I recently renewed my fascination with how the powers of 3…
3, 9, 27, 81, 243...
relate to harmonics and scales.
I’ve revisited this subject several times since I first picked up an instrument and a book on music. This time I seem to have reached a nicely rounded summary of my thoughts. I’m not sure what I am ever going to do with this, but having worked through it, I thought I’d write it all out, and sticking it here in my learning log seems at least halfway appropriate.
As most of us know, if you double the frequency of a note, you get the same note an octave higher. This is known as the 1st overtone or the 2nd harmonic. Similarly, if you treble a note you get its fifth. This is the 2nd overtone or the 3rd harmonic.
For example, concert pitch of A4 at 440 Hz (A4) when doubled gives us A again at 880 Hz (A5); trebled, it gives us E at 1320 Hz (E6). The interesting point is that trebling moves us to a new note, rather than just the same note at a different pitch. So what happens if you compound the trebling, such as trebling the already-trebled E at 1320 Hz? The answer is you get a note at 3960 Hz, which happens to be B7.
The sequence A, E, B can be continued by repeated trebling…
A, E, B, F#, C#, G#, D#
Some of you may recognise this as a segment of the circle of fifths. We could continue in this fashion, trebling as we go, until we had completed the circle, but there is an easier way. Since we know that the relationship of a tonic to a fifth (treble the pitch) is equivalent to that of the fourth to the tonic, we can start applying the trebling rule in the opposite direction. Reading from right to left in this case, we get:
D#, A#, F, C, G, D, A
Since we are travelling in the opposite direction, we are dividing the pitch by three, rather than multiplying it. We end up in the same place, D#, the tritone of A, so the circle is complete:
A, E, B, F#, C#, G#, D#, A#, F, C, G, D, A
Using only powers of 3, we have been able to calculate exact pitches for all notes in the 12-note chromatic scale. This would be the moment to celebrate, but there is just one problem, we get two different pitches for D#. How is this possible?
To answer that I’d like to consider the circle of fifths in a less common but more universal form, taking out all references to actual notes, and dealing with only intervals or pitch ratios. Now the sequence is:
tonic, 5th, 2nd, 6th, 3rd, 7th, tritone, min 2nd, min 6th, min 3rd, min 7th, 4th, tonic
Or, rearranged with the tonic in the centre:
tritone, min 2nd, min 6th, min 3rd, min 7th, 4th, tonic, 5th, 2nd, 6th, 3rd, 7th, tritone
Now let’s look closely at the ratios we get by repeatedly trebling, or going through the powers of 3. Remember that the tonic, when trebled, gives us the fifth, which is the 2nd overtone (the 1st overtone being an octave):
tonic
tonic x 2 = 1st overtone = octave above
tonic x 3 = 2nd overtone = fifth above higher octave
This puts the fifth above the octave, but we are more familiar with a scale having the fifth between the tonic and the octave. This is easily accomplished by simply dividing the pitch of the fifth by two, which lowers it by an octave, without changing its role - that is, it is still the fifth. Now we have it in its expected position:
tonic
tonic x 3 / 2 = fifth
tonic x 2 = octave above
This step of dividing by 2 or a power of 2 can be applied to all the other compounded fifths, so that the ratio of pitches is reduced to something between 1 (the tonic) and 2 (the octave above). This gives us:
tonic
tonic x 9 / 8 = second
tonic x 81 / 64 = third
tonic x 729 / 512 = tritone
tonic x 3 / 2 = fifth
tonic x 27 / 32 = sixth
tonic x 243 / 128 = seventh
tonic x 2 = octave above
Note that all these ratios have integer powers of 2 or 3. It’s rather neat, I think.
Reversing absolutely everything to derive compounded fourths gives us:
tonic x 1 / 2 = octave below
tonic x 128 / 243 = minor second
tonic x 16 / 27 = minor third
tonic x 2 / 3 = fourth
tonic x 512 / 729 = tritone
tonic x 64 / 81 = minor sixth
tonic x 8 / 9 = minor seventh
tonic
Now we have all twelve pitches in the circle of fifths, but note that the tritone is calculated as both 729/512 (using fifths) and 512/729 (using fourths). This is not a mistake. Firstly, we are dealing with two different pitches:
tritone below tonic: 512 / 729
tritone above tonic: 729 / 512
Let’s equalise them by multiplying the lower tritone by 2:
1024 / 729 = 1.4047
729 / 512 = 1.4238
But still they disagree by approximately 2%. You may wonder that since the tritone is discordant anyway, does it matter? Well, yes. With each additional octave, the difference only increases, and it is not limited to tritones. Eventually, a strictly fifth-based approach to pitch ratios will lead to built-in biassed dissonance, where some intervals sound fine, but others do not. In recognition of this problem, the twelve-tone equal temperament (12TET) was devised. It may seem counter-intuitive, but the answer to uneven dissonance in the system of fifths is a system that has near universal dissonance, but also has the benefit of limiting its extremes.
Under 12TET, the octave is divided into twelve pitches whose ratios, in all cases, are powers of 2^(1/12), or roughly 1.0595. Yes, unfortunately decimals are inescapable under 12TET. The tritone incidentally, now works out at exactly the square root of 2, or 1.4142. Compare this to the two figures we derived by compounding fifths and fourths, and you will see the 12TET tritone is in the middle; a compromise, in other words, but a useful one.
To extend and summarise all this, I created a table in which pitches generated by three different methods are compared. The methods are compounding fifths, compounding fourths, and 12TET. The 12TET pitches are shown in red, down the centre of the table. The compounding fifths, extended using further powers of 3 until they fill all twelve tones in the circle, both above and below the tonic, are on the left. The compounded fourths, similarly extended, are on the right. Differences between each of these systems and 12TET are in grey columns. The cells with an orange background are seed values, all other cells are derived through the compounding process.
Line 14 contains the tonic (in bold face). All other lines relate to this one.
To the right are two supplementary tables where I use simple integer ratios to generate all the pitches - except for the tritone, which is an irrational number and therefore not amenable to this treatment.
- Blue, for descending intervals, starting with a denominator of 256 for the tonic, six lower pitches are precisely calculated using integers between 128 and 243. The ratios match those of the compound fifths in the main table.
- Green, for ascending intervals, starting with a denominator of 243 for the tonic, six higher pitches are precisely calculated using integers between 256 and 486. The ratios match those of the compound fourths in the main table.
I’m surprised that I’ve not seen intervals shown like this before, with a common denominator.
