sounding too far off. A few years ago I suggested a 12 note subset of 34edo that tempers out the diaschisma rather than the syntonic comma, so it is not too conventional. Lately I have been looking at just tuned diatonic scales. The diaschismic tuning avoids perfect fifths that are too far off, so I had the idea that it could provide a form of well temperament, supporting all twelve key signatures with reasonable diatonic scale shapes.

C Major

An algorithmic example.

G Major

An algorithmic example.

D Major

An algorithmic example.

A Major

An algorithmic example.

E Major

An algorithmic example.

B Major

An algorithmic example.

F# Major

The shapes of the diatonic scales now start to repeat, so their characters will be the same as those already given.

C# Major

Ab Major

Eb Major

Bb Major

F Major

Hanson[19]

A couple days ago I posted a piece in a 19 note scale, using 55edo. This was a classic meantone scale, built from a chain of perfect fifths with the syntonic comma tempered out.

Here is a new piece in a different 19 note scale, using 53edo. This is a Hanson scale, built from a chain of minor thirds with the kleisma tempered out.

I thought it was a nice coincidence that both meantone scales and Hanson scales work with 19 notes per octave: one can shift the scale along the chain of generating intervals by just sharpening or flattening one note of the scale. Initially I just tweak my code to switch the tuning system and scale, but what the software generated was mostly just a single note repeated again and again. Exactly what combinations of parameters will result in anything reasonable musical... I can usually guess the general ballpark, but generally I have to sift through at least a few trials to find something that sounds plausible. So the parameters here have wandered quite a bit from last Saturday's!

1/6-Comma Meantone

Here is a new piece in 55edo. This piece uses a 19 note subset of the 55 notes per octave of 55edo. This subset is generated from a chain of perfect fifths, which is the way conventional scales are generated. So the scale can be mapped to conventional note names:

A perfect fifth is 32 microsteps of 55edo, which is 698.18 cents. A just tuned perfect fifth is 701.96 cents, so the perfect fifth of 55edo is about 3.77 cents flat. A syntonic comma, the difference between a just tuned major third 5:4 and a pythagorean major third 81:64, is about 21.5 cents. Since 3.77 cents is quite close to 1/6 of 21.5, 55edo is quite close to 1/6-comma meantone. This is a tuning that would have been familiar to musicians in the 18th Century.

Hanson[34]

Scales generated by a minor third support traversing the kleisma, a comma tempered out by 53edo. Larry Hanson built keyboards that exploited this fact, so the scales are called Hanson scales. Hanson[19] is a common size such scale. Starting from pitch class 0, a sequence of eighteen minor thirds generates a scale with 19 notes per octave. The last pitch class in the series is 40. The whole sequence can be shifted by a minor third, by ommitting the starting pitch class, 0, and adding one more pitch class at the end, pitch class 1. Thus the shift is accomplished by sharpening 0 by a single step of 53edo.

With scale sizes of 7, 11, 15, and 34, shifting the sequence ahead a minor third is accomplished by flattening the 0 pitch class. For a scale size of 7, the 0 pitch class would be flattened to pitch class 45. For a scale size 11, 0 is flattened to 48; for size 15, to 51; for 34, to 52. Hanson[34] is largest such scale, a scale that can be shifted by sharpening or flattening a pitch class to an adjacent pitch class.

Here is an algorithmic piece in Hanson[34].

Yet More Ways to Tune a Piano

I've been exploring the possibilities for just intonation for the twelve notes per octave on a piano. One can pick a subset of the intervals to tune exactly, and then the other intervals will be have even worse errors than with conventionl equal temperament. I counted 41,844 ways to choose which intervals to tune exactly!

But much of the fun and fascination of tuning has to do with temperament. By relaxing the requirement for intervals to be tuned to exact simple frequency ratios, one can increase the number of intervals that sound acceptably. Of course equal temperament pushes this to the limit, but at the cost of thirds sounding rather rough. There are many other possible choices.

Another way to expand the range of choices is to tune unconventional intervals between the piano keys, or to let go entirely of the seven white and five black keys of the piano, to change the layout of the keyboard. Historically, with meantone tuning, keyboards often enough had split black keys, e.g. seperate black keys for G# and Ab.

