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Sorry if it's some trivial/already answered question, but I could not find a suitable answer online.

I am a beginner at music theory, and learning through the chords and scales made me think "Why is A4 fixed as 440hz?", and also "Why divide the scales into 12 pitches even though it is not completely natural?" (since dividing into 12 pitches will make irrational approximate pitch differences, not perfect rational differences..).

So I wonder why the current musical scale system is so popular.

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  • "dividing into 12 pitches will make irrational approximate pitch differences" ...are you getting into tuning systems like equal temperment versus just intonation? Jan 23, 2020 at 18:24
  • that's a good question. if our music would have been for centenaries built on scales of 12 semitones or even quarter tones this other scale system would be "so popular". That in our western hemisphere the minor major system has become popular may be the result of the natural overtone series, the history (harmony and polyphony) and the dominance (cruelty) of western culture. The winner takes it all! Jan 23, 2020 at 20:12

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Those seem to be 2 different questions and the answer for both are really hard.

why A4 is fixed as 440hz

You might be surpried that A4 is not fixed as 440Hz! Some recordings use A 446Hz instead! (apparently it's a bit more commercial and some thing it has a brighter sound). For more info: https://en.wikipedia.org/wiki/A440_(pitch_standard)

When you have different musicians playing different instruments, you need to tune them all to each other so that the sounds go together nicely. Some instruments (like guitar) are easily tuned, but others like wind instruments are not that easy to adjust. At the end of the day, it makes sense to have everybody agreeing on a single pitch, and A440Hz was the agreed convention. It is just a convention, in the same way that we agree that 1h is made up of 60 minutes instead of 100.

"why divide the scales into 12 pitches even though it is not completely natural?" (since dividing into 12 pitches will make irrational approximate pitch differences, not perfect rational differences..).

The 12 tone scale is the accepted standard in Western music. Historically, other cultures have used scales made up of more than 12 notes (what we call "microtonal scales").

Now, I don't think that you can achieve perfect rational differences. Many phenomena in physics are logarithmic in nature. Even if you chose a different separation (like the 17-interval scale known as 17-TET) you would end up with irrational pitch differences.

At the core of our scale likes the octave, which corresponds to a double frequency. I once heard that given that our inner ear is shaped like a Nautilus, 2 frequencies of the same octave are aligned with each other in the same angle. Unfortunately, I don't have a source for that.

If we take A4 = 440, this means that A5 is 880, and A3 is 220.

Now, that interval is always a different number of Hertz, so no matter in how many intervals you divide it, you will end up with a logarithmic scale of some sort. To visualize this, imagine that we divide the scale in 6 tones instead (a really bad idea, but just humour me).

Visualize the fretboard of a guitar, and now remove every other fret. Still, by having only 6 intervals to an octave, you will end up with logarithmic/irrational intervals.

At the end of the day, it boils down to how our inner ear works, whereby we perceive logarithmic frequency differences to be "equal".

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  • I get the first answer, but yet I don't get the second one. If we get a suitable starting pitch (256hz, for example) and let the octave as times 4 per one octave and pitch per octave as 1, then we will have perfect rational (in terms of hertz) pitches. I heard that we are using 12 pitches (in Western culture) since Aristotle, but at the time it was rational fractions, and because of the convenience of extending notes recently we have changed into root 12 systems. Now my question was "why 12"? Is it just a convention or there is certain benefits of number 12. (I upvoted your answer anyway :) ) Jan 23, 2020 at 18:00
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    Which cultures use 17 notes? I know that 17edo is a pretty "popular" microtonal scale among microtonalists, but I haven't heard of its use in "world music".
    – awe lotta
    Jan 23, 2020 at 19:20
  • The fact that an octave is 2x the original frequency is not a "decision", but rather a consequence of how our ear hears the sounds. If you visualize the sine waves, all the "nulls" (Where the wave crosses the "0" mark) are aligned, so there is something inherently similar between A4 and A5.
    – mkorman
    Jan 24, 2020 at 9:30
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    Why break it down into 12 and not, say 14? That's another question altogether. The next obvious interval is 3x original frequency, which is a 12th (higher 5th). Let's bring it down 1 octave to play with more easily, and you get the 5th. If you start jumpling following the circle of 5ths, you get a pentatonic after 5 jumps, a diatonic after 7 and the chromatic after 12.
    – mkorman
    Jan 24, 2020 at 9:30
  • @awe Iotta - I actualy don't know of a culture that uses it! I read that some Eastern / Middle Eastern use scales that we considered microtonal, and I've heard that 17 TET is one of the most frequent, and I put 2 and 2 together. Bu tI don't know anyone for sure. I will edit my answer, thanks.
    – mkorman
    Jan 24, 2020 at 10:30
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The 12 equal changes in an octave is the result of equal tempered tuning system. You are correct that it is irrational but it's not the difference in pitch that is standardized but the ratio of consecutive half steps. This was done to make other aspects of just tuning go away. In particular if you work out the frequency ratios of the 2nd to 1 and the 3rd to 2nd (both whole steps) you get two different ratios, 9/8 for (1, 2) and 10/9 for (2, 3). So the whole steps are not quite equal. This means that (among other issues) what you play for an E in the key of C might be different than E in the key of D (assuming that "D" is really they same as the 2nd note of the C scale). The "notes" need slight altering from key to key to maintain these ratios. Also, one has the result that a sharp 2 might not be enharmonic with a flat 3 (not necessarily a bad thing). Our ears can hear more finely that 1/2 steps. We can hear 1/4 steps and even finer differences. The ratio approach in just tuning comes from the natural harmonics in a vibrating system such as a string. The 5th/1st is 3/2 and 3rd/1st = 5/4, from this all other ratios can be determined.

Granted the 12th root of 2 is impossible to exactly calculate and there is possible some dissonance between the ET tones and the harmonics of the instrument (and ear) but for all practical purposes this is not noticeable by most people (or we've been brainwashed to accept it). I have been told by people with perfect pitch that they can perceive this and consider 12TET "out of tune". But most of us don't experience this.

The 12TET major scale is very close to the Just major scale. A more basic question in my opinion is why we don't like 1/4 tones in the Western tradition. Turkish folk music has 1/4 tones and they make fretted instruments with a 1/4 tone fret here and there.

As for 440Hz, this has evolved over the centuries. It isn't always 440. But we set a standard convention so we are all on the same page. It's like choosing a definition for a meter, or kg at the international bureau of standards.

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A isn't necessarily fixed at 440Hz. In some parts of the World it's different, and years ago it wasn't even a fixed Hz. It's become an 'industry standard', where if a figure is given to it, anyone anywhere can tune to that standard, and it'll be fine for playing along. Bit like measurement systems work because most people recognise and use them.

The 12tet you ask about is a compromise. It's not exactly in tune for any key, but what it does is make it possible to play a 12tet tuned instrument in any key, and it won't sound too out-of-tune. Other tuning systems are better in tune, and sound more musical, but essentially work in one specific key. If all the playable notes are tuned to that one key, everything is hunky dory, but play in other keys, and pieces will sound like some notes from them are slightly off.

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    You should clarify that your point about “other tuning systems...essentially work in one specific key” is rather a point about equal-tempered vs unequal ones. Higher ET systems don't have the “essentially work in one specific key” problem either. Problem is just, 31-edo is the first one that's really better than 12-edo as a meantone temperament, and 31 is already rather too many steps to seem practical. Jan 23, 2020 at 18:06
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The current scale of 12 chromatic tone with equal temperament is based on a much older systems of a 7 tone diatonic scale ABCDEFG.

There is a lot of history involved with the development of harmony and tuning systems, but you can simplify (hopefully not too gross an oversimplification) it by saying: all tones ABCDEFG can be the starting tone of a scale (a tonic) provided you include a tone one half step below each (a leading tone.)

BC and EF already have that half step below relationship. You need to add five more tones to get half steps below the remaining tones. All together that makes 12 tones.

Music often moves through various tonic centers so all of those 12 tones is needed in practice. These changes of the tonic can be called modulation or transposition depending on the particulars of the music.

This leading tone view is a minimal explanation for the 12 chromatic tones. In actuality many other tone can be involved with modulations and transpositions.

So, the current 12 tone chromatic system is popular because it accommodates 7 tone diatonic scales and their modulation/transposition.

NOTE:

You can also see how both the 7 and 12 tone scales can be created from a series of perfect fifths:

F C G D A E B orders the 7 diatonic tones by perfect fifths, but it doesn't repeat.

Gb Db Ab Eb Bb F C G D A E B F# orders the 12 chromatic tones by perfect fifths, it repeats in a circle where F# and Gb are enharmonically the same tone.

That is a point about the historic development of music, but it's an interesting way to see the chromatic system as a kind of extension of the diatonic system.

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  • I am confused. How is it that B does not have a 5th?
    – user50691
    Jan 23, 2020 at 19:36
  • @ggcg, do you mean a fifth above the B after F C G D A E B? Jan 23, 2020 at 20:20
  • the point is simply to arrange the letters of the gamut in ascending fifths Jan 23, 2020 at 20:22
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    You seem to be implying that you cannot write a 5th above B (if I am misunderstanding sorry). But this tower of 5ths goes on forever in theory. There is nothing special about the equal tempered scale that allows this. Unless you mean that there is no unambiguous name for it in terms of letters and accidentals.
    – user50691
    Jan 23, 2020 at 20:30
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    it not that there isn't a perfect fifth above B - of course that would be F# - it just that F# isn't part of the basic gamut, it requires accidentals or you can place it the chromatic series Jan 23, 2020 at 20:52

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