Pythagorean Tuning

: The Harmonic Overtone Series

There are charming historic myths about Pythagoras discovering principles of music theory from observations of metal anvils and single string harps, but this isn't a book of music history.

Pythagorean tuning, regardless of how and when it was discovered, is a simple system for defining music. Our starting observation is that resonant objects which produce clear tones - from flutes to harps to metal anvils struck with hammers - all of these have have "harmonics": faint higher frequencies on top of the low and clearly identifiable one, which together give the instrument a characteristic sound other than that of a pure sine wave. The low and most obvious frequency is called a "fundamental" or "fundamental frequency" and is usually also called a harmonic for mathematical convenience. 

The harmonics of an ideal harmonic resonant instrument occur at one times, two times, three times, ... the fundamental frequency, but they occur with different magnitudes. Real instruments have some inharmonicity, and the upper harmonics will go off their integer placement or just devolve into a wash of high frequency spectral energy - no peaks.

People can hear the lower and stronger harmonics with a little bit of training, or you can dampen lower vibratory modes of some instruments to make the upper ones more apparent. You can lightly touch a string at different points along the neck of string instrument to dampen some vibratory modes and allow other ones shine. The second harmonic (or octave) of a the string happens to occur halfway down its length, and can be sounded either by fretting or by light-touch-dampening. Every harmonic has its own places you have to touch to elicit it. For many people, "harmonics" means the noises you can elicit from a guitar by plucking a string in one place while touching the string lightly in another. But harmonics are so much more than that.

I think harmonics are even more obivous on brass instrumnets than on strings. On a Bb trombone, with the slide all the way in, you can play a sequence of notes just by changing the pose of your lip muscles and the flow of your breath. The sequence goes

[Bb1, Bb2, F3, Bb3, D4, F4, Ab4, Bb4, C5, D5, E5, F5, G5, Ab5, A5, Bb5, B5, ...]

with a few of these being a little too flat or a little too sharp. These are exactly the harmonics of the instrument! They occur at 1x, 2x, 3x, 4x, ... the fundamental frequency. Sounding the harmonics isn't some different and seldom used technique that requires you to find special magic spots to dampen on a string - it's the method by which all brass players make noise.

I've shown pitch names for harmonics 1 to 17 above, but a novice trombone player might only be able to play harmonics 2 through 8. If you don't see what those letters have to do with those 1x, 2x, 3x, .... scalar multiples, that's no trouble - we'll be working with this sequence a lot in future chapters.

The second harmonic we hear, i.e. the one directly above the fundamental frequency, is called the octave (or perfect eighth or P8) and has twice the frequency of the fundamental. The third harmonic has three times the base frequency and so on. The third harmonic is conventionally called "the perfect 12th". We'll learn what interval names go with what frequency ratios in good time.

Pythagorean tuning uses only the second and third harmonics to generate an infinite family of sounds for mathematically precise music making. There were many other sounds used in ancient Greek music - they had geometric constructions defined on the neck of string instrument to give all sorts of complicated frequency ratios - but Pythagorean tuning was the first regular tuning system where every frequency had an associated pitch name, and all the names and notes were simply and consistently related.

: Tuning With Harmonics

Let's dive in! Here's how to find infinitely many notes using the first and second harmonics of a reference note, in the Pythagorean way.

You have one sound, maybe a plucked string in tension, and you can find its second and third harmonics. Or you have a trombone and you can play Bb2 and F3. Great! We're on our way. We tune another string by changing its tension until it matches the third harmonic of the first string. New note! Great! Now find the third harmonic of this second note. Now find the third harmonic of that note. And so on. These notes get really high really fast. At every step we're multiplying by 3, so it goes [1, 3, 9, 27, 81, ...] times the fundamental reference frequency. This is where the second harmonic comes in! Our next operation is to tune a new string by tensioning so that its 2nd harmonic (i.e. its octave) matches a note we've already found. If you have a note 3x the fundamental frequency, then this gives you a note that is 3/2 times the fundamental. By the same trick, you can find frequencies that are 1/3 of another one - tune a string until its third hamronic matches the one you want to compare to.

By tuning a bunch of resonators, maybe strings, maybe choir students, maybe tuning forks, in this interrelated way, we can get frequences that are are separated from our fundamental by any ratio of the form (2^x * 3^y) for integers {x} and {y} - ratios like 4/3 and 81/64 and 246/243 and 1/8 and 36/1 and infinitely many more.