Still, it is a nice exercise to stick with the twelve piano keys and their conventional intervals, to start with a choice from among the 41,844 just intonation possibilities, and then to introduce temperament to add a few more acceptably tuned intervals.

This is a tonnetz diagram showing the Pythagorean tuning of the twelve piano keys. The piano is tuned to a chain of perfect fifths. There are no just tuned major thirds available in this tuning.

When this tuning is mapped to 53edo, the tuning system that divides the octave into 53 equal steps, the chain of perfect fifths gets flattened slightly, which brings some of the major thirds into a good approximation. These new relationships change the topology of the tuning: instead of a line segment, the tuning has been wrapped into a circular shape, a loop. The schisma is one of the commas that is tempered out by 53edo. This scale supports traversal of the schisma.

Here is an example of this tuning. This piece is built from a traversal of the schisma, looping around 64 times. A scale built from a chain of perfect fifths, and tempering out the schisma: this is a quite conventional way to tune. This piece does not sound too terribly exotic, at least to my ears!

Here is another of the 41,844 just tunings of a piano. This is built from four chains of major thirds. There are not many perfect fifths in this tuning! It's a more more exotic tuning.

When this tuning is mapped to 53edo, an additional major third is added, which forms a loop that traverses the semicomma.

Here is an example of this tuning, built from 64 traversals of the semicomma.

This exotic tuning does not have a very neat structure: for example, there are three sizes of steps between the notes. By adding one more note per octave, and shifting a couple of the other notes, a more neatly structured tuning can be created:

This tuning does not fit well on a piano keyboard. It's not just that there are thirteen notes per octave. This tuning is built from chains of major thirds of length four and five. Conventional tuning does not allow such chains!

Here is an example of this unconventional and exotic tuning, again built from 64 traversals of the semicomma.

Cyclic Paths in Tuning

This diagram shows relationships between the sixteen ways to tune a diatonic scale using just intonation. Each arrow in the diagram represents moving from one tuning to another by shifting a single note by a syntonic comma. The arrows point in the direction of raising the pitch of the note. The diagram has a loop: once all seven notes have been raised by a syntonic comma, one has returned to the same tuning structure that one started with, just a tad higher.

I've made diagrams for each of the sixteen tunings, showing the just tuned perfect fifths, major thirds, and minor thirds. In this first tuning, for example, there is no arrow from G to D. In conventional equal tempered tuning, every interval of seven half steps is the same. In just intonation, not all similar intervals can be tuned the same. In this first tuning, the G-D interval is tuned to a 40:27 frequency ratio, and will sound rather harsh.

Here is an example of tuning 1. I used 87edo to create these examples, rather than just intonation, because my algorithmic composition software works mainly with edo. This software uses weighted random choices to decide what pitches to play. The weights are computed based on the consonance or dissonance of intervals between related notes. So with tuning 1 for example, the program will not very often put a G near a D. It will much more often put A and D near each other.

Here is an example of tuning 2.

Here is an example of tuning 3.

Here is an example of tuning 4.

Here is an example of tuning 5.

Here is an example of tuning 6.

Here is an example of tuning 7.

Here is an example of tuning 8.

Here is an example of tuning 9.

Here is an example of tuning 10.

Here is an example of tuning 11.

Here is an example of tuning 12.

Here is an example of tuning 13.

Here is an example of tuning 14.

Here is an example of tuning 15.

Here is an example of tuning 16.

Diatonic Scale in Just Intonation

I am continuing to explore conventional scales, like the 12 notes of a piano or the 7 white notes, tuned with just intonation.

This interval graph is a simple way to tune a piano - just a little bit out of the ordinary. Here is an algorithmic example using this tuning.

Just toying with possibilities, I came up with a tweaked version:

Here is an algorithmic example in this tuning. This network still has a diatonic scale as a connected subgraph, but this subgraph does not appear as any of the seven tuning modes I listed a few days ago. This got me wondering: how many ways are there to just tune a diatonic scale?

It was a pretty simple tweak to the code I wrote that counted 41844 ways to tune all twelve notes of the piano with just intonation. Looking just at the seven white notes, and requiring these seven notes to be all interconnected by simple just ratios - there are 16 ways to tune a diatonic scale! Here is a list.

