[Smt-talk] Harmonics

[Marcel de Velde]

I have researched just intonation full time for the past 9 years or so, 
and can share the following on the major and minor triad and harmonics.
First of all, these days we have computers which can reproduce perfectly 
the tuning of 5/4 "major third" and 6/5 "minor third" time and time 
again without error and with a variety of acoustical (sampled), and 
synthetic sounds with perfect harmonic spectrum etc.
I've tuned hundreds of common practice period pieces to just about every 
form of "just intonation" ever written about and several more invented 
by myself. A luxury non of the music theorists and writers on the 
subject in the past had.
And I've come to a very clear conclusion. 5/4 is NOT the "just" major 
third and 6/5 is NOT the "just" minor third.
We distinguish intervals solely according to a chain of perfect 3/2 
fifths and 2/1 octaves.
Though playing / singing a major third as 5/4 in certain places will 
give a nice acoustic effect with most timbres because of many 
interlocking harmonic overtones, and more rarely practised the same goes 
for the 6/5, it is ultimately "out of tune".
And it is true that many string quartets, trombone quartets and choirs 
(especially barbershop quartets) will often play a major third as 5/4, 
many others do not, and those that do often get themselves into trouble. 
I have analysed a lot of polyphonic music with Melodyne Direct Note 
Access which allows one to detect the pitches of individual instruments 
within a polyphonic recording.
There is a great difference between overtones and chords as has been 
mentioned in this discussion before. The confusion is and potential for 
error is made even bigger in part because one can play for instance a 
dominant 7th chord as 1/1 5/4 3/2 7/4 in isolation without trouble and 
most ears will not hear it as out of tune instead only marvelling in 
it's highly synchronous sound. Yet once one starts to tune actual music 
this way, for instance in a piece by Beethoven or Bach or Chopin it will 
sound very out of tune to most ears.
In addition there are mathematical problems with tuning major thirds as 
5/4 etc that cannot be overcome. For instance circle progressions. It is 
because 4 perfect fifths (or fourths) reduced by octaves do not make a 
5/4 major third, but an 81/64 major third instead (this 81/64 major 
third is in fact the correct tuning for the major third).
The 5/4 major third is functionally completely incompatible with western 
music theory and western notation.
Western music theory and notation functions according to a chain of 
perfect fifths and octaves, in other words Pythagorean tuning (which one 
can call Pythagorean just intonation).

As for the fundamental nature of the major and minor thirds I agree.
But I can add that the pentatonic scale is arguably as fundamental. 
Especially in world music where polyphony hasn't developed.
The pentatonic scale is of course 5 tones connected by 4 perfect fifths 
and contains both the major and minor triad within itself (whereas the 
western major and natural minor scales are of course 7 tones connected 
by 6 perfect fifths).
I don't see how one can sell the idea of a pentatonic scale based on 
harmonic overtones.
For an easy demonstration let's play for instance the chord E-A-D-G-C. 
Surely one would tune these fourths as 4/3, a wolf fourth would sound 
very out of tune.
Now play in addition a C in the bass, we have contained herein a C - E - 
G major chord tuned as 1/1 81/64 3/2. It doesn't matter if we play the 
additional A and D it's still a major triad in there.
Same if we play an additional A in the bass, we get an A - E - C minor 
chord tuned as 1/1 32/27 3/2.

The question then remains why is a 1/1 81/64 3/2 major triad perceived 
as "consonant" and as more consonant than a 1/1 9/8 3/2 suspension even 
though the latter has more simple ratios.
There are several reasons for this I belief but one of them is to be 
found in the "roughness" theory, which very roughly states that the 
closer 2 pitches (and one should take into account the most prominent 
harmonics of the timbre as well like the 2nd and third harmonics) the 
more "dissonant" they become. The 81/64 major third and 32/27 minor 
third are simply the only simple intervals that are of maximum distance 
between the 1/1 unison and 3/2 perfect fifth.

Hope this is of help to some here.

Kind regards,

Marcel de Velde
Zwolle, Netherlands
marcel at justintonation.com