One more 12-tone tuning . . .

[Richard Moody]

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> From: Paul H. Erlich <PErlich@acadian-asset.com>
> To: 'Richard Moody' <remoody@easnet.net>; pianotech@ptg.org
> Subject: RE: One more 12-tone tuning . . .
> Date: Wednesday, September 08, 1999 3:19 PM
> 
> >In Meantone, to keyboard tuners, for
> >at least  the past 400 years there has been no C--D#.
> 
> Not in all cases. For example, Handel's organ was set up to produce 16
> pitches of meantone, including D#. The interval C--D# could occur in A minor
> in an F augmented sixth chord.

Yes! pardon my omission due to lapse, I don't have first hand information about
Handel's organ, , but some organs (presumidly in Meantone) did have split keys, 
comming at Eb to give D#, and G# to give Ab.  etc. Harpsichords also. If you haven't
seen the Chapman translation of Mersenne, _Harmonie Universelle_  with the pictures
and schemes for "split keyboards" it is quite an eye opener.  
> 
> When you tune a 6:5 ratio, the 5:4 is theoretically louder and beating
> rapidly. Yet that does not disturb the consonance of 6:5. The situation is
> similar with 7:6, it's just that 7:6 is an unfamiliar sound to most.

You are raising some interesting points. I am wondering if you have explored
musically the series of what  is called superparticular ratios...3:2, 4:3, 5:4, 6:5,
7:6, 8:7, 9:8, 10:9    So yes the "justness" of the coincident ratios does become a
question the closer these intervals get.  If we take your example of a 6:5 and
compare it to a 5:4 we find at A220,  a just minor third (6:5) up would give C 264. 
Here the 6th and 5th partials are at 1320.    Comparing the 5:4 we have 1100 and
1056.  1100 - 1056 = 44 bps.   Isn't it interesting that 44 * 5 = 220 and 44*6 =
264.   In other words the beat frequency, 44,  is a lower harmonic of both 220 and
264.  Or 220 is the fifth harmonic of 44 and 264 is the 6th harmonic of 44. So when
you say the 5:4 is louder and beating rapidly but does not distrub  the consonance
of the 6:5 this may be the reason why.  An unexpected observation here is that 44
cps is two octaves and a Major third below 220, I was expecting a minor third.  
  
> >If you would like me to tune it on a piano and post the
> >results I certainly will.
> 
> My piano is in standard meantone temperament, which has two 4:5:6:7 (German
> augmented sixth) chords: Bb D F G# and Eb G Bb C#. These chords sound
> absolutely convincing and beautifully resonant to me. The "subharmonic"
> equivalents, Bb C# E G# and Eb F# A C#, are also beautiful in their own way.
> The tuning I described has three 4:5:6:7 chords, and three more in the
> "subharmonic" configuration. That's the point of it.

At the moment my piano happens also to be in meantone but I would call it "Historic
Meantone", specifically what is called "quarter comma Meantone" where all the fifths
are flattened by 1/4 comma  (21.5cents/4) execpt G# and Eb. 

When you say 4:5:6:7 do you mean that Bb--D is pure, and D--F is pure and F--G# is
pure? This chord at Bb4 sounds on my piano, "beautifully resonant" to me also.  ,
but an octave lower not as good. However my tuning was aural, and I have no way of
knowing how close it comes to the cents projection.  Unless there is a tuner who was
taught by his grandfather how to tune Meantone, the technique of tuning Meantone is
lost, so machines are the only judge.  
 
> >ps, You might be interested in Helmholtz and Ellis on resultants and
> combination
> >tones, and perhaps  experiments to obtain audible tones from beat
> frequencies of
> >supersonic pitches. 
> 
> I'm well aware of combination tones, which I mentioned in my last message.
> I'm afraid Helmholtz and his translator are now well over 100 years out of
> date. The psychoacoustical field, like most areas of psychology, has
> progressed immensely in the 20th century, so much so that pre-20th century
> psychology is about the equivalent of ancient Greek physics.

But, ancient Greek physists (Pythagorus, Didymus, Aristoxenes) have given us what we
are talking about right now.  Where else does 4:5:6:7 come from? And if two
supersonic tones yield an audible resultant then Helmholtz et al are up to date. 
Well anyhow it is an intersting discourse. I am anxious to see if we are talking
about the same Meantone ---ric