Non-Diatonic

I hadn't realized how many ways the 12 conventional notes could be tuned in just intonation!There are so many possibilities with temperaments, with scale sizes... but even with this very restricted approach, there is a lot of room for exploration!

Many of the tunings will be oddly shaped with few options for harmonic movement. Many will be based on conventional diatonic tuning, with the usual seven note pattern connected by close harmonic relationships. The above tuning network does not fit the diatonic pattern. The core of the pattern consists of the two short chains of perfect fifths, C#, Ab, Eb, and E, B, F#. Either A or Bb could be added to make a diatonic scale, but both A and Bb are not directly related.

Here is an algorithmic example of this non-diatonic tuning.

This piece was created in 53edo, which is quite close to just intonation. The table above shows just tuning and also the 53edo approximation for this interval network.

It's simple enough to move A and Bb in the network so that diatonic scales are supported. Here E major and Ab minor will be tuned properly:

Here's a piece in this more conventional tuning.

41844 Ways to Tune a Piano

Here's the big list I generated! This gives twelves pitches per octave, as fractions and as cents values. The twelve pitches in the octave all have to be related to each other by just intervals. In each tuning, there is a tree of simple intervals that relates the twelve pitch classes.

One could certainly extend the notion of simple intervals, e.g. to include ratios like 8:7. How exactly these intervals should appear on the piano keyboard, I don't know. With this list of 41844 tunings, the intervals appear on the keyboard in their conventional way: minor thirds are three half-steps, etc.

Just Intonation

Here is a puzzle: how many ways are there to tune a piano using just intonation? The answer will of course depend on the exact rules.

• All octaves are perfect: C5 is twice the frequency of C4, etc.
• A4 is fixed to 440 Hz.
• Each note must be tuned to at least one other note by a just interval, one of
  • an octave 2:1
  • a perfect fifth 3:2
  • a perfect fourth 4:3
  • a major third 5:4
  • a minor third 6:5
  • a major sixth 5:3
  • a minor sixth 8:5
• these interval relationships must correspond to convention. E.g. if E is linked to C by an just interval, that interval must be a major third.
• there must be a path of these just intervals connecting any two notes

I think these rules are enough to define the puzzle.

The diagram above provides a hint that the number of ways to tune a piano with just intonation is likely quite large. Writing a bit of software to enumerate the possibilities shouldn't be too difficult...

Interval Networks

I am continuing to explore the tuning modes I posted about yesterday. The table of fractions I posted then is compact, but difficult to interpret in terms of simple relationships. So I have made graph or network diagrams for a few of the modes.

Above is a diagram for mode 1. As with the diagrams in the post from a couple days ago for diatonic modes, the green arrows represent perfect fifths, the blue arrows are major thirds, and the red arrows are minor thirds.

This is a diagram for mode 7.

This is a diagram for mode 5. If you remove all the black keys, you can see that it corresponds to the 5th diatonic mode that I posted two days ago. There are seven diatonic modes and twelve dodecatonic modes, so they don't line up exactly. But each diatonic mode will appear as a sort of spine inside at least one of the dodecatonic modes, and their orderings are consistent with each other.

There are many ways to use just intonation to tune the twelve notes on a piano, and to step along a path through these options. I chose this particular approach because it is consistent with the diatonic modes I described.

Yesterday I posted algorithmic examples for modes 1 and 7. Here is an example for mode 5. I've been tweaking my code to work better with these examples. Usually I am experimenting with temperament, working to traverse commas that a tuning tempers out. Temperament creates a non-trivial topology for the interval networks. This topology combines with the non-trivial topology of the rhythmic structure to create knots, so the order that emerges from the thermodynamic simulation doesn't collapse into triviality. These just tuned interval graphs have a trivial topology, i.e. there are no cycles, which means there are no knots that prevent collapse. My approach with these tunings is mostly just to keep the temperature higher. With this piece, I gradually lowered the temperature, watching for some pitch class to start to dominate. So this piece is more about order just starting to emerge, which will happen before the phase transition, i.e. at a higher temperature than most of what I post. Anyway, it still sounds fun enough for me!