on Meantone

[Richard Moody]

>
> << 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