Audible numbers – Chiel Zwinkels

Introduction

It may be hard to imagine in the 21st century, where music and mathematics appear on opposite sides of the curriculum, but for much of human history, these two subjects were thought of as pretty much the same thing, and with very good reason. In this essay, we will explore the fundamental relationships between numbers and sounds. We will do so through a series of interactive illustrations, making it easy to follow for everyone, even musicians with a severe fear of numbers (or musically challenged mathematicians).

This page is part of my project for designing a new musical instrument, the Arithmophone, and is meant to explain the principles upon which that instrument works. Because these principles are very general and apply to all kinds of music, I am hoping that this essay will be of interest to anyone seeking a better understanding of the basic principles underlying melody and harmony.

A little background

When I went to school in the 1980s, music and mathematics were presented as two completely separate subjects. In fact, I went through a university education without ever learning of any relation between the two. After university, I went on to study guitar at the Amsterdam Conservatory and there I learnt a bit more about music theory, but again nothing that even hinted at the deep relationship between numbers and sounds. It would take me many more years before I would get to the heart of this relationship. First, I had to learn to ask the right question. The right question turned out to be:

“Why can’t I get this damn ukulele in tune?”

As a guitarist, I had known for a long time that it was impossible to get all the strings on my instrument perfectly in tune relative to each other. I even had an inkling that this had “something to do with harmonics”, but that was about the extent of it, and as it was always possible to get the guitar tuned well enough, I never gave it much further thought. On the ukulele though, with its shorter scale and lower string tension, this inherent ‘out-of-tune-ness’ is much more prominent, and I could never quite get to the point of ‘well enough’. Wanting to really understand why this was so led me down a rabbit hole that resulted in my exploration of many different tuning systems and eventually designing my own instruments which incorporate these tunings.

Some of the most elemental things I learned in the process are presented in this essay. I believe that these things deserve to be much wider known than they currently are. I certainly wish I could have learned them in school. Not only could they have helped me to get a better understanding of music theory through some simple mathematics, I think they could have also helped me develop intuitions for some fundamental mathematical concepts through sound and music. And most importantly, they could have taught me that science and art, music and mathematics, are all deeply connected at the fundamental level.

I hope you will enjoy this trip through musical mathematics wonderland. Let’s start with some definitions.

Tones and notes

Tones are patterns in sound that repeat over time. The faster the pattern repeats, the higher the frequency of the tone. This frequency, also called the pitch, is measured in Hertz (Hz), the number of repetitions per second. Human ears are sensitive to frequencies from about 20 to 20000 Hz, but at the extremes of this range it becomes very difficult do identify pitches. In musical terms, most people will be able to recognise tones with a base frequency between about 50 and 5000 Hz.

Adjust the slider to change the frequency, then click to the button to hear the tone. You probably won’t be able to hear the very highest frequencies because of the limits of your hearing. The very lowest frequencies cannot be reproduced by computer speakers but should be audible on headphones.

Tones of different frequencies can be recognised as different musical notes. The ratios between these notes are called intervals. The simplest musical interval is called the octave, which is the ratio of 1 to 2. If you play a tone with a certain frequency, then a tone with double that frequency will sound an octave higher, while a tone with half that frequency will sound an octave lower. Even though these tones have different frequencies, in musical terms, they are all the same note. Other simple ratios, such as 2 to 3 and 4 to 5, produce different intervals and also different notes. Together, these intervals function as the basic building blocks for melody and harmony.

The harmonic series

When you pluck a string, it starts vibrating at a certain frequency. For example, if you strike the open A string on a Cello, the frequency is around 220 Hz. This frequency is called the fundamental or the first harmonic. But a vibrating string doesn’t just produce a fundamental tone, it produces overtones as well. These overtones, also called ‘harmonics’, are simple multiples of the fundamental frequency. So, when you pluck a string that is tuned to 220 Hz, the sound you hear will also contain the harmonic frequencies of 440 Hz, 660 Hz et cetera. This is a natural consequence of the physical properties of resonating bodies, and with some slight modifications, the same principle applies to other methods of generating sound, like blowing on a wind instrument.

The fundamental frequency is usually the loudest and in general, the higher the harmonic, the less noticeable it becomes. Our ears experience this combination of frequencies as a single note with a certain sound colour or timbre. The relative loudness of the harmonics determines the timbre. This is one of the reasons that we can easily distinguish a note played on a trumpet from the same note played on a guitar: they have different harmonic characteristics. Some sounds are very low in harmonic content and contain mainly the fundamental: whistling is a good example of this. However, most musical instruments are very rich in harmonics.

