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That's the first thing that the author states. I'll bet you know more music than me, but maybe that puts me more into the target audience. It's a very interest

That's the first thing that the author states. 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.

kazinator 8y ago
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.
- bradford 8y ago
  > The problem is that the twelve-tone system was designed for a very specific "use case" 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>
  - kazinator 8y ago
    Sorry, I made a mistake: I specifically wanted to make that remark about the equal temperament tuning, not twelve tones in general.
  - posterboy 8y ago
    > wavelengths of two notes form simple mathematical ratios, they have a stability that sounds nice O.o that is called resonance and it sounds harmonious.
  - yesenadam 8y ago
    Twelve is used because it has certain properties of divisibility that make life easier for the 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.
    - crazynick4 8y ago
      > 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.
      - yesenadam 8y ago
        Hmm that's not quite what I was saying - symmetry in scales in actually unpleasant, asymmetry pleasant. It seems that with these symmetrical scales/chords, the ear suffers vertigo, cognitive dissonance, because it can't hear a tonal centre. With e.g. a major scale, the asymmetry makes it obvious what key you're in.
        crazynick4 8y ago
- soundwave106 8y ago
  The specific use case for equal temperament is more or less how do make a fixed tuning instrument sound "good enough", without resorting to a very large amount of keys or buttons to adjust the tuning so the chords sound right. Although it was developed around Western music, I don't think that Western scales per se are necessarily the constraining factor. 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) 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.
elihu 8y ago
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.

kazinator 8y ago

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.

bradford 8y ago
> The problem is that the twelve-tone system was designed for a very specific "use case" 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>
- kazinator 8y ago
  Sorry, I made a mistake: I specifically wanted to make that remark about the equal temperament tuning, not twelve tones in general.
- posterboy 8y ago
  > wavelengths of two notes form simple mathematical ratios, they have a stability that sounds nice O.o that is called resonance and it sounds harmonious.
- yesenadam 8y ago
  Twelve is used because it has certain properties of divisibility that make life easier for the 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.
  - crazynick4 8y ago
    > 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.
    - yesenadam 8y ago
      Hmm that's not quite what I was saying - symmetry in scales in actually unpleasant, asymmetry pleasant. It seems that with these symmetrical scales/chords, the ear suffers vertigo, cognitive dissonance, because it can't hear a tonal centre. With e.g. a major scale, the asymmetry makes it obvious what key you're in.
      - crazynick4 8y ago
soundwave106 8y ago
The specific use case for equal temperament is more or less how do make a fixed tuning instrument sound "good enough", without resorting to a very large amount of keys or buttons to adjust the tuning so the chords sound right. Although it was developed around Western music, I don't think that Western scales per se are necessarily the constraining factor. 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) 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.

elihu 8y ago

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.