A new tonality

(2022) Sense over science
Our western tonal system, our music, has some serious flaws, many of which are the result of decisions made by theorists centuries ago. Most of us are unaware of the consequences and the negative impact they have on our music to this day: what we hear is not how it should or could sound.

With modern digital techniques we can now change that, and we should. Beter still, there is a whole new musical world waiting for us to discover, which could augment and evolve our music and that of generations to come.

Equal Temperament has ruined our music
When we look at the most basic form of a tone, a sine wave, we see that is has a base frequency: the number of times the wave fluctuates per second. In the western world, we generally take the A as our basic note, at 440 Herz, meaning the waveform, or actually the air, fluctuates 440 times per second and when it reaches our ear, we think we hear a tone which we call A.

Generally a tone not only consists of a base frequency but of many other frequencies on top of this frequency as well. These are called harmonics. They are what defines the character of the note, and why for instance an A on a piano sounds like a piano and not like a flute.

This is a natural, physical phenomenon, it is very logical and straight forward: harmonic frequencies are devisions and multiplications of the base frequency (root). In sync, and aligned with other notes and frequencies, this forms what we call harmony.

When we play a second note (interval), with a frequency not quite in range with the first, harmonics clash and cause a beat or pulsation.

When we look at the natural devision of frequencies, and build a 12 tone scale based on an A of 440 Hz, you will get this just chromatic scale of C:

TonicCT264 Hz
Minor secondDb(16/15)T281,6 Hz
Major secondD(9/8)T297 Hz
Minor thirdEb(6/5)T316,8 Hz
Major thirdE(5/4)T330 Hz
FourthF(4/3)T352 Hz
Diminished fifthGb(45/32)T371,1 Hz
FifthG(3/2)T396 Hz
Minor sixthAb(8/5)T422,4 Hz
Major sixthA(5/3)T440 Hz
Minor seventhBb(9/5)T475,2 Hz
Major seventhB(15/8)T495 Hz
OctaveC2T528 Hz

With these notes, we can form the 8-note scale of C major we are so familiar with: C – D – E – F – G – A – B – C.
The notes making up this scale are called diatonic. Because the notes have pure, natural frequencies, this is called a just scale.

As long as we play in this key, there is no problem. However, when we transpose to another key, or add notes or intervals which do not originally belong to this key, we will hear that they are out of tune.

When we play two major triads G and C in the just key of C, and see them as dominant and tonic in the key of C (V-I), their frequencies should be:

Dsecond(9/8)T297 Hz
Bseventh(15/8)T495 Hz
Gfifth(3/2)T396 Hz
Ethird(5/4)T330 Hz
CtonicT264 Hz
Gfifth(3/2)T396 Hz

As we can see, these frequencies all fit within the frequencies of our just key of C. This will sound perfect.

However, if we were to transpose to D, play two major triads A-D and see them as dominant and tonic in the key of D (V-I), their frequencies should be:

Esecond(9/8)T334,1 Hz
C#seventh(15/8)T278,4 Hz
Afifth(3/2)T445,5 Hz
F#third(5/4)T371,3 Hz
DtonicT297 Hz
Afifth(3/2)T445,5 Hz

As we can see, when we play these triads in our just key of C, only the D and the F# would fit, all others will sound out of tune.
What’s more, when we transpose to D, its frequency relative to A should arguably be 293,3 Hz, not 297 Hz. The right frequencies should therefor be:

Esecond(9/8)T330 Hz
C#seventh(15/8)T275 Hz
Afifth(3/2)T440 Hz
F#third(5/4)T366,7 Hz
DtonicT293,3 Hz
Afifth(3/2)T440 Hz

Which ever way you look at it, most transposed notes do not fit in one just key. Remember that the harmonics of notes, intervals and chords depend on the basic frequencies and interact with each other, determining the overall sound and harmony. These chords will not sound the way they could but less beautiful at best or more likely horribly out of tune.

If we want to use pure scales, which will give us the most characteristic, natural sounds, we will have to change scale whenever we change key. This means that if we want to play in the key of C, we will need to tune our instrument accordingly. If we want to transpose to another key, we will have to re-tune the entire instrument. Or bring a second one. And another one. And another. Both solutions are unpractical, and so were most others people came up with.

Bad solution
As of the 17th century, after centuries of struggling with and fighting over opinions and solutions and thanks to mathematical progress, the powers that were forced upon the world not a musical but a scientific solution, the equal temperament (ET).

In stead of using the natural frequencies as a starting point, the octave was to be divided into 12 equal parts. This decision still affects our music today, which since then has been and still is out of tune:

TonicC264 Hz261,1 Hz
SecondD297 Hz293,7 Hz
Major thirdE330 Hz329,6 Hz
FourthF352 Hz349,2 Hz
FifthG396 Hz392 Hz
SixthA440Hz440 Hz
SeventhB495 Hz493,9 Hz
ActoveC528 Hz523,2 Hz

Notice how messy the ET frequencies are compared to the just frequencies.
These equally divided frequencies of course have a major effect on all intervals and chords, most notably the third which in ET is too wide and the fifth which is too small.

Example: C, F and G triad, first in ET, then in just key

At first sight it may seem like a sensible solution to divide everything up evenly, but it is far from that. To avoid some notes sounding very wrong some of the time, now all notes sound a bit wrong all of the time.

In doing so, all characteristics of notes and keys vanished, every key now sounds the same. Harmonics no longer match, resonance is disturbed, and every note is out of tune, always.

Granted, finding a solution is not easy, but there was another reason for ET: to standardize instruments in order to produce them cheaper, so more people could buy one. It’s quantity over quality. Capitalism avant la lettre.

More decisions to question: why twelve?
Our octave was divided into twelve notes. Many other cultures use more than these twelve, for good reason, and funnily enough, major music genres like blues or jazz are most know for the use of notes outside these twelve, which are then called blue notes, or blues notes.

One good reason for a wider than twelve division is the fact that we are forced to enharmonise. There is a difference between chromatic and diatonic halftones — in general, a C# is a fraction lower than a Db — but we cannot make this difference on most instruments: on a piano, the C# and the Db have the same key.

Example: the difference between D# and Eb, alone and in Cm

My brave new world: 55/12/Just
With modern day computers and digital musical instruments and interfaces, new possibilities arise. We can now play and hear music with the proper frequencies. But what’s even more interesting: a vast new musical world is opening up.

Searching for a way to explore this unknown world of micro-tuning, I’ve created a tonal system based on pure scales, with the octave divided into 55 in stead of 12 notes.

I am not the first one to use 55 notes, several theorists since the 18th century have done so, for instance Pier Francesco Tosi (1654-1732), who also differentiates between chromatic and diatonic half notes by using 4 steps for chromatic and 5 for diatonic half notes, making a whole note 9 steps. In doing so, C-C# is 4 steps, C-Db 5 steps, C#-D 5 steps, Db-D 4 steps. This makes 55 notes together:

I use 12 different scales of 55 notes, all starting on a base note derived from the just scale of A. This means that my 55 tone system does not consist of 55 equal steps, but frequencies and steps depend on the key you are in at any moment in time.

It also means that I can choose at any time whether I want to tune an interval or chord as just or otherwise, or change key. I have either 4 or 5 steps for a half note and 9 steps for a whole note at my disposal.

I now have not 12, but several hundred notes available per octave. One could compare this to making a painting with hundreds of colors in stead of 12.

Gefördert durch die Beauftragte der Bundesregierung für Kultur und Medien im Programm NEUSTART KULTUR Modul D – Digitale Vermittlungsformate