4 ms·

The explanation of the origin of major scale in the article is pure voodoo. Minor third is not a simple fraction - is that the reason to exclude it from the sca

The explanation of the origin of major scale in the article is pure voodoo. Minor third is not a simple fraction - is that the reason to exclude it from the scale? How do you explain minor scale then? Maybe it should be excluded, too? Here're my thoughts on the subject. For some reason no one can explain, Western music settled on a system of 12 tones with equal temperament, This system emerged as a result of long evolution of Western music, and experimentally proven to be very rich in possibilities. Scales used in Western music (of which jazz is a part of) are built on two simple principles: 1) interval between adjacent notes of the scale is either tone or semitone 2) there's no two semitones in a row. It's easy to check that all scales that satisfy these 2 rules are: major scale and its modes (7-note scales; 7 modes) melodic minor scale and its modes (7-note scales; 7 modes) diminished scale and its modes (8-note scales; 2 modes) whole-tone scale (6-note scale, single mode). (Whole tone scale is not used very often, except by T.Monk) But even after we "explain" scales, we need to figure out how to use them, what their role is, what the properties of each mode are. There's no hard science behind this, the properties just "emerge", and you have to experience them - theorizing is not of much help, math formulas don't explain anything, just lead to confusion. In short: you have to play AND think; thinking alone won't help. It's an experimental subject. Edit: forgot to say: scale is a very useful notion, but in some contexts, it's more convenient to think in terms of triads and interpolation. I know this all sounds hand-wavy, and it is! Unfortunately, without piano, it's impossible to to illustrate what it all means. The subject doesn't easily lend itself to verbalization.