So even if we hear just one note, we already hear many different frequencies contained within that note. These frequencies are simple multiples of the fundamental tone. From this, we can begin to see where different musical notes are coming from. If we start with a frequency of 220 Hz, then the frequencies of the first 8 harmonics are:

220 Hz, 440 Hz, 660 Hz, 880 Hz, 1100 Hz, 1320 Hz, 1540 Hz, 1760 Hz

In terms of ratios, taking the fundamental frequency as 1, this can be rewritten as:

1/1, 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1

Or simply as:

1, 2, 3, 4, 5, 6, 7, 8

Click the red button to hear a single note containing many harmonics. Click on the blue buttons to hear the first 8 harmonics in isolation.

Octave reduction

It is a central fact of music that the simplest and smallest harmonic ratios represent very large jumps in frequency. The third harmonic is 3 times the fundamental frequency, so it sounds about an octave and a half higher than the fundamental. The fifth harmonic of 5 times the fundamental frequency is another octave higher up, and so on. These simple ratios function as basic underlying principles of music, but in melody and harmony they are rarely heard in their ‘naked’ form.

The harmonic series produces different notes across many octaves, but we can bring these notes closer together by dividing their frequencies by 2 as many times as we like, because that will just give us the same note in a lower octave. For the first 8 harmonics, that leaves us with:

1/1, 2/2, 3/2, 4/4, 5/4, 6/4, 7/4, 8/8.

After simplifying the fractions, this becomes:

1, 1, 3/2, 1, 5/4, 3/2, 7/4, 1

So even though we have 8 different harmonics, we only have 4 different notes. Of these, the 1 is very strong: it is both the fundamental, which is usually the loudest, and it is also present 4 times in the sequence, which further reinforces it. The next strongest note is the 3, because it appears twice and also relatively early in the sequence (remember that the higher the harmonic, the softer it sounds). The 5 is weaker, appearing only once, and the 7 weaker still, appearing later in the sequence. Of course, this pattern continues after the first 8 harmonics. Every odd number produces a new overtone, but every new harmonic is weaker than the ones before. So even though in theory the harmonics go on to infinity, in practice their relevance quickly diminishes. In fact, the whole system of Western music, with 12 notes to the octave, can be constructed just by using the first five harmonics, which is to say, the octave, the third and the fifth.

Click the buttons to hear the first 8 notes of the harmonic series, brought within close range by octave reduction.

Moving up and down

Because any interval is a relationship between two notes, it functions in two directions. If we start with a tone of 100 Hz and then take the third harmonic of 300 Hz, the second stands to the first in a ratio of 3 to 1, or 3/1. But at the same time, the first stands to the second in a ratio of 1 to 3, or 1/3. If the third harmonic is an overtone of the fundamental, we might just as well call the fundamental an undertone of the third harmonic. In naturally occurring sounds, such as an oscillating string, the fundamental is the lowest frequency produced and only overtones are present as harmonics. But as far as musical intervals are concerned, there is a great symmetry between overtones and undertones. This is because we can experience any pair of notes from the ‘perspective’ of either the lower or the higher note, depending on the musical context.

Click the buttons to hear the symmetry between overtones and undertones. You can switch from simple ratios to octave reduced ratios with the buttons at the top to make the similarities easier to recognise. For example, the interval from 4/5 to 1/1 is the same as the interval from 1/1 to 5/4. The central note 1/1 is contained within the undertone 1/5 in exactly the same way that it itself contains the overtone 5/1.

Moving step by step

When we apply octave reduction, the order of the notes (from lowest to highest) changes. For example, 5/1 is a higher harmonic than 3/1, and therefore it has a higher frequency. But 5/4 is closer to 1/1 than 3/2, so after octave reduction their order is reversed. Once we bring our notes within a single octave range and put them in sequence, familiar-sounding melodies will start to emerge when we move through the notes step by step.

Click the buttons from left to right to hear a simple melody that is ‘hidden’ within the simple ratios of the harmonic series but becomes apparent after octave reduction.

Combining harmonics

Some of the most commonly used notes in music are not direct over- or undertones but are the result of combined ratios. For example, if you go up from 1/1 to 3/1 and then down by 1/5, you end up with the ratio 3/5. This musical ratio is known as the just minor third and is used all the time in many different kinds of music.

Click the buttons to hear how combined ratios produce new musical notes. Note that 5/3 relates to 1/3 in exactly the same way that 5/1 relates to 1/1, et cetera. Switching to octave reduced ratios makes these relationships easier to hear.

