Experimental 12-TET tuning systems

Equal tempered systems

The defining property of equal tempered tuning systems is that transposing a melody will preserve the ratio’s of the note’s frequencies. So let’s try and create an equal tempered system. Any tuning system is just a mapping from note numbers to frequncies, so let’s represent our equal tempered tuning system with a function T(n):

$T:$ Note number $\rightarrow$ Frequency(Hz)

Then the property of equal temperment is given by

$\displaystyle\frac{T(n)}{T(n+d)} = \displaystyle\frac{T(m)}{T(m+d)} \forall n,m,d \in \mathbb{Z}$

In less mathematical terms, this just translates to saying that the ratio of frequencies of an interval of size d is the same no matter which note you start from. So now choose m=1 and d=-1. Then substitute in, and we get:

$\displaystyle\frac{T(n)}{T(n-1)} = \displaystyle\frac{T(1)}{T(0)}$
Which gives:
$T(n)= \displaystyle\frac{T(1)}{T(0)} T(n-1)$
Hence, by induction, we have:
$T(n)= T(0)\left( \displaystyle\frac{T(1)}{T(0)} \right)^{n}$
Let
$\alpha = \displaystyle\frac{T(1)}{T(0)}, K = T(0)$

The above section isn’t really that important, the whole point is that all equal tempered tuning systems must be of the form:

$T(n) = K\alpha ^{n}$ where $\alpha$ is some real number, and K is some tuning constant usually set to 440Hz. We won’t focus on the value of K today because it’s not very consequencial.

So the only freedom we have is how we choose alpha, which represents how much we go up in frequency between consecutive notes. Standard 12TET chooses $\alpha = \sqrt[12]{2} \approx 1.05946309436$ . This has the nice property that octaves are exactly a 2:1 ratio. However, other intervals are slightly detuned. For example, 5ths are slightly flat at a ratio of 1:1.498… instead of 1:1.5. In fact, every interval except the octave is slightly out of tune. So what can we do about this in an equal tempered system? One idea might be to choose $\alpha = \displaystyle\frac{16}{15}$ , because that’s the exact interval a semi-tone should be. However, this quickly makes every interval VERY out of tune as we go up the scale.

7-12-TET System

So 12-TET picks the octave as it’s favourite interval and then goes from there. But what if we chose the 5th as our favourite interval. We still want 12 tones in this system, so that would mean dividing the 5th into 7 equal parts. Hence the name 7-12-TET. In practice this means that $\alpha = \sqrt[7]{1.5}$ . In this system we have sacrificed exact octaves for exact 5ths. Let’s do a quick comparison to 12-TET:

Interval	“Exact” ratio	12-TET	7-12-TET	Winner
Perfect unison	1	1	1
Minor second	1.066666667	1.059463094	1.059634023	7-12-TET
Major second	1.125	1.122462048	1.122824262	7-12-TET
Minor third	1.2	1.189207115	1.189782789	7-12-TET
Major third	1.25	1.25992105	1.260734323	12-TET
Perfect fourth	1.333333333	1.334839854	1.335916983	12-TET
Tritone	1.40625	1.414213562	1.415583086	12-TET
Perfect fifth	1.5	1.498307077	1.5	7-12-TET
Minor sixth	1.6	1.587401052	1.589451034	7-12-TET
Major sixth	1.666666667	1.681792831	1.684236393	12-TET
Minor seventh	1.777777778	1.781797436	1.784674184	12-TET
Major seventh	1.875	1.887748625	1.891101485	12-TET
Perfect octave	2	2	2.003875474	12-TET

Table showing various intervals and how close each tuning system is. The winner column is the key thing to look at.

4-12-TET System

Okay but 12-TET already had pretty good 5ths. We all know that it really struggles with major 3rds. So what if we calibrated it for that? This would mean $\alpha = \sqrt[4]{1.25}$ . Here’s another quick comparison with 12-TET:

Interval	“Exact” ratio	12-TET	4-12-TET	Winner
Perfect unison	1	1	1
Minor second	1.066666667	1.059463094	1.057371263	12-TET
Major second	1.125	1.122462048	1.118033989	12-TET
Minor third	1.2	1.189207115	1.182177011	12-TET
Major third	1.25	1.25992105	1.25	4-12-TET
Perfect fourth	1.333333333	1.334839854	1.321714079	12-TET
Tritone	1.40625	1.414213562	1.397542486	12-TET
Perfect fifth	1.5	1.498307077	1.477721264	12-TET
Minor sixth	1.6	1.587401052	1.5625	12-TET
Major sixth	1.666666667	1.681792831	1.652142599	4-12-TET
Minor seventh	1.777777778	1.781797436	1.746928107	12-TET
Major seventh	1.875	1.887748625	1.84715158	12-TET
Perfect octave	2	2	1.953125	12-TET

4-12-TET isn’t looking too hot…

Audio samples

So, what do these tuning systems actually sound like?
First some chords:

C -> F -> Dm -> Cmaj7 -> F. First up is normal 12-TET, then 7-12-TET and finally 4-12-TET

Now a little song:

Same order as above

In conclusion…

Not bad but also not great. Not having exact octaves means that it will only get worse as you go further from the bass notes. I do kind of like 7-12-TET tuning and maybe it could have some uses, but 4-12-TET is too out of tune for me. There are probably more ways to choose alpha to create a twelve tone equal tempered system, but that’s for another time.

Try it yourself!

I created my own plugin with JUCE to get these samples. If you want to play around with 7-12-TET or 4-12-TET then you can download my plugin here: https://github.com/AugsEU/Augs-Synth