rspeer 10y ago
> Minor third is not a simple fraction - is that the reason to exclude it from the scale? The minor third is a 6:5 ratio. Does that change your mind? Where do your tones and semitones come from? You've just rejected the explanation for why we have a 12-tone scale (it is a local optimum of tuning, closed under the operation of transposition, that satisfies lots of nice ratios), so what do you propose instead? Saying "no one can explain" is a cop-out. To follow up: Major and minor scales are 7-tone subsets of the 12-tone scale that were discovered first. They allow for many of the same possibilities, but they're not closed under transposition, which is why we now have the 12-tone scale that includes all of them.
- lohankin 10y ago
  Please read wikipedia article on temperament. 7-note scales were "discovered" long before equal temperament was introduced. Equal temperament is a relatively recent invention, initially was very controversial, and remained so for a long time. In other cultures, there's still a variety of scales and temperaments that have nothing to do with 7-note scales at all. (Jazz uses variety of scales, too - some even with 2 semitones in a row). There's only one thing people seem to agree on: the role of fifth (3/2).
  - rspeer 10y ago
    This is exactly what I'm telling you. Why are you being condescending to me about it? You're the one claiming that the semitones of the 12-tone scale are some sort of fundamental axiom. Why didn't you know where minor thirds come from? Why do you seem to be denying that simple harmonic ratios are where harmony comes from? (EDIT: he's not, it's fine) Why are you promoting a theory of music with absurd axioms, which manage to explain the octatonic scale (which almost nobody uses) better than the pentatonic scale (which almost everybody uses)? That's not a sign of a good theory.
    - lohankin 10y ago
      Please read the part of the article starting with "The twelfth root of two may be irrational, but it turns out to almost create several nice ratios". The author excludes minor third from the set of "nice ratios". I referred to this while calling it a voodoo.
      - sverige 10y ago
        Dude, if you can't hear it, it doesn't matter. You'll never listen to Coletrane's "Night Train" and have your head explode when you realize he's jumping from Dorian to Mixolodian to whatever the fuck was that!?
      - rspeer 10y ago
        A simplified explanation is not voodoo. Yes, he chose to list only the intervals in the major scale. Showing the intervals in the major and minor scale at the same time would have been very confusing. I understand now that you weren't trying to say that minor thirds were a hole in the theory he presented, but that they were a hole in the way he presented it. I took issue when I thought you were criticizing the theory itself, because what he presented is a simplified view of generally accepted foundations of harmony. I apologize for mis-characterizing your position there. There is more that could have been said -- and it sounds like we agree on what could have been said -- but I think it's not necessary to go into all the details of the construction of scales, including temperament, just to write a blog post. Describing temperament accurately, without handwaving, requires a book.
haberman 10y ago
I agree very much with your post's thesis (you have to play AND think; thinking alone won't help. It's an experimental subject.) Just noticed one thing: > For some reason no one can explain, Western music settled on a system of 12 tones with equal temperament It doesn't seems surprising to me. If you start from a pitch and go upwards in both octaves and perfect fifths (2:1 and 3:2, the two most fundamental intervals), the perfect fifth sequence will land on 11 distinct tones before (nearly) meeting the octave sequence. Mathematically, (3/2)^12 ≈ 2^7. So 12 semitones works out nicely because you can follow perfect fifths out in any direction as far as you want and never go outside the set of semitones. And most of the small-ratio'd intervals can be represented with pairs of notes inside this set.
- lohankin 10y ago
  Interesting idea indeed. I need to think about it. Edit after thinking: still, it doesn't explain the number 12 IMO. It could be 17 or something else. Probably, it's a long chain of coincidences at play: Western music settled on 7-note scales long time ago (long before equal temperament was invented), and we should start looking for explanations from here. Another edit: one of the important coincidences is that number 12 makes possible the existence of diminished scale, which serves as a "universal glue" due to 2 tritones. (There's not enough space here to elaborate, but you probably know what I mean). And maybe tritone itself is one of factors leading to number 12.
  - fenomas 10y ago
    Quora (of all places) has some really good articles about this. Check for example the second answer in this page: https://www.quora.com/Why-are-there-only-12-pitch-notes-C-C-B-in-the-world https://www.quora.com/Why-are-there-only-12-pitch-notes-C-C-...
  - jacobolus 10y ago
    The number 12 is just a coincidence, that 3^12 = 531441 ≈ 524288 = 2^19 This means that 3/2 ≈ 2^(7/12) [accurate to about 0.1%] And also 4/3 ≈ 2^(5/12) [also accurate to about 0.1%] And also 9/8 = (3/2) / (4/3) ≈ 2^(2/12) [accurate to about 0.2%; putting two factors of 3 in makes the approximation only half as good] You also get another nice coincidence, that 5^3 = 125 ≈ 128 = 2^7 This means that 5/4 ≈ 2^(4/12) [accurate to about 1%] There are some people who have written music in a 41-note equal tempered scale, because then you get an even better approximation to the 3/2 ratio: 3^41 = 36472996377170786403 ≈ 36893488147419103232 = 2^65 (3/2) ≈ 2^(24/41) [accurate to about 0.03%]
midgetjones 10y ago
I think you're being unfair on the whole tone there, it's the intro to 'Take the A Train', and people will often substitute it in a dominant chord when improvising.

rspeer 10y ago

> Minor third is not a simple fraction - is that the reason to exclude it from the scale? The minor third is a 6:5 ratio. Does that change your mind? Where do your tones and semitones come from? You've just rejected the explanation for why we have a 12-tone scale (it is a local optimum of tuning, closed under the operation of transposition, that satisfies lots of nice ratios), so what do you propose instead? Saying "no one can explain" is a cop-out. To follow up: Major and minor scales are 7-tone subsets of the 12-tone scale that were discovered first. They allow for many of the same possibilities, but they're not closed under transposition, which is why we now have the 12-tone scale that includes all of them.