Not only do we have a procedure for defining notes, but it turns out that closely related notes sound good together! The note at frequency F1 and the note at 3/2 * F1 are so closely related, when you play them together, it barely adds any texture. It's rock solid harmony. They have lovely consonance together. You can really hear the small integers of the frequency ratio in your ears.

Here are some intervallic names associated with some Pythagorean frequency ratios

1/1 - the unison (or perfect first)

256/243 - the minor second

9/8 - the major second

4/3 - the perfect fourth

3/2 - the perfect fifth

2/1 - the octave (or perfect eighth)

3/1 - the perfect 12th

Intervals aren't quite the same as frequency ratios, but they're intimiately related, and we can think of them being in 1-to-1 correspondence within the system of Pythagrean tuning.

: The Additive Psychoacoustics Of Intervals

It turns out that we hear frequencies logarithmically. What feels, looks, and sounds like "going up" one linear amount on an instrument, like one key up on a piano or one fret up on a guitar, is more like going a factor of 2^(1/12) up in frequency space. When frequency ratios multiply, intervals add. This is one great conveience of communicating about music in terms of intervals - they have subjective psychological realism.

Intervals add, while frequencies multiply. Intervals have long names like "perfect fifth" and also short names like "P5". Let's look at a bunch in order to get familiar with them:

P1: unison (or perfect first)

P8: octave (or perfect eighth)

P5: perfect fifth

P4: perfect fourth

m2: minor second

M2: major second

m3: minor third

M3: major third

m6: minor sixth

M6: major sixth

m7: minor seventh

M9: major ninth

d5: diminished fifth

A4: augmented fourth

d2: diminished second

AA2: augmented augmented second (or twice augmented second)

dd0: diminished diminished zeroth

You may soon consider these to be close friends, if you don't already. If you're familiar with sharps and flats in staff notation, then you're more familiar with intervals that you might think - sharpening a pitch is the same as augmenting a related interval, like (perfect fifth plus augmented unison equals augmented fifth)

P5 + A1 = A5

and likewise flattening a pitch is the same as diminishing a related interval, like

P5 + d1 = d5.

We'll talk about it more in a future chapter.

: Generating A Chromatic Scale

If we start at the unison and keep on adding perfect fifths on top to get a larger frequency ratio, and then we drop the frequency ratio down an octave when the ratio is greater than (2/1), then we get a family of frequency ratios that lie between the unison and the octave. We can also start at the unison, subtract a perfect fifth repeatedly, and add an octave any time we get a frequency ratio less than (1/1). Up and down, new friends to be found.

You might wonder why we'd use the perfect fifth, with a tuning of 3/2, instead of the 3rd harmonic, with a frequency ratio of 3/1. Weren't we going to define everything in terms of the second and third harmonics? Well, you might notice that 3/1 is larger than an octave, so everytime we go up by 3/1, we also have to go down by 2/1. Using the perfect fifth just saves us some time. The ratios we get won't be any different.

*Now* we're ready for the historic Pythagorean myth as I like to tell it. *clears throat* Once upon a time, Pythagoras wrote this table down, exactly as it appears below, on a goat skin parchment after eating some bad spanakopita: 

Coordinates : Frequency Ratio :: Interval Name

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

...

(-6, 4) : 1024/729 :: d5

(-5, 3) : 256/243 :: m2

(-4, 3) : 128/81 :: m6

(-3, 2) : 32/27 :: m3

(-2, 2) : 16/9 :: m7

(-1, 1) : 4/3 :: P4

(0, 0 ) : 1/1 :: P1

(1, 0) : 3/2 :: P5

(2, -1) : 9/8 :: M2

(3, -1) : 27/16 :: M6

(4, -2) : 81/64 :: M3

(5, -2) : 243/128 :: M7

(6, -3) : 729/512 :: A4

...

That's the whole myth. The coordinates in the first column tell us how many perfect fifths and how many octaves, respectively, you need to add up or subtract to get the frequency ratio on that line, starting from a reference frequency.  We can think of this list as starting at the bold line in the middle - at the unison - and working outward in two directions. The next column of each line is the frequency ratio of course, and the last column has an intervallic name for that frequency.

For example, look at the line above the bold unison line in the table. I'll even reproduce it for you:

(-1, 1) : 4/3 :: P4

If you go a perfect fifth below a unison and then an octave up, such that our coordinates are (-1, 1), then you get a frequency ratio of 4/3 relative to the base frequency. We call this interval the perfect fourth or P4.

We can write this in interval space as

- 1 * P5 + 1 * P8 = P4

We can also write this multiplicatively in frequency space:

(3/2)^(-1) * (2/1)^(1) = (4/3)

Here P5 is replaced with its frequency ratio of (3/2), and the octave is replaced with (2/1)  and what were multiplicative coefficients for added intervals have now become exponents for multiplied frequency ratios. But they're basically the same statement, at least while we're working in Pythagorean tuning.

