### Timbre and Tuning

I was going to write something very profound and intellectual, about tuning systems, starting from the beginning to maybe arrive at some new very clever system which hasn't been used, or has fallen into disuse or whatever. Something with interesing intervals, interesting chords, at some time in the process of studying I was also imagining some most harmonious system. At a minimum I was hoping to put some personal mental order into the already existing, and extensive, information available about tuning systems in general.

And then i started thinking about chords, harmonic content and timbre, at which point it dawned on me that of course every instrument that is not a sinewave generator has a timbre which is characteristic and determined by the presence of the overtones of the played note. Of course we all know, but often forget, that the frequencies of the overtone series above a given fundamental do not correspond with the notes of a tuning system unless that tuning system is based on the overtone series above the same fundamental frequency. The overtone series as a basis for a chromatic tuning system is clearly not feasible as it is tied to a particular fundamental frequency. It does play an important role in music accompanied by a drone and that plays with the harmonics above that drone through modifications of timbre - harmonic singing, didgeridoo, jaw harp being some examples. In the discussion of tuning systems one deviates from the overtone series by making compromises that allow some sort of chromaticity - the possibility to change chords and tonalities following the cycle of fifths while maintaining a consistent minimal dissonance. Actually the discussion of tuning systems often begins with the Pythagorean system constructed only from the fundamental and first two overtones, the rest of the notes obtained by following the cycle of fifths. The cycle of fifths does not close after 12 steps however. Thus you need to correct, introduce commas, slightly modify intervals, go away from rational proportions for nice intervals all the way to the equal temperament where you need to introduce intervals determined by various roots of 2, the modern western scale built from 12th roots of 2.

After absorbing much of what is written about tuning systems I arrived at an impasse when it dawned on me that every note that you play on an instrument (that is not a sine wave generator), actually corresponds to a chord based on the overtone series - the timbre of the instrument. You already have a collection of notes playing and no matter how clever you want to be, some of them, maybe lots of them, will be in tension with your interesting interval creation ideas and tuning system. This is actually well known but it was new to me and maybe also to others who read this, so I will keep on going.

To make my thoughts more precise let's start with the overtone series. The overtone series is built up from all frequencies that are integral multiples of the fundamental frequency. These frequencies are present to varying degress in the sound of a musical instrument when it plays that particular frequency. Let's explore the intervals of the overtone series and see how they match with the notes of the 12 tone equal temperament (12TET) tuning (that of the modern pianoforte and of the major part of all music that we listen to today). This choice is for concreteness, the same comparison can be made with any tuning system, there is always some interference.

Below we will look at the successive intervals between the notes of the overtone series above a fixed fundamental frequency.

1st interval: Octave. A ratio of 1:2. The only interval common to all tuning systems.

2nd interval: Fifth. A ratio of 2:3. Common to various tuning systems based on simple ratios, but does NOT agree with the fifth that you get from a piano which is 2^(7/12) = 1.498 and not 1.5. As already noted it also does NOT agree with the cycle of fifths according to which twelve consecutive intervals of a fifth should be precisely 7 octaves.

Aside: It is said that the human ear can discern a difference between notes played consecutively corresponding to a frequency ratio of 81:80 - the synotic comma. When the two notes a played simultaneously we can discern interference for frequency ratios much closer to 1.

3rd interval: The complement of the second, a fourth. A ratio of 3:4. Complementary as a fifth followed by a fourth gives an octave. (2/3x3/4) = 1:2. This is slightly smaller than a piano fourth, just as the 2nd interval is slightly larger than a piano fifth.

4th interval: A slightly flat major third. A ratio of 4:5

5th interval: A slightly sharp minor third. A ratio of 5:6

6th interval: A slightly flat minor third. A ratio of 6:7

7th interval: A slightly sharp second. A ratio of 7:8

8th and 9th: divide the slightly flat major third asymmetrically 8:9 and 9:10

and things go on like this.

The harmonic content that creates the timbre of a musical instrument goes at least up to the 8th harmonic, but often goes much higher, but we can stop here as the point has been made. However you tune your instruments the tension and interference created between them when they play two different notes is determined obviously by the notes played, but also largely by the interaction between the two different overtone series (timbre) that they produce. Maybe that is what makes music with several instruments so beautiful, this intricate and unavoidable tension between harmony and interference.

I realise in hindsight that for anyone who has studied orchestration all of this is obvious, but for me it was a revelation that I wanted to share in my own personal way.

Thanks for listening.