lohankin 10y ago
Please read wikipedia article on temperament. 7-note scales were "discovered" long before equal temperament was introduced. Equal temperament is a relatively recent invention, initially was very controversial, and remained so for a long time. In other cultures, there's still a variety of scales and temperaments that have nothing to do with 7-note scales at all. (Jazz uses variety of scales, too - some even with 2 semitones in a row). There's only one thing people seem to agree on: the role of fifth (3/2).
- rspeer 10y ago
  This is exactly what I'm telling you. Why are you being condescending to me about it? You're the one claiming that the semitones of the 12-tone scale are some sort of fundamental axiom. Why didn't you know where minor thirds come from? Why do you seem to be denying that simple harmonic ratios are where harmony comes from? (EDIT: he's not, it's fine) Why are you promoting a theory of music with absurd axioms, which manage to explain the octatonic scale (which almost nobody uses) better than the pentatonic scale (which almost everybody uses)? That's not a sign of a good theory.
  - lohankin 10y ago
    Please read the part of the article starting with "The twelfth root of two may be irrational, but it turns out to almost create several nice ratios". The author excludes minor third from the set of "nice ratios". I referred to this while calling it a voodoo.
    - sverige 10y ago
      Dude, if you can't hear it, it doesn't matter. You'll never listen to Coletrane's "Night Train" and have your head explode when you realize he's jumping from Dorian to Mixolodian to whatever the fuck was that!?
    - rspeer 10y ago
      A simplified explanation is not voodoo. Yes, he chose to list only the intervals in the major scale. Showing the intervals in the major and minor scale at the same time would have been very confusing. I understand now that you weren't trying to say that minor thirds were a hole in the theory he presented, but that they were a hole in the way he presented it. I took issue when I thought you were criticizing the theory itself, because what he presented is a simplified view of generally accepted foundations of harmony. I apologize for mis-characterizing your position there. There is more that could have been said -- and it sounds like we agree on what could have been said -- but I think it's not necessary to go into all the details of the construction of scales, including temperament, just to write a blog post. Describing temperament accurately, without handwaving, requires a book.

haberman 10y ago

I agree very much with your post's thesis (you have to play AND think; thinking alone won't help. It's an experimental subject.) Just noticed one thing: > For some reason no one can explain, Western music settled on a system of 12 tones with equal temperament It doesn't seems surprising to me. If you start from a pitch and go upwards in both octaves and perfect fifths (2:1 and 3:2, the two most fundamental intervals), the perfect fifth sequence will land on 11 distinct tones before (nearly) meeting the octave sequence. Mathematically, (3/2)^12 ≈ 2^7. So 12 semitones works out nicely because you can follow perfect fifths out in any direction as far as you want and never go outside the set of semitones. And most of the small-ratio'd intervals can be represented with pairs of notes inside this set.

lohankin 10y ago
Interesting idea indeed. I need to think about it. Edit after thinking: still, it doesn't explain the number 12 IMO. It could be 17 or something else. Probably, it's a long chain of coincidences at play: Western music settled on 7-note scales long time ago (long before equal temperament was invented), and we should start looking for explanations from here. Another edit: one of the important coincidences is that number 12 makes possible the existence of diminished scale, which serves as a "universal glue" due to 2 tritones. (There's not enough space here to elaborate, but you probably know what I mean). And maybe tritone itself is one of factors leading to number 12.
- fenomas 10y ago
  Quora (of all places) has some really good articles about this. Check for example the second answer in this page: https://www.quora.com/Why-are-there-only-12-pitch-notes-C-C-B-in-the-world https://www.quora.com/Why-are-there-only-12-pitch-notes-C-C-...
- jacobolus 10y ago
  The number 12 is just a coincidence, that 3^12 = 531441 ≈ 524288 = 2^19 This means that 3/2 ≈ 2^(7/12) [accurate to about 0.1%] And also 4/3 ≈ 2^(5/12) [also accurate to about 0.1%] And also 9/8 = (3/2) / (4/3) ≈ 2^(2/12) [accurate to about 0.2%; putting two factors of 3 in makes the approximation only half as good] You also get another nice coincidence, that 5^3 = 125 ≈ 128 = 2^7 This means that 5/4 ≈ 2^(4/12) [accurate to about 1%] There are some people who have written music in a 41-note equal tempered scale, because then you get an even better approximation to the 3/2 ratio: 3^41 = 36472996377170786403 ≈ 36893488147419103232 = 2^65 (3/2) ≈ 2^(24/41) [accurate to about 0.03%]

midgetjones 10y ago

I think you're being unfair on the whole tone there, it's the intro to 'Take the A Train', and people will often substitute it in a dominant chord when improvising.