I was trying to construct the pythagorean scale using fifths. You know by multiplying the tonic with 3:2 ratios and then that with 4:3 ratios and so on. Now in this way the ratios keep on getting bigger. But isn't the pythagorean scale all about simple rational numbers? At least for the fourths and fifths. Well if I proceed this way the fourth isn't turning out to be 4:3 as it should be. But rather something quite unfathomably huge and obnoxious. What's happening here. Can somebody please help me out with a step by step construction of the scale?
Edit: alright so basically suppose I start from a C and I go up a fifth to a G. So the G is in a 3:2 ratio from the C. Now if I go up another fifth from the G I'll go upto a D. Now the ratio of this D to G is 3/2 * 3/2 = 9/4. Since this 9:4 > 2:1, the D exceeds the higher C and is beyond the current octave at hand. So if we step it down an octave by multiplying it by 1/2, it comes to a 9/8 which is our major second. Now multiplying this 9/8 with 3/2 gives me 27/16 which is the major sixth. Likewise if I continue by the time I reach a fourth it's quite a big ratio. But pythagoras also said, a perfect fourth is supposed to be a 4:3 with the tonic. So what's going wrong here?
If you go up twelve Pythagorean fifths, starting from C, and down seven octaves, you don't end up at the C from which you started, but at a $\text{B}^\sharp$ higher than that C by a Pythagorean comma. If instead, you go up eleven Pythagorean fifths and down five octaves, you don't end up at the F a fourth above the C, but at the $\text{E}^\sharp$ an augmented third above it. Unlike in the scales based on equal temperament, in those based on Pythagorean tuning, $\text{B}^\sharp$ does not have the same pitch as C, $\text{E}^\sharp$ does not have the same pitch as F, and an augmented third is a strictly larger interval than a perfect fourth. The notes of the Pythagorean scale that you get by travelling eleven steps upward and nine steps downward from C around the spiral of fifths are given in the following table, along with the ratio of their frequencies to that of the C from which you started. \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \text{C}&\text{B}^\sharp&\text{D}^\flat&\text{C}^\sharp&\text{D}&\text{E}^\flat&\text{D}^\sharp &\text{F}^\flat&\text{E}&\text{F}&\text{E}^\sharp\\ \hline 1&\frac{3^{12}}{2^{19}}&\frac{2^8}{3^5}&\frac{3^7}{2^{11}}&\frac{3^2}{2^3}&\frac{2^5}{3^3}&\frac{3^9}{2^{14}}&\frac{2^{13}}{3^8}&\frac{3^4}{2^6}&\frac{2^2}{3}&\frac{3^{11}}{2^{17}} \end{array} \begin{array}{c|c|c|c|c|c|c|c|c|c|c|c|} &\text{G}^\flat&\text{F}^\sharp&\text{G}&\text{A}^\flat&\text{G}^\sharp&\text{A}&\text{B}^\flat&\text{A}^\sharp&\text{C}^\flat&\text{B}\\ \hline &\frac{2^{10}}{3^6}&\frac{3^6}{2^9}&\frac{3}{2}&\frac{2^7}{3^4}&\frac{3^8}{2^{12}}&\frac{3^3}{2^4}&\frac{2^4}{3^2}&\frac{3^{10}}{2^{15}}&\frac{2^{12}}{3^7}&\frac{3^5}{2^7} \end{array} Note that the pairs $\ \text{E}/\text{F}^\flat$, $\ \text{B}/\text{C}^\flat$, $\ \text{F}^\sharp/\text{G}^\flat$, $\ \text{C}^\sharp/\text{D}^\flat$,$\ \text{G}^\sharp/\text{A}^\flat$, $\text{D}^\sharp/\text{A}^\flat$,$\ \text{A}^\sharp/\text{B}^\flat$,$\ \text{E}^\sharp/\text{F}$,$\ \text{B}^\sharp/\text{C}$ do not have the same pitch like they do in equal temperament scales. In practice, a twelve note Pythagorean scale would typically comprise either the notes A, B, C, D, E, F, G, $\text{C}^\sharp$, $\text{F}^\sharp$, $\text{G}^\sharp$, $\text{B}^\flat$ and $\text{E}^\flat$, or A, B, C, D, E, F, G, $\text{C}^\sharp$, $\text{F}^\sharp$, $\text{A}^\flat$, $\text{B}^\flat$ and $\text{E}^\flat$.