Full scale harmonics

With the inclusion of the combined ratios introduced above, we now have all the ingredients we need to make the most familiar chords and melodies. For example, the commonly used major scale is built from these ratios:

Click on the buttons to hear each note, or play them from your computer keyboard, starting on the letter “z” (click on the image first to activate this feature). The 1/1 note is tuned to 261.63 Hz, commonly known as “Middle C”.

From melody to harmony

Chords are simply multiple notes sounding at once. The example below presents two of the most basic chords: the major and minor triads (a triad is a three-note chord). The closer the notes in a chord are to simple ratios, the more ‘harmonious’ the chord sounds.

Click the buttons on the left to hear the notes of the major and minor chords one by one, click the buttons on the right to hear all three notes simultaneously.

A note on names and numbers

We have seen that we can divide or multiply any frequency by 2 as many times as we like, without changing the note it represents. It simply becomes the same note in a different octave. The harmonics of a note stretch out over many octaves. But when we construct a scale, we ‘reduce’ the ratios so that all the different notes land in the same octave. As we have seen, this influences the order of the notes. 1, 3, 5 becomes 1/1, 3/2, 5/4 and since 5/4 is less than 3/2, the notes appear in the scale in the order 1, 5, 3.

In the common major scale “do, re, mi, fa, sol, la, ti, do”, the third note “mi” is based on the ratio 5/1 and the fifth note “sol” is based on the ratio 3/1. So the third note of the scale is actually the fifth harmonic and the fifth note of the scale is the third harmonic. This switching around of 3 and 5 can be more than a little confusing. To make things as clear as possible, I have included the table below. It is good to keep in mind that ratios always refer to harmonics, whereas intervals described in words or in roman numerals, like “a major third” or “a flat VII” generally refer to scale position.

Note (key of C)	C	D	E	F	G	A	B	C
Note name (Solmization)	Do	Re	Mi	Fa	Sol	La	Ti	Do
Note name (Sargam)	Sa	Re	Ga	Ma	Pa	Dha	Ni	Sa
Scale degree	I	II	III	IV	V	VI	VII	I
Harmonic ratio (pure)	1/1	9/1	5/1	1/3	3/1	5/3	15/1	1/1
Harmonic ratio (octave reduced)	1/1	9/8	5/4	4/3	3/2	5/3	15/8	2/1

The 3-5 Tone Grid

As long as we limit ourselves to the first five harmonics, we can arrange all possible notes in a two-dimensional grid. Multiplying or dividing by 1 does not change a note and multiplying or dividing by 2 and 4 just gives the same note in a different octave. So, we only need two axes to produce any note: one for the 3-ratios and one for the 5-ratios. For instance, this can be done in the following way: any step to the right is always a multiplication by 3, any step to the left a division by 3, any step up a multiplication by 5 and any step down a division by 5. In this way, any ratio within the 5-limit system can be produced.

How this works can be seen and heard in the illustration below. To keep things practical, I have limited the grid to a distance of 3 harmonic steps from the centre. In principle, it could be extended infinitely in all direction, but even with this restriction we already get 25 different notes, more than twice as many as you’ll find on a regular piano, which has only 12 different notes (a standard piano has 88 keys, but these are just the same 12 notes repeated again and again over 7 octaves).

Click the buttons to play the notes. You can adjust the legend to show ratios or note names. The central frequency is around 294 Hz, the musical note “D” and the ratio 1/1 in this example. All other ratios are frequency reduced to bring them closer to the central frequency. For example, the ratio 3/1 is divided by 2, so that the sounding frequency is 294 *3/2 = 441 Hz, the musical note “A”. For clarity, the pure (unreduced) ratios are shown in the illustration.

You may notice that some note names appear more than once on this grid. For example, both the ratio 5/9 and the ratio 9/1 are labelled “E”. After octave reduction, 5/9 becomes 10/9 and 9/1 becomes 9/8. These two ratios are very close together (10/9 is approximately 1.111 and 9/8 is 1.125) and that is the reason they end up on the same note name, but they are not identical. On a piano there is only one “E” but in harmonic tuning there are at least two.

So, which “E” is the right one? The answer is: it depends on the context. If you want to play an A chord, the 9/1 ratio sounds more in tune, because that is the third harmonic of the 3/1 A. But if you want to play an C chord, it makes more sense to use the 5/9 ratio, because that one combines with the 1/9 C and the 1/3 G to form a perfectly tuned major triad. You can play around with the 3-5 tone grid above to get a feeling for these subtle differences.

