\n", "\n", "

\n", "Following Excercise 1.10 and Section 5.1.1.2 of [MÃ¼ller, FMP, Springer 2015], we discuss in this notebook the tuning system introduced by Pythagoras as well as the Pythagorean comma.\n", "

" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Pythagorean Comma\n", "\n", "The oldest known tuning system was introduced by the Greek philosopher and mathematician Pythagoras (sixth century BC). **Pythagorean tuning** is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio $3:2$ as found in the harmonic series. This ratio is also known as the **perfect fifth**. We now construct a scale starting with the center frequency $\\omega$ of a root note (corresponding to the frequency ratio $1$). Then, we successively multiply the frequency value by a factor of $3/2$, and if necessary, divide it by two such that all frequency values lie between $\\omega$ and $2\\cdot\\omega$ (corresponding to frequency ratios between $1$ and $2$). We repeat this procedure to produce $13$ frequency values (and $13$ frequency ratios). The last (the 13$^\\mathrm{th}$) frequency ratio is also known as the **Pythagorean comma**, which indicates the degree of inconsistency when trying to define a twelve-tone scale using only perfect fifths.\n", "\n", "In the following code example, we construct the thirteen frequency ratios. Furthermore, these frequency ratios are compared with the one obtained from equal-tempered scale (the difference is specified in cents). " ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "m = 0, note = 0, num3 = 0, num2 = 0, ratio = 1: 1 = 1.0000, diff = +0.00\n", "m = 1, note = 7, num3 = 1, num2 = 1, ratio = 3: 2 = 1.5000, diff = +1.96\n", "m = 2, note = 2, num3 = 2, num2 = 3, ratio = 9: 8 = 1.1250, diff = +3.91\n", "m = 3, note = 9, num3 = 3, num2 = 4, ratio = 27: 16 = 1.6875, diff = +5.87\n", "m = 4, note = 4, num3 = 4, num2 = 6, ratio = 81: 64 = 1.2656, diff = +7.82\n", "m = 5, note = 11, num3 = 5, num2 = 7, ratio = 243: 128 = 1.8984, diff = +9.78\n", "m = 6, note = 6, num3 = 6, num2 = 9, ratio = 729: 512 = 1.4238, diff = +11.73\n", "m = 7, note = 1, num3 = 7, num2 = 11, ratio = 2187: 2048 = 1.0679, diff = +13.69\n", "m = 8, note = 8, num3 = 8, num2 = 12, ratio = 6561: 4096 = 1.6018, diff = +15.64\n", "m = 9, note = 3, num3 = 9, num2 = 14, ratio = 19683: 16384 = 1.2014, diff = +17.60\n", "m = 10, note = 10, num3 = 10, num2 = 15, ratio = 59049: 32768 = 1.8020, diff = +19.55\n", "m = 11, note = 5, num3 = 11, num2 = 17, ratio = 177147:131072 = 1.3515, diff = +21.51\n", "m = 12, note = 12, num3 = 12, num2 = 19, ratio = 531441:524288 = 1.0136, diff = +23.46\n", "Pythagorean comma: 1.0136 (+23.46 cents)\n", "\n", "Sinsoid of 440 Hz (A4):\n" ] }, { "data": { "text/html": [ "\n", " \n", " " ], "text/plain": [ "Note | ET Freq. (Hz) | ET Sinusoid | Pyt. Ratio | Pyt. Freq. (Hz) | Pyt. Sinusoid | Difference (Cents) | |
---|---|---|---|---|---|---|---|

1 | C4 | 261.63 | 1:1 | 261.63 | 0.00 | ||

2 | C$^\\sharp$4 | 277.18 | $2^8:3^5$ | 275.62 | -9.78 | ||

3 | D4 | 293.66 | $3^2:2^3$ | 294.33 | 3.91 | ||

4 | D$^\\sharp$4 | 311.13 | $2^5:3^3$ | 310.07 | -5.87 | ||

5 | E4 | 329.63 | $3^4:2^6$ | 331.12 | 7.82 | ||

6 | F4 | 349.23 | $2^2:3$ | 348.83 | -1.96 | ||

7 | F$^\\sharp$4 | 369.99 | $3^6:2^9$ | 372.51 | 11.73 | ||

8 | G4 | 392.00 | $3:2$ | 392.44 | 1.96 | ||

9 | G$^\\sharp$4 | 415.30 | $2^7:3^4$ | 413.43 | -7.82 | ||

10 | A4 | 440.00 | $3^3:2^4$ | 441.49 | 5.87 | ||

11 | A$^\\sharp$4 | 466.16 | $2^4:3^2$ | 465.11 | -3.91 | ||

12 | B4 | 493.88 |