If you want your coordinates to be in terms of the perfect twelfth and the octave (which Pythagoras associates with 3/1 and 2/1) instead of P5 and P8, those coordinates are very easy to figure out from the frequency ratio. For example, a major sixth has frequency ratio 27/16 in Pythagorean tuning:

(3, -1) : 27/16 :: M6

If we factorize this ratio, we get

27/16 = 3^3 * 2^-4

so we could say that the coordinates for M6 in the (P12, P8) representation are (3, -4), just as the coordinates were (3, -1) in the (P5, P8) representation. We can also convert between representations like (P5, P8) and (P12, P8) using linear algebra. We'll do that in a different chapter.

Back to the goatskin! You can keep adding on perfects fifths to extend the list downward from unison, and you can keep subtracting perfect fifth to extend the list upward. In doing so, the frequency ratios keep getting crazier, with larger numerators and denominators. 

Most interval names don't start with "perfect". For example, this line

(2, -1) : 9/8 :: M2

tells us that we can stack two perfect fifths over unison and then drop it down an octave to get a new interval {x} in the range 

P1 <  x < P8

with frequency ratio 9/8, which we call a major second, M2. 

Another example: two perfect fifths below unison, raised by two octaves to put us in range again, are called a minor seventh, m7.

(-2, 2) : 16/9 :: m7

This procedure of stacking P5s and sometimes adjusting by P8 won't get you all possible intervals or frequency ratios: for example, it won't get you an octave, P8. So this game isn't showing us all Pythagorean frequency ratios, it's just showing us a bunch of notes between P1 and P8. It will show us infinitely many ratios in that range, but still not all ratios of the form

2^x * 3^y

Now, I stopped extending the table where I did for a good reason: the notes circled around! Kind of. The diminished fifth interval at the top, d5,  with a frequency ratio of 1024/729 ~= 0.4 is basically the same as the augmented fourth interval at the bottom, with a frequency ratio of 729/512 ~= 0.4. They're definitely not identical to human perception, but they're decently close.

With this set of frequencies (or the related intervals), we have ourselves a chromatic scale! Its members are these: 

(P1, m2, M2, m3, M3, P4, d5, A4 ~=~ P5, m6, M6, m7, M7).

The notes of this scale are fairly evenly spaced and they have small spaces between. I tried making a diagram to show how much they are not-quite evenly spaced and came up with this:

The figure on the top is a Pythagorean chromatic scale, with both d5 and A4 shown straddling the middle (with A4 being the larger frequency ratio (the more rightward of the two). The figure on the bottom is an evenly spaced tuning system (12-EDO), in which d5 merges with A4. The far left edge is zero steps in 12-EDO (the unison, P1) and the far right edge is 12 steps of 12-EDO (the octave, P8). There are lots of ways to define 12-EDO, but a common one is by saying that it is a tuning system which makes the interval between d5 and A4 disappear. We'll talk about this more in future chapters.

It turns out that even-spacing (on the bottom) and small-integer-consonance (which Pythagorean tuning has) are both quite nice psycho-acoustically, and a lot of music theory is about finding ways to make our brains think we're getting both at once.

Nothing about Pythagorean tuning says that you have to stick to the chromatic intervals I listed above (and their octaves): it's convenient to have a known and fixed chromatic scale, but a composer more comfortable with microtonal music will use more of the interval space. For example, the medieval Iranian music theorist Safi al-Din al-Urmawi analyzed middle eastern microtonal music as being made of up notes from a 17-note scale made by continuing the Pythagorean spiral of fifths past the chromatic scale. And you don't even have to have a fixed scale. You could compose on a two-dimensional lattice where one direction is multiplying/dividing by 3 /1 (or 3/2) and the other direction is multiplying/dividng by 2/1. There are many options.

: Fixed Pitch Scales With Rational Frequency Ratios Aren't Great For Modulation

Unfortunately, the chromatic scale of Pythagorean tuning is kind of hard to use in many musical keys. 

The chromatic scale [P1, m2, M2, m3, M3, P4, d5/A4, P5, m6, M6, m7, M7] , if we start it on C natural, corresponds to these pitch classes [C, Db, D, Eb, E, F, F#/Gb, G, Ab, A, Bb, B]. Even if you have a split key to play both F# and Gb, this is still simply missing lots of basic familiar pitch classes. You can't really modulate to the key of D major, because there's no C#. For the key of E major, you'd need a C# and a G# and a D#, and they're all missing. 