Infinite possibilities

The 3-5 tone grid above has 25 different notes, more than twice as many as in standard piano tuning. But it is by no means complete. We can create many different melodies and harmonies starting from the central 1/1 ratio, but when we take one of the other notes as a starting point, we run out of options pretty soon. Let’s say we would like to make a major chord starting on the 45/1 “G#”. To make this a well tuned chord, we would need a third and fifth harmonic above the starting note: a 135/1 “D#” and a 225/1 “B#”. Neither of these notes is present in our grid. Of course, we could extend the grid further to add more notes but this will quickly become unpractical, and what is more, we will still never get a complete set of notes, because there are infinitely many of them.

If anyone ever told you that there are only 12 notes in music, they were lying, even if they probably didn’t realise it. In reality, there aren’t 12 different notes, or 25 or any other number: there is an infinite number of notes. Our 3-5 grid can be extended forever in any direction, and even then we wouldn’t have covered any note that isn’t within the five limit system, like the simple 7/1 ratio (the seventh harmonic).

This may seem like a big problem for making music, but in practice, it usually isn’t much of an issue. Most kinds of music use only a very modest selection of notes and stick pretty close to a single tonal centre (like the 1/1 “D” in the example above). Even when more notes are required, as long as your instrument is flexible in pitch (like the human voice or the violin), you can simply adjust the tuning for each note as you play it. This often happens when people sing in close harmony or play in a string quartet: they adjust the pitch to gently slide between one “E” and the other, as circumstances require and often without being consciously aware of it.

There is however one scenario in which this infinity of notes presents a very practical problem and that is when you want to play music that is able to modulate freely from one tonal centre to the next, on an instrument with fixed pitch notes. This is what happened in Western Music from the Renaissance era onwards, when keyboard instruments (organs, harpsichords and later on pianos) became prominent and the musical style demanded a lot of harmonic mobility. It is in this context that the system of 12 tone equal temperament, by now completely ubiquitous in Western music, first evolved.

The circle of fifths

If you’ve ever had any music theory lessons, chances are that you’ve encountered the circle of fifths, a pretty simple diagram that explains the relationship between different notes and keys. These fifths form a perfect circle of exactly twelve notes. But we have already seen in our 3-5 tone grid that repeatedly jumping the same interval keeps generating new notes, an infinite amount of them. So how does this work?

For starters, we should be aware that the word “fifths” in this context refers to the fifth note of a scale, not the 5/1 ratio. So, an example of a fifth is the jump from “D” to “A” (skipping over “E”, “F” and “G”). You can also think of it as DO (re mi fa) SOL. This is in fact the ratio 3/1. If you’re still somewhat confused by this reversal of 3 and 5 (and understandably so), please reread the paragraph “A note on names and numbers” above.

If we jump a fifth from “D” (or 1/1), we end up on “A” or (3/1). If we then jump another fifth, we go from “A” to “E” (or 9/1, because that is equal to 3/1 times 3/1). We can keep going in this way, and our sequence of notes will look like this:

D → A → E → B → F# → C# → G# → D# → A# → E# → B# → F## → C##

Now at this last note, something interesting happens. This “C##” is a new note just like the ones before it, but its frequency is very close to the “D” we started on (after we’ve reduced the octaves to bring it in the same range). It is in fact a bit higher, but so little that to our ears it may sound more like an out of tune version of “D” than like an entirely new note.

So “C##” is almost, but not quite, the same as “D”, or to put it more generally: 12 jumps of a fifth are almost equal to an octave. In mathematical terms: 2 to the power of 7 almost equals 3/2 to the power of 12. This simple fact is the key to 12 tone equal temperament. By tuning every fifth to a ratio that is slightly smaller than 3/1, we can make it so that our “D” and “C##” notes do in fact become perfectly identical. Because the tuning difference is spread out equally over all 12 notes, all our fifths still sound almost perfectly in tune.

In order to accomplish this, we have to switch from a rational tuning scheme, where notes are related to each other via simple harmonic proportions, to an irrational tuning scheme. Dividing the octave into twelve equal parts means that the distance between any note and the next must be 2 to the power of 1/12, or the twelfth root of 2 (which is another way of saying the same thing). In 12 tone equal temperament, none of the notes (except for the octaves) are related to each other via rational intervals, because all of them are related through fractional powers of 2, which are always irrational. This would have been abhorrent to the rationally inclined music theorists of ancient times, like Pythagoras, but it has served composers and musicians quite well over the past few centuries.

