Ramirez wrote: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.