An E major chord is spalled [E, G#, B]. If you have a fixed pitch keyboard based on a Pythagorean chromatic scale over C, you might be tempted to use the available pitches [E, Ab, B], instead. You will immediately discover that what should be a major third is actually a diminished fourth, at a frequency ratio of 8192/6561 instead of 81/64.  At 384 cents, this diminished fourth interval is 24 cents flat of the Pythagorean major third (81/64 at 408 cents) and 16 cents flat of 12-EDO's major third (at 400 cents even). This is what we know in the business as "jank" or "scunge" or "hot garbage".

So, the tuned chromatic intervals are evenly spaced enough that we can call the set chromatic, but there are still some really bad sounding intervals between nearby notes that composers naturally reach for. I think this is more a critique of fixed pitch instruments, or composers who misuse their fixed pitch instruments, but it's certainly frustrating that things don't work simply.

Another weakness of Pythagorean tuning is its complicated frequency ratios, like the minor second at 256/243. It turns out that humans like listening to fractions with powers of 5 in them just fine, and introducing powers of five will make lots of the frequency ratios simpler, only changing the consonance a little. We'll talk about it in the next chapter.

Let's talk a little bit more about what we can do well with Pythagorean tuning. By selecting elements of the Chromatic scale, we can make some other famous scales. For example,

Major scale: [P1, M2, M3, P4, P5, M6, M7, P8] 

Minor scale: [P1, M2, m3, P4, P5, m6, m7, P8] 

And many hundreds more, but that's a good start.

If you keep extending the table by stacking perfect fifths and adjusting octaves, you start to see that lots of intervals are tuned about 23.5 cents away from another one. The interval difference that shows up repeatedly is the augmented zeroth, A0, with coordinates (12, -7) in the (P5, P8) basis. It has a frequency ratio of 531441/524288. This ratio shows up so much that people gave it a name: "the Pythagorean comma". 

Here are some ways to generate the A0 interval besides stacking P5s:

A0 = M2 - d3

A0 = AA1 - M2

A0 = m3 - dd4

A0 = M3 - d4

A0 = P4 - dd5

A0 = A3 - P4

A0 = d5 - dd6

A0 = A4 - AA3

A0 = AA4 - P5

A0 = m6 - dd7

A0 = M6 - d7

A0 = AA5 - M6

A0 = m7 - dd8

A0 = A6 - m7

It's everywhere!

These should be read in the following way, taking the last line as an example: "The augmented zeroth is an augmented sixth minus a minor seventh".  In the next post, we'll talk a lot about this kind of interval arithmetic. The main principle is that adding and subtracting intervals is analogous to multiplying and dividing their frequency ratios. It's almost as though intervals are the logarithms of frequency ratios in this small way, though this analogy quickly falls apart - since different tuning systems will assocaite an interval with different frequency ratios, and since intervals are multi-dimensional objects, unlike a scalar logarithm.

: But How Does It Sound?

I think human aesthetics for melodic intervals (those from one note to the next in time) are much looser than human aesthetics for harmonic intervals (those between notes sounded concurrently). Melodically, monophonic music in Pythagorean tuning is a little hard to tell from 12-EDO music or music with factors of 5 in the frequency ratios. I won't be shocked if you can do it even as a novice, but there's more melodic between-person variation in a choir of untrained family members singing happy birthday than there is between these tuning systems.

It's when you starting layering notes that you really notice the sound of a tuning system. If the music you've heard all  your life has been in 12-EDO, then Pythagorean tuning might sound a little bit off. You've been conditioned your whole life to hear equally-spaced notes as normal, and not to hear simple harmony as normal. But even if it's a litle off, it's still not bad. And sometimes it's better. In so much as harmony is about concordance between the overtones of two sounds, and in so much as the sounds are harmonic, then of course harmony will sound good between notes that are separated by small integer and rational frequency ratios.

Learning to appreciate real harmony is a key step of becoming a microtonal music lover. And I do mean real harmony. 12-EDO might sound normal to most western ears, but some of its harmonic intervals are badly mistuned, producing auditory beating or auditory roughness. That's fine - you can make music out of harsh sounds if you want, but as a matter of fact, it is less harmonious.

: A Fuller Spiral of Fifths

You won't have to work with microtonal music very long before you'll come to want a bigger goat-skin table than the mythic one that Pythagoras wrote down. in 12-EDO, you migth know it as the circle of fifths, but in Pythagorean tuning, the notes don't circle back on themselvs, so it's either a chain of fifth or perhaps a spiral of fifths, if you want to embed it in two dimensions to hint at the similarity of frequency ratios modulo 2/1. 