Click the buttons to hear the circle of fifths in rational and tempered tuning. The orange inner circle is tuned to pure ratios. All fifths are perfectly in tune with their neighbouring notes, but 12 fifths add up to slightly more than an octave. The further away a note is from the tonal centre (the “D” in this example), the more its frequency differs from the equal tempered tuning. Because G# and Ab are on opposite sides of the tonal centre, they do not have the same frequency. The green outer circle is tuned to 12 tone equal temperament. In this system, each fifth is tuned slightly flat so that 12 fifths add up to exactly one octave. Because of this, the tempered versions of G# and Ab actually do have the same frequency.

Musical modes

Some of you may have wondered why I used the musical note “D” as the starting point for my circle of fifths, rather than the more commonly used “C”. I have a compelling reason for doing so, because a lot of music theory is much easier to understand (and explain) from the perspective of a “D-centred” system. To see why, we should have a look at the musical modes.

We now have a system of 12 equally spaced notes at our disposal, but in general, musical scales don’t use all of these. Many scales, like the common major scale described above, use a total of 7 different notes. This is the reason our musical notation system uses 7 letters and also that a piano has 7 white keys (per octave).

When you play only the white keys on a piano, from low to high, on some steps, you have to skip a black key, but on some steps there is no black key in between. Because all 12 notes are equally spaced in terms of their pitch, this means that by playing only the white keys, you create a particular pattern of skips. If you start on the note “A”, the pattern will look like this:

[A] [skip] [B] [C] [skip] [D] [skip] [E] [F] [skip] [G] [skip] [A]

Or to make it more general, with “N” for a note and “x” for a skip:

N x N N x N x N N x N x N

Now, this pattern is what gives the scale its ‘personality’. In this case, we get a scale called “natural minor”, which is also known as “Aeolian”. You can always get a minor scale by starting on “A” and playing only the white keys. Or, and here comes the important bit: by starting on any key and following the pattern “N x N N x N N x N x N”. If you start on a different key than “A”, one or more black keys will be involved but you will get the same scale, only shifted up or down by a number of notes.

The scale that has become the standard for Western music, is called the major scale. In 12 tone equal temperament, it follows the pattern “N x N x N N x N x N x N N”. There is only one way to play this pattern using just the white keys, and that is by starting on the note “C”. This is the reason that “C” pops up so often as the default starting point for scales, chords and circles-of-fifths. This reason is cultural and historical, but there is no structural or logical reason inherent in 12 tone equal temperament (or in music in general) for preferring the “C” over any other note.

In ancient Greek music, the “standard” scale was what we now call the Phrygian mode, the pattern attained by playing the white keys starting on “E”: “N N x N x N x N N x N x N”. In many kinds of folk music, the default mode is Dorian, the white-key pattern starting on “D”: “N x N N x N x N x N N x N”. These folk musicians may have been on to something because what sets “D” apart from the other notes, is that it functions as a symmetry point. Not only is the letter “D” right in the middle of the sequence a-b-c-D-e-f-g, but the Dorian scale is also its own mirror image: the pattern “N x N N x N x N x N N x N” is the same going from left to right as it is going from right to left.

If all of this sounds a bit abstract and theoretical to you, I encourage you to play around with the illustration below to get a feel for the different modes and their ‘personalities’.

The diatonic modes (7 note scales starting from each of the “white keys”). Try playing each row from left to right to get a feel for the character of each mode.

In the illustration above, we have taken a fixed set of 7 notes (A B C D E F G) and changed the starting point to create different scale patterns. We can go the other way as well: if we always start on the same note but change which notes we skip, we can produce the same patterns as before. If we do this with “D” as our fixed starting note, the pattern becomes especially clear:

The diatonic modes on “D”: instead of using a fixed set of notes, we now have a fixed starting point, but each scale has a different set of notes. Each scale also differs exactly one note from the scales above and below it. Try playing the scales and comparing them to the scales in the previous illustration to get a better understanding of their structure and sound.

The Dorian mode is the central mode. It is symmetrical by itself, but it also functions as the symmetry point for the other modes. By raising 1, 2 or 3 notes, we reach the Mixolydian, Ionian and Lydian modes. By lowering 1, 2 or 3 notes, we reach the Aeolian, Phrygian and Locrian modes.

Tempered tuning versus just intonation

Nature has provided an infinite spiral of notes, and with some clever retuning, humans turned it in to a much simpler circle of only 12 notes. The benefits of equal temperament are enormous. It gives us a practical way of constructing mechanical keyboard instruments that are highly flexible and sophisticated. It also gives us a musical framework that allows for enormous harmonic freedom, because all notes are now equally suitable to function as the tonal centre. However, 12 tone equal temperament does have its downsides and limitations.

We’ve seen that musical notes have their origin in under- and overtones, and that these can be expressed as simple numerical ratios. We can use these rational intervals to tune our instrument, and this kind of tuning is usually called ‘just intonation’. There are infinitely many musical ratios, so if we only have a limited number of keys / frets / organ pipes at our disposal, we can only use a small selection of these. With just 12 different notes, we can still make a very musically useful selection, but only from one tonal centre. As long as we stick to our ‘home’ key, we are perfectly in tune, but as soon as we want to modulate to a different key, our tuning goes sour.

12 tone equal temperament does not have this problem, but instead it sounds slightly out of tune in all keys. The difference is not so noticeable in pure 3-ratios, because these are very closely approximated in this tuning. But for 5-ratios like 5/1 and also for combined ratios like 15/1 or 3/5, the tuning is actually pretty far off. So, while the jump from “D” to “A”, for example, is almost perfectly in tune in equal temperament, the jump from “D” to “F#” leaves quite a bit to be desired.

A piano tuned to just intonation with the “C” taken as the 1/1 ratio will sound perfectly in tune as long as you play in the key of C, but if you start playing in the key of F# for example, it will sound quite unpleasant. A piano tuned to equal temperament will sound a bit more out of tune in the key of C, but better in any other key. The difference between just intonation and equal tempered tuning is quite subtle, but definitely audible, especially once you start playing more than one note at a time. Try it out in the illustration below:

Click on the buttons to hear each note, or play them from your computer keyboard, starting on the letter “z” for the lower keyboard and “q” for the upper keyboard (click on the image first to activate this feature). The lower keyboard is tuned to rational intervals, starting from the leftmost note. The upper keyboard is tuned to 12 tone equal temperament. The first and last notes of both keyboards are identical, the other notes differ. For some notes, the difference is bigger than for others. For example, the “G” and 3/2 notes sound nearly identical, but the “A” and 5/3 notes are quite different. Headphones are recommended for hearing these differences as clearly as possible.

The ‘out-of-tune-ness’ of 12 tone equal temperament is most apparent when several notes are played simultaneously. Most melodies by themselves sound quite similar in both 5-limit just intonation and 12 tone equal temperament, but chords really do get a different character depending on the tuning that is used:

Click the buttons to hear some common chords in just intonation (JI) and 12 tone equal temperament (ET) and notice the difference. The ratios of the notes in the just intonation chords are spelled out above their respective buttons. The last chord is a bit of a special case because it contains the 7th harmonic in just intonation, a note that isn’t really represented in 12 tone equal temperament. This just intonation chord is however the closest JI equivalent of the dominant 7th chord (C – E – G – Bb in this example) that plays a very important role in equal tempered music. Headphones are recommended again for this example.

Comparing notes

Musical notes come in many different flavours. Rational tuning is arguably the purest expression of these flavours, but it is not always the most practical option. 12 tone equal temperament is a fantastic invention and its many advantages have led it to dominate Western music, perhaps so much so that we hardly even notice its out-of-tune-ness anymore. But it is not the be-all and end-all of music.

Just intonation still has its place in many forms of traditional music and can be put to excellent use in electronic music as well. Not only does it offer better tuning and stronger resonances for the familiar notes and chords of standard tuning, but by accessing ratios based on higher overtones like the 7th and 11th harmonics, whole new sonic worlds can be opened up.

As far as temperaments go, 12 tone equal is also not the only option. Many other temperaments have been developed, and some of these have distinct advantages over the familiar 12 tone system. For example, 31 tone equal temperament provides a much better approximation of 5-ratios than 12 tone equal temperament, while the 3-ratios are only slightly less in tune. It also offers a very good representations of the 7th harmonic which is almost completely lost in 12 tone equal temperament.

The Arithmophone project explores some of these options and puts them side by side for easy comparison. So if you’d like to see and hear more, please continue reading about the Arithmophonic keyboard designs or just try playing the Arithmophone.

For further reading, this excellent article gives a great account of the cultural significance of different tunings, particularly in relation to electronic music. And for a more in-depth exploration of the topics discussed in this essay, I can highly recommend the following books: Harmonic Experience by W.A. Mathieu and Temperament by Stuart Isacoff.