SOUND ON SOUND

[Sam Spoons]

Well, Dm (the m2) in the key of C would be different to Dm (6m) in F. And we still haven't explored the variations in concert pitch over the years, Dm in A=440 would be the same as D#m for A=415 (commonly used when playing Baroque instruments)...

[merlyn]

I don’t see how one interval can be more ‘natural’ than another.

In this context I'm assuming 'natural' means 'derived from the harmonic series'.

With just intonation a perfect fifth, for example, has a frequency at a ratio of 3/2 of the root note.

So a just intonated perfect fifth is derived from the harmonic series. It is the third harmonic (multiply by three) brought down an octave (divide by two).

This is regardless of key. As such, the relationship between the pitches is the same in all keys, regardless of any absolute frequency values

Yes, but with just intonation there are more than twelve keys. For example F# is a different note from Gb and so a scale could be built on either. In tempered tuning or 12 tone equal temperament (12 TET) Gb and F# are the same note. This produces the circle of fifths :



It is possible to derive 12 notes by repeatedly using the ratio 3/2 and this is called Pythagorean tuning. But there is a problem. Powers of 3/2 never come out as a power of 2. Starting on C and going up 12 fifths should bring us back to the note C transposed 7 octaves up. But it doesn't.

(3/2)^12 = ~129.75
2^7 = 128

This is saying B# is a different note from C. In Pythagorean tuning the circle of fifths isn't a circle anymore -- it's a spiral of fifths. In my picture of history Bach took the spiral of fifths, heated it in a furnace, and hammered it into a circle. The gap between B# and C is called the Pythagorean comma, and it is this error that is distributed evenly across all notes in 12 TET.

[Murray B]

Merlyn,

Your post is a wonderful piece of work and explains the issues of intonation, intervals and the compromises of standard tunings beautifully. Thank you for taking the time and effort to compose it.

It also helps me justify to myself all the time I spend subtly changing the tuning of my guitar until I can make it sound as right as I can for the key I'm in :-) It's a form of slight OCD madness of course that usually get dropped as soon as I'm with a full band, but it works well when I working with the fiddle players probably cause they are playing with their ears rather than with frets.

[shufflebeat]

Beautifully expressed, Merlin, explains perfectly why I can't sing in tune and how it's not my fault.

[merlyn]

You're welcome. I'm glad it was helpful. :thumbup: