Why 12 semitones (using math)

While I'm here, I was wondering if anyone would find this helpful (i do) its kind of like a falling 5th chord progression chart nested in the circle of fifths that I made. Its REALLY helpful with jazz improvization because you can switch keys easily

http://abiscus.com/music/MusicTheory/harm_prog_chart.pdf

-Nate

Curious why you credit Ptolemy with the idea of mathematical ratios between scale tones - most music history works credit Pythagoras with the idea, about 600 years earlier.

Although we don't have any surviving works by Pythagoras, Aristoxenus (500 years before Ptolemy) in his work "Elements of Harmony" criticized the mathematical approach to music that the Pythagoreans held - so Pythagoras is probably a better bet. That's further supported by the fact that other mathematical ideas credited to Pythagoras - like the triangle stuff, or the ratios of periodic orbits - show a fasination with numerical ratios.

If you take the ratio of a perfect fifth, 3:2, and apply it to any frequency 12 times (running all the way around the circle of fifths), you'll find that the frequency you get isn't seven octaves above your starting point - it's too high by about 1.3%. This mathematical discrepancy is still called the 'Pythagorean comma'.

really?? thats interesting. The stuff that I did was just me trying to find a pattern with the numbers.

So is the reason that we use 12 semitones due to the fact that this produces a very close 4th and 5th to the just scale (fraction of a percent off) or does this 'Pythagorean comma' come into play?

To add to Noteboat's comments, if you apply the logic of a semitone in this way it only works on an ascending scale - the anomalous size of a semitone means that C# and Db (or any other sharp/flat note) are not the same note, which lead to keyboards being built with up to 32 notes in one octave. Equal temperament gets round this by having each note 100 cents apart and only one sharp/flat note in between - hence the layout of the modern piano.

http://www.medieval.org has a lot of stuff about Pythagorean tuning and the comma, and explains a lot about why 12th and 13th century music sounds the way it does.

Best,

A :-)

Pythagoras' comma just shows that the relationship of 3:2 isn't dead on - it's an approximation like 3.14159 is for pi. In practical terms, it had no real affect, since the range required to find it - seven octaves - is pretty close to the entire range of human hearing (about 10 octaves)

The system known as Pythagorean tuning works by tuning 3:2 intervals - so octaves are actually slightly out of tune. This is still used in some types of music, mostly in the middle east - where they carried it out a bit further, and decided using 17 divisions to the octave would eliminate the comma. Safi ad-Din al-Urmawi is usually credited with that idea - he was a 13th century theorist, although several others around that time pursued the same line of thought and came up with 18 divisions instead.

'Just' intonation isn't quite Pythagorean - it recognizes that octaves must be perfect doubles, and that you've got to have 12 divisions. From there it uses low numbers (16 or less) to describe ratios.

Neither one was particularly important - most music was solo voice, fretless strings, or other instruments (like brass) where the natural progression of overtones fixed the pitches. (The reason the Arabs got into the division thing early in theory was because they had fretted lutes.)

Things really started to change with the introduction of harmony around 1000 years ago. The first solutions were to use different instruments - a horn player would pack several to be able to play in different keys. As music became more complex, it was pretty clear that something needed to be done about intonation.

Around the 16th century, folks started playing with the idea of even tempering - splitting up the ratio differences so that each note (except the octave) was off by the same amount. The first efforts are called well temperaments - as in Bach's 'Well Tempered Clavier' - but we didn't really settle on how to do an evenly tempered scale until the second half of the 1800s. Most of the delay was due to the fact we didn't have precise measuring tools for frequency yet.

Plenty of other systems besides x^1/12 have been proposed for tempering - and they're still being proposed. Charles Lucy suggested all fifths be 600+300/pi cents apart (an octave is 1200 cents) right around the time everyone else was starting to use our current standard. Just 20 years or so ago, the Bohlen-Pierce scale was proposed, which uses a 3:1 perfect 12th as its core, instead of a 2:1 octave.

But those are just how we handle the division. The main reason we use 12 tones is traditional, rather than mathematical. It fits the past music of our culture. If you pursue math as a means to a developing a scale, lots of folks take it in other directions - there are 31 tone, 53 tone, 55 tone, and 72 tone tempered scales that keep trying for perfect consonnance between all the note relationships.

Thak you, you've been most helpful!

I think it would be nice if, around the paragraph about how 12EDO fourths and fifths are very close to just ratios, you mentioned that a person who wanted to make thirds, for example, more in tune, might select a different division of the octave, like 31, although their fifths would be slightly less perfect. It's important to know why twelve is good, but it's equally important to know that twelve tones is only one choice, which may not suit all purposes. Also, it's traditional to express deviations from just intervals in cents, rather than as percentages, so you might wish to add another column with those values.

They should be as follows, working from your percentages, and rounded to the nearest tenth of a cent.
unison: 0.0
minor 2nd: 11.6
major 2nd: -4.0
minor 3rd: -15.5
major 3rd: 13.6
fourth: 1.9
tritone: 9.8
fifth: -1.9
minor 6th: -13.6
major 6th: 15.5
minor 7th: -17.4
major 7th: 11.7
8ve: 0.0

And all this time I thought it was just because we have 12 fingers and toes.

oh wait, that's just me. :shock: