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I am reading the Treatise on Harmony and I want to make sure I have this clear in my mind on the 5th

In order to define an octave you need to subdivide a fundamental tone in half, intervals from that define the rest of the notes.

If the fundamental is C' then
C is 1/2 C'
D is 1/3 C'
E is 1/4 C'
F is 1/5 C'
G is 1/6 C'
A is 1/7 C'
B is 1/8 C'

SO G is a perfect 5th since it is composed of two fundamentals D & C at 1/2 * 1/3 = 1/6... and is considered perfect as it is the only note composed of 2 fundamentals?

This sounds correct, but I don't want to go off into the weeds. Before I get to the section on harmony.

This is not correct, clearly seen because 1/3 is lower than 1/2 (not higher), and 1/4 is lower than that. 0.5, 0.33, 0.25 goes down, not up.

For a numerical check, the exact frequencies of equal temperment piano notes are at http://www.vibrationdata.com/piano.htm

A fifth is the ratio 3/2 higher (x 1.5), and a fourth is the ratio 4/3 higher (x 1.33). This has a 2500 year history, and it is not real simple, but search google.com for the terms
piano temperament pythagoras
for more details.

Then modern equal temperament changes this very slighty. One very good site is
http://www.uk-piano.org/edfoote

Thanks for the info, the description in the book was really hard to pull out of it (it was like some sort of LSAT question) There were diagrams used that would indicate the above ratios releative to the fundamental frequency.

The ratios given were the inverse to the tone (like if you cut a string in half according to the diagrams) which if maintained as C of 1 and D of 1/2 (1/2 interval from C?) would produce 3/2 in some combination of tone... I will have into look at that.

Thanks for the links I will figure it out.

Wow! Ok now I know what Bach meant by a "Well Tempered Keyboard"

http://www.uk-piano.org/edfoote/

I am going to see if I can dial my Motif ES into a "Well Tempered" mode.

Pythagorus knew that if you made the string length be half or double, to get the length ratio 2/1, that a very special strong resonance sound occurred, and sounded good. This 2:1 is an octave, which is all important to theory.

Strings longer than half, but less than full length, are the notes within this octave. How those intermediate notes are defined is temperament, so generally these origins are usually discussed under the temperament heading.
There is lots on the web about temperament.

Pinching off the string at the 2/3 length (like a guitar) gives a 3 to 2 ratio which is called a fifth (fifth note in 7 note scale in an octave). They didnt know about frequency in the days of Pythagorus, to know this inverse is 1.5x frequency, but they could hear the resonance of the harmonics matching, and knew the length was special. For example if the note was A at 440 Hz (we know today), then 2/3 length gives the higher E (fifth) at 1.5x440 = 660Hz. (but todays equal temperament calls it 659.25).

The harmonics (2x, 3x, 4x, 5x, etc) are
440: 880 1320, 1760, 2200, 2640, etc
660: 1280, 1980, 2640, etc

The fifth harmonic of 440 is the same frequency as the third harmonic of 660, which simultaneously blends smoothly to make the resonance we hear, which is what makes the fifth a special sound. There are similar relations for the third at 4/3 ratio.

Thanks

I kept reading and what you described was in a figure about 10 pages ahead. The earlier diagrams made it really hard to understand. The physics and math isn't scary (to me), just a really bad description to start heading off into the weeds.

I found a mode on my Motif ES that allows me to dial in a number of "temperments" I hope to get a feel for what other temperments sound like.

google search "Well Tempered Motif ES" brings up a story on how to do this.

I had assumed that this temperament controversy was about done after a couple of hundred years, but digital keyboards have the capability to bring it back with just a simple setting. It is totally amazing how many web pages exist on this topic.

I have a P90, which has five other choices besides Equal. Looks like the Motif has even more. The P90 for example has one combined Werckmeister/Kirnberger choice, which I think the Motif retains as two, and actually, there were variations of each. As I understand it, this is said to be close to what Bach probably used, but his actual details are not known now.

Is it Logarithms or Logarhythms ?

On an ET piano, the ratio of any successive semi-tones should be 2^(1/12) or 1.05946...So if the freqency of the fundamental is C, C#=2^(2/12)*C, D=2^(3/12)*C, D#=2^(4/12)*C ....

C=C
D=2^(2/12)*C
E=2^(4/12)*C
F=2^(5/12)*C
G=2^(7/12)*C
A=2^(9/12)*C
B=2^(11/12)*C
C at Octave = 2^(12/12)*C=2^(1)*C=2*C

Somehow I think the Triangle would be Pythagorus' instrument of choice? Your class sounds interesting -- sorry about the bad jokes and puns...