A study of musical scales (2017)
ianring.com
ianring.com
Musical scales are not just check-box menu choices out of twelve-tone equal temperament.
It caused a long discussion between the people there, and I learned a lot, but my post was eventually deleted. Somebody posted a follow-up question trying to figure out what the terms were really supposed to mean, but it doesn't seem to me they ever managed to agree: https://music.stackexchange.com/questions/66620
I think that a scale and its temperaments are in addition to being an artistic choice, also a technology. Prior to electronic music and professional technicians, a tuning system had to be something that a musician could carry out themselves, possibly quite frequently. The 12 tone system is the simplest consonant system, possibly making it the easiest and quickest to tune.
You can still use the octave as a 'start' and 'end' point for the scale to make things simple (although there's no axiom anywhere that says this has to be the case, it's just a good candidate because it's the simplest and most easily recognizable interval, aside from the unison).
Since you'd be dealing with a potentially infinite number of intervals, you can start with the 'simplest' 20 or so (using lowest possible integer values to create the distinct ratios). Some would argue that the simplest 40 can be considered musical although that would get you into some seriously dissonant territory, not to mention the amount of possibilities you would have available at that point.
I'll bet you know more music than me, but maybe that puts me more into the target audience. It's a very interesting way to explore essentially the whole of western music. I did some similar "modeling" when I was learning, too, mostly because it's an effective way for me to commit things to memory. I remember struggling with using tones/scale degrees or intervals as a basis, just as the author describes.
It might not be universal and suitable for high-level academic use, but I don't think there's a problem with it.
Certainly just intonation sounds more pure on a fixed tuning instrument if you play within the constraints of the tuning ratios chosen. Step outside those constraints, and the sound is usually pretty awful.
12TET also has some "musical usefulness" on its own. The system introduced a sort of tonal ambiguity in modulation -- the commas that normally result in a more just / Pythagorean type style (https://en.wikipedia.org/wiki/Comma_(music)) get "blurred", and this is a characteristic musicians can exploit. Furthermore, in my experience, for sounds with lots of overtones (sawtooth synth waves, distorted guitars, etc.), that slightly detuned third in a 12TET trichord ends up sounding "thicker" for a lack of a better word, with the detuned beating more adding to the texture. This is best heard on a synthesizer where it's easy to flip between two tunings: to me, that, say, 1980s "power chord sawtooth stab" type of sound just doesn't sound as "thick" in just tuning, in my opinion. (In fact, when programming a synthesizer, people often slightly detune oscillators for this exact same reason!)
Nonetheless, it would be interesting if more "adaptive" type just intonation type systems came out that more approximate what non-fixed pitch instruments do (adjusting the notes to a "correct" your tuning during any modulation). Pure sounding triads also are useful, and you really can't do it super exact with 12TET, it's always "good enough". :) Hermode tuning is the only major one I know of, it is implemented in some synthesizers and DAWs. I haven't tried it yet -- my bet is you still would be a bit constrained (no dramatic 20th century classical type of chromatic modulation or the like), but you'll gain some freedom to modulate compared to a fixed just tuning setting.
This isn't really correct though. Twelve is used because it has certain properties of divisibility that make life easier for the musician.
When the wavelengths of two notes form simple mathematical ratios, they have a stability that sounds nice, and this makes it easier to construct intervals and chords.
Wouldn't choosing a non-twelve division be more of a limit?
<disclaimer, I'm merely an amateur musician>
Not true. It's used because (in short) when you try to construct a scale from octaves and fifths, finding a certain number of octaves that are close to another number of fifths (which comes to finding continued fraction approximants to 117/200, i.e. log 3/log 2 - 1) there are very few contenders, and only 12 notes per octave has neither too few nor too many for our tastes. It's based on the coincidence that 7 octaves is very nearly 12 fifths, i.e. 2^7~(3/2)^12.
Also, the idea that symmetry sounds nice is intuitively appealing, but nothing sounds better to us, more pleasing, than the (highly irregular) major scale, or scarier, more ominous than diminished and augmented chords - musically, a square and equilateral triangle. Or more disorienting than a whole tone scale - a hexagon.
Yes, the perceived consonance has less to do with the simplicity/symmetry of the concept used to construct the scale and more to do with the 'simplicity' of the denominators of the ratios that make up the intervals. The smaller (simpler) the denominators involved, the more pleasing/consonant the sound will be (provided that they're integer ratios). Frequency ratios like 3/2 and 4/3 are among the simplest possible and are both present in the major scale but absent in the latter two examples.
The frequency ratios that they represent, 7/5 and 6/5, respectively, don't lend themselves to the same type of symmetry. If you have a tritone, 7/5, you would need a 10/7 ratio to make the octave, which would be a different interval (although close).
With a minor third (6/5), if you were to stack the intervals starting from let's say 500 Hz, you would go to 600, 720, 864, and 1036.8, and you'd be off 36.8 Hz from the octave.
O.o that is called resonance and it sounds harmonious.
Having said that, 15-tet is pretty rancid. 22-tet can be much more consonant.
Try this, which is in just intonation:
https://www.youtube.com/watch?v=1uZUQqOLyPQ
It's contemporary but still sort-of accessible for many people.
I'm sure you've heard about this: https://en.wikipedia.org/wiki/Just_intonation
Most people brush off the concept because of a lack of accessible music, and because a lot of musicians writing the music, well, let's just say they lack a sense of popular sentiment. I don't think it discredits the theory, which is based in physics. Potentially, tuning intervals by ratios can be used to write the same style of music we are used to (without tempered intervals) just as well as providing harmonies that we don't have access to in our prevalent (Western) tuning system.
There is really a lot more that could be gained from this work. I am sensing someone's PhD project.
Anyway, pretty cool generator.
Not that it's a super novel concept, but we both used this for the modes of limited transposition. For me it was an efficient way of generating MOLTs for any equal temperament scale.
Also reminds me a bit of how folks like Jacob Collier think about harmonies and new ways to arrive at different notes. The more you can internalize these kinds of mathematics, the more you can improvise in new and different ways.
Well done! Really enjoyed this.
I should go back and learn to play a musical instrument