> > << diatonic scale created by tuning pure fifths. In such a scale, the > do-re and re-mi intervals, both whole tones, are not equal, the do-re whole > tone > is larger than the re-mi whole tone. > > Greetings, > As you mentioned, a pure fifths, Pythagorean octave would create unequal > size semitones, etc. and it would also have some very harsh thirds. My take on the Pythagorean tuning is that the whole tones would be equal. That the tone between C and D would be 9/8 and the tone between D and E would be 9/8 resulting in an interval between C and E or 9/8 * 9/8, or 81/64. This is known as the Pythagorean 3rd and is a wide or "harsh third". If the third were pure it would be 80/64 which reduces to 5/4. It is in Just Intonation that the whole tones are unequal. Here the C--D is 9/8 and D--E is 10/9 and C--E is 5/4. The arithmetic works out 9/8 * 10/9 = 5/4. Why these particular ratios are formed are because of the way a diatonic octave is gotten through pure intervals. Basically it starts with all of these intervals pure. C--G...G--D...C--F...C--E...G--B...F---A . Now in this scheme D--A is "offensive" as musical interval because it beats so much and all of the other intervals don't. But to say the semitones in a Pythagorean tuning are unequal might not be true execpt for one. Since such a tuning of pure fifths carried out until the octave of the starting note is reached, all tones and semitones should be equal until the last interval which should differ by the comma of Pythagorus.. ---ric > Mersenne mentions that tuners would use pure thirds in their tuning > meantone, so the central location of say, D between C and E, was a > resultant, rather than a goal. > Regards, > Ed Foote RPT Yes, this is an important aspect. The theorists realized that a meantone would result even if 4 pure fifths were tuned or 4 tempered fifths were tuned. The math proves no matter how the 5ths are tempered, a 3rd will result from the fourth successive 5th tuned. The 2nd fifth tuned will be a mean tone between the starting note and the note of the fourth fifth tuned. Now why they called it "meantone" because of a result rather than objective or method is beyond me. When the term was first used and by whom might be telling. Mersenne didn't use it, neither did Aaron to whom the earliest description of meantone is credited. ---ric
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