The beef that I have with you is on the construction of the major scale. You revealed the ratios that gave rise to the scale, but two of those notes aren't like the others. Here's a handy-dandy chart to reference (See also: your book):
| Just Intonation | ||||||||
| The Ratios you gave: | 1/1 | 9/8 | 5/4 | 4/3 | 3/2 | 5/3 | 15/8 | 2/1 |
| Fancy Schmancy: | Tonic | Super- tonic | mediant | Sub- dominant | Dominant | Sub- mediant | Leading Tone | Octave Tonic |
| Sound of Music: | do | re | mi | fa | so | la | ti | do |
| Harmonic: | 1st | 9th | 5th | ?th | 3rd | ?th | 15th | 2nd |
| Equal tempered bastardization: | 261.63 | 293.66 | 329.63 | 349.23 | 392.00 | 440 | 493.88 | 523.25 |
| Their Names: | C | D | E | F | G | A | B | C |
You know the notes I'm talking about. The Subdominant and the Submediant. They are the only ones that don't approximate a harmonic of the tonic. In a way, that sucks. You can pluck a string and tap a node and play any note except for the subs. I started playing around with alternatives and something nifty popped up: The ratio of F to C (349.23HZ/261.63HZ = 1.3348...) comes close not just to the Subdominant (4/3 = 1.3333...) but also to three octaves below the 11th harmonic (11/8 = 1.375).
Now just hold on. Hear me out. I know what your thinking, "But jeez Brian, 1.3348 comes so much closer to 1.3333 than 1.375 does. It's obvious which route is superior!" Well, I'm not really debating that. What I'm leading up to is AN ALTERNATE MAJOR SCALE... or at least an interesting "runner up".
Before we move on, let's talk a little about the submediant. The pure submediant ratio checks in at 5/3, which approximatley equals 1.66666. If you take the ratio of A to C (440HZ/261.63HZ) you get the heavenly number 1.6817. The closest harmonic that comes to that is two octaves below the 7th harmonic. It's ratio is 7/4 a.k.a. 14/8 and comes out in decimal form as 1.75. Now, it's obvious that the 1.75 isn't a very good approximation for 1.66666, but at this point we've stopped trying to copy the major scale and are forging on to uncharted territory. Well... here's the chart, so I guess it's charted now:
| Just Intonation - Part II | ||||||||
| The Ratios you gave: | 1/1 | 9/8 | 5/4 | 4/3 | 3/2 | 5/3 | 15/8 | 2/1 |
| Harmonic approximations: | 8/8 | 9/8 | 10/8 | 11/8 | 12/8 | 14/8 | 15/8 | 16/8 |
Pay close attention to the Harmonic approximations. Notice a break in the pattern? The Just Intonation ratios don't come close to a 13th harmonic. This is what I've been building up to. A NEW NOTE! More appropriatley, a new member to the major scale family. Behold, the table:
| Just Intonation - Part III | |||||||||
| Harmonic: | 1st | 9th | 5th | 11th | 3rd | 13th | 7th | 15th | 2nd |
| Harmonic ratios: | 8/8 | 9/8 | 10/8 | 11/8 | 12/8 | 13/8 | 14/8 | 15/8 | 16/8 |
| Equal tempered bastardization: | 261.63 | 293.66 | 329.63 | 349.23 | 392.00 | 415.30 | 440 | 493.88 | 523.25 |
| Their Names: | C | D | E | F | G | G# | A | B | C |
Here our new note is nestled snugly in it's place.
I would call this scale the "Harmonic Scale" since it approximates the tonics harmonics, but that name is occupied at the moment. To tell the truth, I don"t know if a scale like this exists already, although if it does, now we know the math behind it.
What do you think about all this?
Regards