The design of the Arithmophone

Introduction

In this article, I will give a detailed description of the design of the Arithmophone. First I will discuss the general concepts that went in to the design: 7-limit just intonation tuning, the 2-dimensional keyboard layout, harmonic spines, octave grouping and colour coded keys. After that, I will discuss the specific design of the Arithmophone in its 4th revision. The following sections assume a basic understanding of musical notes and intervals, just intonation ratios and temperaments. If you are not entirely familiar with these things, everything you need to know can be easily learned in this interactive guide.

7-limit just intonation

The Arithmophone keyboard uses 7-limit just intonation tuning. The ‘just intonation’ part refers to the fact that this tuning is based on perfect harmonic ratios. This sets it apart from instruments like piano and guitar, that use a tempered tuning system (12 tone equal temperament).

The ‘7-limit’ part refers to the highest prime factor that occurs in these ratios. In 7-limit tuning, only the first four primes (2, 3, 5 and 7) are used. This includes ratios like 9/8, which can be factored as (3*3)/(2*2*2), but excludes any number that have higher prime factors like 11 or 13.

7-limit just intonation provides a much wider palette of tones than standard tuning. 12 tone equal temperament is essentially an approximation of 5-limit just intonation: it provides decent representations of ratios like 3/2 and 5/4, but it doesn’t really represent any of the basic septimal intervals like 7/4 or 6/7. Expanding the prime limit from 5 to 7 introduces a whole new class of notes, that are not really a part of the standard story of Western music. However, these intervals do appear frequently in many kinds of music, like Blues and Jazz – where they are played on instruments with flexible tuning like slide guitars or saxophones, as well as sung. These intervals are often referred to as ‘blue notes’, a fact I have used as inspiration for the colour scheme of the Arithmophone.

To illustrate the benefits of using just intonation tuning instead of 12 tone equal temperament, here is an analogy: when we store a digital image, we need to quantise the colour value of each pixel to some number of discrete steps. In modern picture formats, colours are usually stored as 24 bit values. This means there are about 16 million colours available. This is a very large number, so to our eyes the transitions from one colour to the next appear completely smooth. An image stored in this way might look like this:

In the early days of computers however, technology was more limited and it was generally not possible to use such a high colour depth. Therefore, images were commonly stored and displayed with 8 bit colour depth. The same picture stored in such a way looks like this:

In the image above, colour values are quantised to 216 steps. This is enough to produce a recognisable image, but most of the intricacies of the original image are lost. Now, when we play music on a fixed pitch instrument like a piano, we also quantise the frequencies of our notes. On a grand piano, the pitch is quantised to 88 equal steps (from the lowest to the highest note on the keyboard). This is less than half the number of steps used for an 8 bit colour image, so it should come as no surprise that here too, many subtleties are lost. Using a just intonation keyboard does not fully resolve this issue, because that would require an unfeasibly large number of keys, but it does allow us to choose a larger and more specific palette so that we can create more intricate melodies and harmonies.

Two-dimensional keyboard layout

On a standard piano keyboard, the notes are arranged from low to high (going from left to right). Although the keyboard itself is a 3-dimensional thing, with its raised black keys, the layout is essentially one-dimensional: all the notes are arranged along a single line.

The Arithmophone on the other hand, use a two-dimensional grid. Arranging the notes in two dimensions instead of one has some interesting advantages, especially for an on-screen keyboard: it can help to make the relationship between different notes more intuitive, it can make certain chords and melodies easier to play and it also makes more efficient use of the available space on the screen.

When we arrange notes in two dimensions, it no longer makes sense to order them simply from lowest to highest, so we need to find some other way of arranging them. This is were harmonic spines come in to play.

Harmonic spines

The Arithmophone keyboard is based on what I call ‘harmonic spines’. A harmonic spine is a set of notes that are directly under- and overtonal to a central note. A basic harmonic spine looks like this:

These numbers represent the ratios of the frequencies: if the 1/1 note has a frequency of 100 Hertz, then the 3/1 note is 300 Hertz (3*100), the 1/5 note is 20 Hertz (100/5), et cetera. The ratios shown in the example above and in all the following illustrations, are in their pure, ‘unreduced’ form. But the pure 3/1 ratio is already a very large interval (a big jump in pitch). In practice, these ratios will almost always be octave-reduced (ie divided or multiplied by 2 some appropriate number of times) to bring them within a musically useful range. So 3/1 might become 3/4, 1/5 might become 8/5, et cetera. However, this does not change the basic ratio of the note it represents, so for clarity and simplicity, unreduced ratios are used throughout this article. (Strictly speaking, this presupposes the use of an octave-repeating tuning scheme and is not applicable to tuning systems that aren’t octave-repeating, such as the Bohlen-Pierce scale, but those are beyond the scope of this article and the Arithmophone project.)

The spine above contains all overtones and undertones of the central 1/1 note up to the seventh harmonic. It could in principle be extended as far as desired in both directions, to include the ninth harmonic, the eleventh harmonic et cetera. Only odd numbers are included, because even numbers don’t introduce any new notes. Those are simply octaves above and below existing notes: 6/1 is an octave above 3/1 and so on.

