5 ms·

The scale generation algorithm described here is really interesting - it starts from the natural core of harmony (using rational relationships between frequenci

by pierrec 12y ago

The scale generation algorithm described here is really interesting - it starts from the natural core of harmony (using rational relationships between frequencies), and gives you the option to keep the resulting natural-tempered notes, or to adapt them to the modern-day 12-tone equal temperament (12TET) and its irratonal, twelvth-root frequencies. The question of whether 12TET is "natural" or not is an interesting one, and in my opinion, it was great for a few centuries, but at some point music will move beyond 12TET. Take a look at one of richest websites exploring this question [1], whose author has a different opinion from mine. As for skipping the 7th harmonic -- i'd say you shouldn't do it. If you scroll down [1] until you get to the "n-tone equal temperament" graph, you'll find one of the nicest ever justifications for 12TET. However, if you added more harmonics to that graph, you'd find that the 7th does not fall anywhere close the 12TET frequencies: that's why we're not used to hearing it, and it sounds the most alien to us. [1]: http://www.geocities.jp/imyfujita/wtcpage004.html http://www.geocities.jp/imyfujita/wtcpage004.html

TheOtherHobbes 12y ago
It's an oversimplification to suggest that scales are based on the harmonic series. Many instruments have slight non-linearities in their overtones, and intervals that sound consonant aren't usually mathematically perfect. E.g. pianos don't even have perfect octaves. The major and minor scales seem to be based on very specific ratio sets, not on the general overtone series. But 12TET has 12 tones for a very practical reason - instruments with more tones are difficult to build and damn near impossible to play, because human hands are only so wide and finger movements have limited precision. There were experiments with more tones per octave back in the middle ages, but the idea never caught on because the musical benefits aren't obvious. So 12TET is a 'good enough' compromise between flexibility and expressiveness. Performers on instruments with an infinite pitch range vary their pitches anyway. Intonation problems drive conductors insane, and the reality is that orchestral and choral pitches tend to drift slightly towards consonance and away from the nominal ET ratios. Of course, with a computer you can do what you like. Here's some recent microtonal music written with SuperCollider: http://composerprogrammer.com/pitchdeviations.html http://composerprogrammer.com/pitchdeviations.html
- pierrec 12y ago
  Yes, scales are not simply based on the harmonic series. What I would suggest, however, is that their emotional value lies within the rational relationships between their frequencies (and I think we agree on this). The physical limitation of instruments with fixed-pitch notes is one reason for the existence of 12TET, but I wouldn't say that it's the main one. The website I linked above gives a good idea as to the mathematical basis of 12TET, and he may go a bit far by calling it "God's temperament", but a logarithmic temperament that has so many close approximations of rational relationships is not just convenient - it's beautiful.
- theOnliest 12y ago
  > E.g. pianos don't even have perfect octaves. This isn't true (at least in theory). Octaves on piano are (or should be) exact, otherwise the 12-tone equal tempering doesn't work out. A4 is 440 Hz, A5 is 880 Hz, and A3 is 220 Hz (using the Acoustical Society of America designations, where middle C = C4). If the octaves weren't pure, the piano would be more in tune in the middle than it would be at either end, which usually isn't the case. > There were experiments with more tones per octave back in the middle ages, but the idea never caught on because the musical benefits aren't obvious. The history of scales and how they came to be in the Western classical system is long and complicated (too long for here, certainly), but I wanted to point out my favorite of these weird old instruments the archicembalo.[1] It was designed by one of the most out there theorists, Nicola Vicentino, in Ferrara, a hotbed of weird music at the time (see Carlo Gesualdo's "Moro lasso" [2], a really chromatic madrigal). 1. http://en.wikipedia.org/wiki/Archicembalo http://en.wikipedia.org/wiki/Archicembalo 2. http://www.youtube.com/watch?v=s_q3EJNUKis http://www.youtube.com/watch?v=s_q3EJNUKis (I'm ABD on a Music Theory PhD, so if anybody has more specific questions I'm happy to help!)
  - analog31 12y ago
    It's been my understanding that the tuning of a piano is "stretched" compared to perfect 2:1 octaves -- at least this is what I heard from one piano tuner. Some details are suggested by all-knowing Wikipedia; http://en.wikipedia.org/wiki/Piano_tuning http://en.wikipedia.org/wiki/Piano_tuning

by pierrec 12y ago

TheOtherHobbes 12y ago

It's an oversimplification to suggest that scales are based on the harmonic series. Many instruments have slight non-linearities in their overtones, and intervals that sound consonant aren't usually mathematically perfect. E.g. pianos don't even have perfect octaves. The major and minor scales seem to be based on very specific ratio sets, not on the general overtone series. But 12TET has 12 tones for a very practical reason - instruments with more tones are difficult to build and damn near impossible to play, because human hands are only so wide and finger movements have limited precision. There were experiments with more tones per octave back in the middle ages, but the idea never caught on because the musical benefits aren't obvious. So 12TET is a 'good enough' compromise between flexibility and expressiveness. Performers on instruments with an infinite pitch range vary their pitches anyway. Intonation problems drive conductors insane, and the reality is that orchestral and choral pitches tend to drift slightly towards consonance and away from the nominal ET ratios. Of course, with a computer you can do what you like. Here's some recent microtonal music written with SuperCollider: http://composerprogrammer.com/pitchdeviations.html http://composerprogrammer.com/pitchdeviations.html

pierrec 12y ago
Yes, scales are not simply based on the harmonic series. What I would suggest, however, is that their emotional value lies within the rational relationships between their frequencies (and I think we agree on this). The physical limitation of instruments with fixed-pitch notes is one reason for the existence of 12TET, but I wouldn't say that it's the main one. The website I linked above gives a good idea as to the mathematical basis of 12TET, and he may go a bit far by calling it "God's temperament", but a logarithmic temperament that has so many close approximations of rational relationships is not just convenient - it's beautiful.
theOnliest 12y ago
> E.g. pianos don't even have perfect octaves. This isn't true (at least in theory). Octaves on piano are (or should be) exact, otherwise the 12-tone equal tempering doesn't work out. A4 is 440 Hz, A5 is 880 Hz, and A3 is 220 Hz (using the Acoustical Society of America designations, where middle C = C4). If the octaves weren't pure, the piano would be more in tune in the middle than it would be at either end, which usually isn't the case. > There were experiments with more tones per octave back in the middle ages, but the idea never caught on because the musical benefits aren't obvious. The history of scales and how they came to be in the Western classical system is long and complicated (too long for here, certainly), but I wanted to point out my favorite of these weird old instruments the archicembalo.[1] It was designed by one of the most out there theorists, Nicola Vicentino, in Ferrara, a hotbed of weird music at the time (see Carlo Gesualdo's "Moro lasso" [2], a really chromatic madrigal). 1. http://en.wikipedia.org/wiki/Archicembalo http://en.wikipedia.org/wiki/Archicembalo 2. http://www.youtube.com/watch?v=s_q3EJNUKis http://www.youtube.com/watch?v=s_q3EJNUKis (I'm ABD on a Music Theory PhD, so if anybody has more specific questions I'm happy to help!)
- analog31 12y ago
  It's been my understanding that the tuning of a piano is "stretched" compared to perfect 2:1 octaves -- at least this is what I heard from one piano tuner. Some details are suggested by all-knowing Wikipedia; http://en.wikipedia.org/wiki/Piano_tuning http://en.wikipedia.org/wiki/Piano_tuning