Let me say that again. In Pythagorean tuning, we have spiral of fifths rather than a circle of fifths because intervallic music theory doesn't collapse enharmonic distinctions like G# with Ab. Here's a large spiral for you:

(-20, 12) 4294967296/3486784401 ddd5 | Gbbb

(-19, 12) 2147483648/1162261467 dd9 | Dbbb

(-18, 11) 536870912/387420489 dd6 | Abbb

(-17, 10) 134217728/129140163 dd3 | Ebbb

(-16, 10) 67108864/43046721 dd7 | Bbbb

(-15, 9) 16777216/14348907 dd4 | Fbb

(-14, 9) 8388608/4782969 dd8 | Cbb

(-13, 8) 2097152/1594323 dd5 | Gbb

(-12, 8) 1048576/531441 d9 | Dbb

(-11, 7) 262144/177147 d6 | Abb

(-10, 6) 65536/59049 d3 | Ebb

(-9, 6) 32768/19683 d7 | Bbb

(-8, 5) 8192/6561 d4 | Fb

(-7, 5) 4096/2187 d8 | Cb

(-6, 4) 1024/729 d5 | Gb

(-5, 3) 256/243 m2 | Db

(-4, 3) 128/81 m6 | Ab

(-3, 2) 32/27 m3 | Eb

(-2, 2) 16/9 m7 | Bb

(-1, 1) 4/3 P4 | F

(0, 0) 1 P1 | C

(1, 0) 3/2 P5 | G

(2, -1) 9/8 M2 | D

(3, -1) 27/16 M6 | A

(4, -2) 81/64 M3 | E

(5, -2) 243/128 M7 | B

(6, -3) 729/512 A4 | F#

(7, -4) 2187/2048 A1 | C#

(8, -4) 6561/4096 A5 | G#

(9, -5) 19683/16384 A2 | D#

(10, -5) 59049/32768 A6 | A#

(11, -6) 177147/131072 A3 | E#

(12, -7) 531441/524288 A0 | B#

(13, -7) 1594323/1048576 AA4 | F##

(14, -8) 4782969/4194304 AA1 | C##

(15, -8) 14348907/8388608 AA5 | G##

(16, -9) 43046721/33554432 AA2 | D##

(17, -9) 129140163/67108864 AA6 | A##

(18, -10) 387420489/268435456 AA3 | E##

(19, -11) 1162261467/1073741824 AA0 | B##

(20, -11) 3486784401/2147483648 AAA4 | F##

Highly useful.

A useful interval that's Pythagorean but not generated by a spiral of fifths is the diminished second, d2. It will show up a lot in future chapters.

(-12, 7) : 524288/531441 :: d2

This doesn't show up in our spiral of fifths because it's out of range: the frequency ratio is slightly smaller than 1/1, at ~ 0.9865. We could have constructed the coordinates and the frequency ratio for d2 by hand by diminishing the minor second, i.e. subtracting an augmented unison from the minor second.

m2 - A1 = d2

which corresponds to division with the frequency ratios:

(256/243) / (2187/2048) = (531441/524288)

In general, diminishing a minor Nth interval or perfect Nth interval will produce a diminished Nth interval.

This diminished second, d2, is also the inverse of the augmented zeroth, A0, that showed up all over the place as a difference of intervals. 

P1 - A0 = d2

What does it mean for one interval to be the inverse of another? One way to think of it is that the interval coordinates cancel out in summation to P1 = (0, 0), i.e. they have opposite signs from each. Compare:

(-12, 7) : 524288/531441 :: d2

(12, -7) : 531441/524288 :: A0

and you'll see why we can make three equivalent statements of interval arithmetic

A0 + d2 = P1

P1 - d2 = A0

P1 - A0 = d2

with total confidence.

Another way to think about inversion is that the frequency ratios are flipped. This has the consequence that multiplying the frequency ratios of d2 and A1 produces 1/1, which is the frequency ratio for P1. Pretty cool.

: Outro

And that's it - our first foray into microtonal music. Not too scary, right? We learned a little about the historic foundations of western music and we got a little bit familiar with intervals and frequency ratios that make up normal scales like the chromatic, the major, and the minor. It's just normal music made from harmonics, because harmonics are an easy way to tune instruments and generate notes. And these notes sound kind of good when played together. Or maybe they sound a little bit bad to you still, but they definitely have something interesting going on, right? Something new and compelling and maybe kind of rad? We're going to chase that feeling.