Pythagorean tuning

This is a diatonic, 7-tone scale, developed by the Pythagoreans, perhaps given to Plato by Archytas of Tarentum ca 400 BCE, and given to us by Plato in the Timeaus in the descending form: HTTTHTT, where T denotes the Pythagorean full-tone descending interval with length ratio 9/8 = 3²/2³, and H denotes the hemitone, 256/243 = 2⁸/3⁵.

Note 1: As these length ratios are larger than one, the intervals represented are falling intervals, and the descending scale indicated, rewritten in ascending mode: TTHTTTH, is the lydian mode of ancient Greece, and (approximately) our modern major scale. (Here, T = 1/T = 8/9, and H = 1/H = 243/256. Again, these are length ratios.)

Note 2: If the 7 descending intervals are sounded consecutively, the larger descending interval resulting is: TTTSTTS = TTTTTSS
= (9/8)⁵ * (256/243)²
= (3¹⁰ * 2¹⁶) / (2¹⁵ * 3¹⁰)
= 2 (a perfect descending octave)

Note 3: If the first 4 intervals are sounded consecutively, the larger interval resulting is: TTTS = (9/8)³ * (256/243)
= (3⁶ * 2⁸) / (2⁹ * 3⁵)
= 3/2 (a perfect descending fifth)

Note 4: If the next 3 intervals are sounded consecutively, the larger interval resulting is: TTS = (9/8)² * (256/243)
= (3⁴ * 2⁸) / (2⁶ * 3⁵)
= 4/3 (a perfect descending fourth)