The problem with this is is that it accepts the twelve-tone equal temperament system as a sort of absolute frame of reference, rather than the compromise/approximation that it is.
Musical scales are not just check-box menu choices out of twelve-tone equal temperament.
When I first learned about music theory, it surprised me how common this is. I liked the idea of scales that made certain intervals sound nice in terms of simple fractions, and wrote a computer program to search for them. I found an interesting scale and posted it to the music stackexchange: https://imgur.com/a/iBwxC4T
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 a tuning is a set of intervals which an instrument is configured to play. A tuning is also a temperament if it is a compromise of another tuning in order to make the configuration technically feasible (or for other convenience reasons). A scale is then some combination of notes within a tuning or temperament?
I don't know, it seemed like they were more concerned with nitpicking words than answering your actual question, which was interesting and not hard to understand.
Exactly. I would like to see this done using integer ratios to define the frequency intervals.
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.
The problem is that the twelve-tone system was designed for a very specific "use case", in the context of a certain kind of music: classical/Baroque revolving around major and minor scales, modulating around the cycle of fifths. For anything else, it is basically a misuse. Mass produced equal temperament instruments are used all over the world and basically turn the local music into shlocky pop.
The issue I have is that, by ignoring the just scales that equal temperament is an approximation of, you ignore the fundamental mathematical foundations of tonal music. Without that, there are a lot of things you can't say about equal temperament, and the things that remain are kind of superficial.
I tried listening to a 15-tone ET piano piece ones. Sure, maybe I'm not worthy, and too limited in my musical understanding (although I have been a musicians since the age of 6 and i'm now 43), but it just sounded like a piano out of tune. 12-tet is more than enough to experiment with for the rest of my life
Part of it could be due to conditioning but part of it is due to the fact that 15 TET is just another type of compromise, one that actually puts the perfect fifth even more out of tune than 12 TET, even though some of the other intervals are less tempered.
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.
WOW. As a former Music Theory 2 and Music Theory 3 student in college this is awesome. I am also very disappointed in myself for not trying to do something like this and failing earlier in my life.
There is really a lot more that could be gained from this work. I am sensing someone's PhD project.
Nice work! If I could make one suggestion it would be to use hex instead of decimal. It would make the numbers easier to remember (shorter) and I think it would be easier to recognize patterns (Ie. whole scale is 0x333 and major scale 0xAB3).
Nearly jumped out of my seat when I saw this! Look at how similar our approaches are: http://welliam.github.io/molts/ (apologies for how poorly written it is, it's been a few years!)
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.
It's been awhile since I've had the chance to dig into more advanced music theory but this is such an interesting framing of all of the possibilities present in both named and unnamed scales and the relationship of notes. The idea to lay it out mathematically like this and how you frame your overall thinking about scales is great. Bach would appreciate this work.
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.
Absolutely amazing. I'm a coder and a musician and I've been complaining to myself about a certain tendency to repetition in my compositions lately, this is amazing stuff to break patterns.
Readit News
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
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.
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.
http://tones.wolfram.com
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.