Pythagorean Scale Part 2

Long before the 12 different chromatic notes that we know of on the piano, Pythagoras had developed his own scale with much less notes. It could be compared to our C Major scale, but with a few differences.
When I began researching the Pythagorean Scale and how it was constructed, something didn’t sit well with me. The most common description is that it is a projection of Perfect Fifths. A Perfect Fifth is starting on one note and going to the fifth note of that scale (C,D,E,F,G). It is also seven half steps in equal temperament (C,C#,D,D#,E,F,F#,G), or the ratio of 3/2 in the overtone series (C,C,G). A projection of fifths is to keep going up Perfect Fifths until you have all the notes of the scale, for example, C,G,D,A,E,B,F#. This brings us to our first problem. If we put this scale in order, we get C,D,E,F#,G,A,B,C, but the C Major scale that we know of has an F, not F#. Most explanations I’ve read say that you have to stop at B (C,G,D,A,E,B) and then for the last note, go to a fifth BELOW C (F). Why? I agree that this fits into what we know as the C Major scale, but it seems to me that whomever thought of this was more concerned with reverse-engineering the major scale, than putting him/herself in Pythagoras’ shoes.
Disclaimer: What follows has not been extensively researched, but is only based on my educated guesses.
Pythagoras experimented with a monochord, or put more simply, a string. By holding this string at certain specific points, he could make different notes when he plucked it (we know these today as overtones). If he held it ½ way, he got an octave higher than the fundamental. If he held it 1/3 of the way in, it was an octave and a fifth above the fundamental, ¼ of the way, it was two octaves, etc.
After checking with my Solar Harmonic Spectrum tuners, I developed a theory of my own. The frequencies of the notes are as follows:
C=256, D=288, E=320, F=341.3, G=384, A=426.7, B=480, C=512. I found that the projection of fifths wouldn’t work. When I went up a fifth from C (256 * 3/2 = 384) and got G: OK. A fifth up from G (384 * 3/2 = 576. 576/2 = 288 (down an octave)) is D: OK. A fifth up from D (288 * 3/2 =432) DID NOT equal the frequency of the A in the tuning fork scale (427.7). So where did they get this frequency for the A? After some trial and error I found that one way to get that frequency for the A is to go a Perfect Fifth BELOW E (320 * 2 = 640 (up an octave). 640 * 2/3 = 426.66666), or a Perfect Fourth above E (320 * 4/3 = 426.66666). So the relationship of the fifth is still important.
Then it occurred to me. OK, I’m Pythagoras and I’ve got a string (C). I find that if I hold the string in the middle, I get a new note, so I get another string and tune it to this new note. I hold the string away from the center, dividing it in thirds and I get another note (G). I get another string and tune it to this new note. I move a little further away from the center, dividing the string in quarters and get yet another note (C). Get a new string, tune it, you get the idea. I do this until these new notes get kind of hard to hear (probably at the 7th partial. That’s where I start to lose them on my guitar anyway). So my “Harp” looks like this:
C C G C E G
After some time of experimenting with the harmonics on the new strings, Pythagoras probably notices the similarities in the C’s and G’s and acknowledged the significance of the ½ and 2/3 ratios. He also probably noticed by using his voice that there was a huge gap in between the C’s where many notes could fit. He probably noticed that as the strings went higher, there were smaller and smaller gaps where he could fit less and less notes. If he wanted to fill in these gaps with new strings, he would have wanted to have them be mathematically related to the original strings. Since the first overtone or 1:2 ratio (the Octave) produces the “same note” it wouldn’t have gotten him very far in filling in the gaps. So he went to the next overtone, the 2:3 ratio or Perfect Fifth. He already had the Perfect Fifth above C, so if he did the Perfect Fifth above G and E, he would get:
C C G C E G D B
The D and B are an Octave higher because the overtone is actually an OCTAVE and a Perfect Fifth above the fundamental. Pythagoras could then use his knowledge that the 1:2 ratio is the “same” note and use that to find the notes in the lower octave and upper octave, giving him this:
C C G C D E G B C D E G B C
Now He probably wouldn’t want to do a Perfect Fifth above D and B, because that would take him too far away from the original notes. But what if he went back to the original notes and went DOWN a fifth instead of up? This would be a lot trickier, because it requires tuning the harmonic of the new string to the existing string open, as opposed to tuning the new string open to the existing harmonic. The P5 below C gave him our F, the P5 below E gave him our A and the P5 below G is C which he already had, giving him this:
C C F G A C D E G B C D E G B C
After finding the upper octaves of F and A, his harp looks like this:
C C F G A C D E F G A B C D E F G A B C
Now that there are two octaves of a full scale with no gaps, it was at this point that he stopped looking for more notes and used the lower octaves to fill in the lower range of his harp:
C D E F G A B C D E F G A B C D E F G A B C D E F G A B C
Let’s apply this to the tuning forks. If C is 256 Hz, to get G, we multiply by 3, then divide by 2, to bring it down an octave and we get 384 Hz. To find E, we multiply 256 Hz by 5, then divide by 4 to bring it down two octaves and we get 320 Hz. Now we have our triad. We already have our fifth above C, so we do a fifth above E by multiplying 320 Hz by 3 then dividing by 2 and we get 480 Hz. To get D from G, we multiply 384 Hz by 3 but then divide by 4 to bring it down two octaves because in the scale it is below G, not above G. Again we already have a fifth below G, so we skip to our F below C. To get a fifth below, we multiply by 2 then divide by 3. 256 Hz times 2 divided by 3, then times 2 again to bring it above C instead of below it, is 341.33333etc. We do the exact same thing to E to get A (because we also want A to be above E), and from 320 Hz we get 426.66666etc. This doesn’t necessarily prove my theory, but it provides interesting discussion.
There are also many more relationships to the overtone series in the tuning fork frequencies. If we look at some of the other intervals in the overtone series, we see that the exact ratios work in other places. For example, the Major 3rd ratio of 5/4 works for F to A and G to B. The minor 3rd ratio of 6/5 works for A to C and B to D, but not D to F. This show that the F Major chord and the G Major chord are the exact same quality as the C Major chord. The Major 2nd ratio of 9/8 works for C to D, F to G and A to B, but not for D to E or G to A. The minor 2nd ratio of 16/15 works for E to F and B to C.
Another important point to mention is that Pythagoras probably didn’t call this a “C” scale. Why would he call the first, main note he worked with after the third letter of the alphabet? Actually, in the Greek alphabet, it’s alpha, beta, gamma, delta, etc. There isn’t even a “C”. I believe that where the “C” originated was from the monks doing Gregorian Chant music. The clef they used to designate the main note, or “Do”, looked like the letter C. I believe that this led to people calling it C and the C Scale grew from there. I also believe that C wasn’t always the exact same frequency. I believe that the “Do” of a particular chant was always the same pitch as the resonant frequency of the church or building that they were chanting in. That way, when they ended a chant, the last note would resonate the entire room, creating a huge amount of energy. If you don’t know what I’m referring to, try this. Next time you are in a public bathroom, or some other big room with echo, start humming a note. Gradually slide this note up and down in pitch. When you get to a certain pitch, the note will get louder and you will feel this vibration all through the room. This is because the note you are humming is the same as the resonant frequency of that room. This is a very powerful sound.