The 1/1 note forms the center of the keyboard, but a harmonic spine does not have to start on this note, it can be constructed from any note. The example below shows a secondary harmonic spine starting on the 3/1 note.

In this way, many musically useful intervals arise. The ratio 3/5 does not appear as a direct over- or undertone of the 1/1 note, but it is present in the column of the 3/1 note and in this way it is strongly (though indirectly) related to the 1/1 as well. 3/5 is in fact the note known as the just minor third, a very common note in almost all kinds of music.

When we extend this principle and use it to make a square of harmonic spines, we obtain the following ratios:

However, as a musical keyboard this layout is not very efficient. It generates a lot of duplicates: the same ratios appearing in different places. To remedy this problem, we can start by folding the central harmonic spine around the 1/1 ratio like this:

If we then start filling in the secondary harmonic spines, we will see that most of the duplicates land in the same place and therefore become a single key. First we fill in the spines in the inward direction:

And then we can extend it in the outward direction:

Now we have all the same intervals as in the square layout, but we only need slightly more than half the keys. This intertwining of harmonic spines forms the basic harmonic structure of the Arithmphone.

Octave grouping

In order to create a keyboard that is musically useful, we need a range of at least several octaves (for reference, the range of a normal singing voice is about 2.5 octaves, while the range of an acoustic guitar is about 3.5 octaves). We could accomplish a multiple octave range by placing multiple copies of our harmonic spine structure side by side, but once again this would not be very efficient. Luckily, there is another solution: grouped octaves.

Unlike on most instruments, the physical distance between notes on the Arithmophone keyboard generally does not correspond to their distance in pitch, but to their harmonic distance. Because the shortest possible harmonic distance between to notes (that aren’t identical) is the octave, it made sense to me to group together different octaves of the same notes. This also solves a big problem inherent in using scales with more than 12 notes per octave, namely that their layout can become very unwieldy. Using grouped octaves allows for a clean and relatively simple keyboard design that still spans a range of several octaves.

Colour coding

On the Arithmophone keyboard, the colours of the keys reflect the nature of the interval they form with the root note. The colour scheme (which was also used for the illustrations above) works as follows:

Factors of 2 have no influence on the colour, so different octaves of the same note always have the same colour. The other primes (3, 5 and 7) each get their own colour, and in-between colours are used for ratios that contain multiple primes, such as 5/7. This colour scheme does not just serve an aesthetic purpose, it also provides a strong visual aid for navigating the keyboard and understanding the relationships between different notes.

The hexagonal keyboard

The layout of the Arithmophone keyboard is based on harmonic spines, as described above. In the final design, these have been placed in a hexagonal grid. The basic pattern looks like this:

While this produces a nice selection of notes, some of the ratios in the bottom row of keys, the ones that are shown dimmed in the illustration above, are not very useful. 1/9 and 9/1 already appear elsewhere on the keyboard, as do 3/9 and 9/3 (because these are identical to 1/3 and 3/1). So including those ratios would add nothing new to the selection of notes.

7/9 and 9/7 are unique ratios, but they are very close to 1/25 and 25/1, respectively. These notes are separated by what is known as the septimal comma, the ratio 225/7. After octave reduction, this ratio becomes 225/(7*2*2*2*2*2), or 225/224, which is very close to 1. Since a factor of 1 means that two notes are identical, a factor of 225/224 means that they are extremely close in pitch. The difference between these two intervals is about 7.7 cents (a cent is 1/100th of a standard semitone). This is close enough that our ears tend to perceive these notes as functionally equivalent.

In fact, every 7-limit ratio has a 5-limit ‘twin’, and vice versa. For example, if we start on 9/1 and want to add a major third (5/1 ratio) above, we would normally use 45/1 (because 9*5 is 45). This ratio is not directly present on the Arithmophone keyboard, but it’s ‘twin’ 7/5 is, and it produces a very useable major third on 9/1. In fact this major third is much more in tune than the major third of 12 tone equal temperament, which is off by almost 14 cents, nearly twice as much as the tuning error introduced by the septimal comma. Considering this fact, I decided that the optimal selections of notes for the Arithmophone keyboard should contain no more than one of the ratios from each of these ‘twin’ pairs.

So, I replaced these ‘superfluous’ ratios with other ratios that provide a more musically useful addition to the selection of available notes, like this:

I selected these particular intervals by first organising the existing notes in a different way, namely on a quasi 3-5 grid. A regular 3-5 grid is a way of organising 5-limit intervals: you start with the 1/1 ratio and for each step right, you multiply it by 3/1. For each step left you multiply by 1/3, then for each step up you multiply by 5/1 and for each step down you multiply by 1/5. This creates a 2-dimensional grid that shows each ratio surrounded by its closest neighbours. Chords and scales are generally built from groups of notes that are directly adjacent in such a grid. I added the septimal intervals in the places of their 5-limit siblings, ie 7/1 in the place of 225/1 et cetera, to create what I call a quasi 3-5 grid. Arranged in this way, the original note selection of the Arithmophone looks like this:

From the image above, it can be seen why 21/5, 27/1 and 27/5 (and their inverses) would be useful additions to the selection: they optimally increase the number of available scales and well tuned chords. In this form, the final note selection looks like this:

Back to the keyboard layout, the final structure looks like this:

The illustrations above contains a lot of useful information, but they are not the easiest to process. When actually playing the Arithmophone, it may be more useful to think of note names instead of ratios. By default, the root note (1/1 ratio) of the Arithmophone is tuned to D, which makes the note map look like this:

The reasons I chose D as the default central note are detailed here. The keyboard can be transposed to a different root note, either with the modulator keys (see below) or with the global transpose slider, and of course that will change the note names as well. An important thing to keep in mind, is that with just intonation tuning, there is no simple 1 on 1 mapping of notes to names as there is in 12 tone equal temperament. With D as 1/1, both 1/9 and 9/5 will result in a C, even though these notes are not identical. You can think of these as two ‘flavours’ of the same note. In 12 tone equal temperament these two are ‘flattened’ down in to a single note, but in just intonation, they are two distinct things. The difference between these notes is about 22 cents, or one fifth of a standard semitone, quite small but large enough to be musically relevant (to my ears at least).

In the illustration above, I have chosen to also represent the 7/1 ratio as C or, to generalise to any note center, as a flat seventh. I have found this way of thinking about it the easiest: in 12 tone equal temperament, there is just one flat seventh, in 5-limit tuning, there are two flavours and extending to 7-limit tuning we gain a third flavour. However, it should be noted that there are also good theoretical reasons to think of the 7/1 ratio as a raised sixth (or B# in this case), like its ‘septimal twin’ 225/1. In 7-limit just intonation, the naming of notes is not quite as clear cut as in 5-limit tuning or in 12 tone equal temperament, but it is still a very useful aid for navigating around different scales and chords.

In order to be musically useful, the final design also needed to have a range of multiple octaves. Because the central 1/1 ratio recurs several times (as 1/1, 3/3, 5/5 and 7/7), these keys could be repurposed as different octaves of the root note. All other ratios occur only once in the layout, so each of those hexagons is split in to lower, middle and upper octave keys. The illustration below shows a small section of the keyboard with the actual (octave reduced) ratios of each individual key:

With these hexagons divided into triple octave keys, the keyboard covers a full three octave range. Not nearly as much as grand piano (which covers more than 7 octaves), but comparable to a guitar or a saxophone, so certainly enough to be musically expressive, especially considering the facts that the range can be extended quite a bit further using the modulator keys and the transpose slider.

Modulators

The modulator keys are not part of the Arithmophone keyboard per se, but they are an important extra feature. In just intonation, the number of possible ratios is essentially infinite, so even though the Arithmophone features 29 different notes, there are still many potentially useful intervals that are not available directly from the keyboard. The modulator keys transpose the entire keyboard by a set ratio. This has the same effect as shifting a capo on a guitar neck: you can still play the same shapes and patterns, but they will sound in a different key.

The yellow modulator keys use the 2-ratio: in the center position, the keyboard is not transposed, in the left position every ratio is divided by 2 and in the right position every ratio is multiplied by 2. This modulator functions exactly like the octave down/up keys that can be found on most electronic keyboard instruments. The orange modulator uses the 3 ratio, octave reduced to 2/3 and 3/2. This results in a shift of a musical fifth. The red modulator uses the 5 ratio as 4/5 and 4/5, which is a musical major third, and finally the blue modulator uses the 7 ratio as 4/7 and 7/4, resulting in a shift of a septimal flat 7. Modulator keys can be combined, so for example by putting the orange modulator in the right position and the red modulator in the left position, the keyboard is transposed by 3/2 * 4/5, which is 6/5, a musical minor third. Altogether, the modulator keys expand the note selection of the Arithmophone keyboard to 179 distinct ratios, each of them available across a range of 5 octaves. for the sake of completeness, the illustration below shows all of these ratios, arranged in 3-5 grids around 1/49. 1/7, 1/1, 7/1 and 49/1.

Function keys

Finally, there are four more keys below the Arithmophone keyboard. The outer two are used for navigational purposes, switching in and out of fullscreen mode (MPE version), between full and simplified interface (WA version) and to a second screen with additional settings (both versions).

The inner keys can be used to sustain and release notes: while the red key is held down, any note that is played will be sustained even after its key is no longer directly held down. Sustained notes will remain lit up on the keyboard to indicate their status. Individual notes can be released by tapping them once more when the red key is not held down. The orange key will release all sustained keys at once. (This feature is available on the MPE version of the Arithmophone only, on the WA version these keys are used as shortcuts for changing sound parameters.)

I hope this article explains some of the workings of the Arithmophone keyboard and that it will encourage you to play around with it and experiment freely. If you have any feedback, corrections or suggestions, please let me know!