[Smt-talk] Question About The First 16 Partials Of The Harmonic Overtone Series

[Marcel de Velde]

Dear Martin, Brian and list,

Sorry for my very late reply.
I was distracted for a while (by something in music theory) and only now 
handling my emails.

Martin, you wrote:
> Hallo Marcel and others,
> You wrote:
> "We are from birth hard-wired to "quantize" the interval space by a 
> chain of perfect fifths and octaves.
> That is, we identify unique intervals according to this "algorithm". 
> Music is built upon this."
> You are right about the octave. Today there is a biology of the 
> octave. But there is no biology of the fifth. From our knowledge of 
> the auditory brain that we have today -- as little as that may be -- 
> it seems unlikely that we will ever see a biology of the fifth.
> Many humans can learn the chain of fifths as something apparently 
> natural. But the main reason may be that in harmonic sound spectra the 
> group of fifth-related partials is the second strongest group after 
> the group of octave-related partials.
>

I think it is amazing you have found a biology of the octave.
And I think the field of the neurobiology of music holds all the 
promises for the future.
But, while I'm not an expert in the field, the things I've read so far 
make me believe that it is likely a very distant future where this field 
can say things about how we actually quantize the pitch space and 
internally represent melody and harmony as we perceive them in music.
I see many things come out of research into the auditory cortex and 
auditory midbrain that point clearly to pitch detection from harmonics 
and things like that. But this is primarily the field of timbre and it 
seems to me of a certain aspect of the "colour" of intervals and chords 
and related things.
In my opinion this is not the same as the intervals of music. These are 
2 very distinct things.
Again, I am not an expert in neurobiology, but it seems to me the field 
currently has a hard time distinguishing the two?
So far I've not read any evidence that the "musical processing" of the 
intervals of melody and harmony even takes place in the auditory cortex 
or auditory midbrain.

What I can tell you and which is my field of expertise is that there is 
most certainly a quantization of the pitch space based on octaves and 
perfect fifths.
And that it is this very quantization that distinguishes intervals from 
others and is the very foundation for melody and harmony in music. From 
my own research I consider this proven.
And while I can't guarantee the following, it seems very likely to me 
this is (or comes naturally from) a system we're born with. There are 
many things which indicate this, one of them is that we simply do not 
have a functional difference between 5/4 and 81/64 as a major third 
(even though we can perceive the difference under good circumstances). 
The consonant major triad in our music is proven to me to be pure and 
functional as 81/64. If this is not the place for a 5/4, then what would 
be? And how would we distinguish this functionally? It is certainly not 
in western music. And I've searched in world music but find no good 
reason to see it there either.

I can give you this audio example as well:
http://youtu.be/FNGZE8GHvtE
It is an accessible but still clear tuning demonstration thanks to the 
chromatic nature of the music and remote intervals in some of the chords.
As you can hopefully tell the Pythagorean (3/2 fifths) version is 
perfectly in tune and shows the "radical" nature of the Tristan chord. 
(don't mind the not perfect sampled piano timbre though)
For those with good ears you can hear, especially in comparison, that 
the 12-tone equal temperament is slightly out of tune.
The more one flattens or stretches the fifths the more out of tune the 
music becomes. Where flattening gives a "sweeter" timbral effect and 
stretching gives a "harsher" timbral effect in general (this effect is 
harmonics related it seems to me).
In the quarter comma meantone version the major thirds are all 5/4, and 
the fifths fairly flat.
To put it in a 5-limit or higher limit rational intonation system (often 
wrongly called 5-limit "just intonation"), would sound in ways similar 
to the quarter comma meantone version, and in other ways worse as it 
will give either many comma shifts (this means a note will suddenly 
while being held shift up or down a Syntonic comma of 22 cents, sounds 
horribly unnatural and out of tune), and / or "wolf fifths" that are 
famous for their out of tune "howl", and / or comma drifting where the 
whole music shifts up and down by commas like a drunken sailor.
There is NO solution to these problems of 5-limit or higher limit 
rational intonation. And roughly 95% of classical music will give these 
sorts of problems, that cannot be overcome.
5-limit rational intonation and similar tuning systems were a nice 
theoretical toy for earlier theorists, but they simply do not match at 
all how we perceive music in practice.
I have spent many years researching this full-time and retuned several 
hundred of short classical pieces to many systems thought of as "just 
intonation" at some point. (all unpublished, though I plan on writing a 
book with a new and improved theory of harmony ;-) )
I believe I am the only person to have undertaken such a thorough 
practical test for tuning systems (and it is only since the coming of 
computers that such a thing is possible in such scale and with such 
precision and repeatability). And my conclusion is very clear. Western 
classical music functions according to the chain of fifths and octaves, 
and nothing else.
The 5/4 major third is nothing but a colouration of the 81/64 just major 
third. And has proven to be the biggest trap for music theorists of the 
past (and tuning enthusiasts of the present as well).

I also think that Pythagorean tuning can have a positive effect in 
education and further research in music theory. As it is possible to 
hear the enharmonic difference.
I am currently researching how to harmonize the more exotic maqam modes. 
Something which would be much more difficult to do on an equal tempered 
instrument.


Brian, you wrote:
> I'm generally inclined to agree that European classical melodic 
> thinking tends toward Pythagorean tuning. However, one should take 
> care not to extend this notion to other repertoires without empirical 
> support.
>
> A couple of years ago, I played along with Blind Willie Johnson's 
> classic recording "Dark was the night, cold was the ground," using a 
> slide guitar with the fingerboard marked according to Harry Partch's 
> 43-one system of 11-limit just intonation. Johnson's flattened version 
> of scale degree 3 very clearly and consistently represents a 6/5 
> relation (i.e., a 5-limit "minor third") to the tonic pitch, not 32/27.
>
> Yes, this recording represents just one data point, but it's precisely 
> the sort of information that one ought to accumulate before making any 
> broad claims about what blues musicians were or weren't (or are or 
> aren't) trying to do.
>

I've analysed music from all over the world.
Indian music, Arabic / Persian / Turkish maqam music from various 
regions (they differ in tuning practice per region), African music 
including Pygmy music, etc. And also american blues.
Where possible and practical (not that often due to strong pitch 
variations of the human voice etc often making good stable pitch 
detection difficult) I've analysed music with Melodyne DNA (direct note 
access), which allows one to analyse the pitches of individual voices in 
recordings of polyphonic music.
All music fits the Pythagorean system perfectly. With two exceptions. 
Some maqam practice I'm not sure yet how to analyse.
In some regions a maqam like Rast for instance fits Pythagorean logic 
perfectly in tuning, C-D-Fb-F (and some Persian regions practice it as 
C-D-D#-F), but some other regions play that Fb up to a Pythagorean comma 
lower which would in theory make it C-D-Gbbb-F. I'm not sure how to see 
this, is it an exaggeration of the Fb to make it more distinct from an 
E? Or is it indeed indicating another distinct interval to our brain, a 
Gbbb? Very remote. I can't figure it out yet. And while I can harmonize 
a C-D-Fb-F (which to do naturally requires quite a different way of 
harmonizing than we do in the west btw), I have great trouble making 
sense of how to harmonize something like C-D-Gbbb-F.. perhaps use that 
Gbbb only as a nonchord tone? And even then, I'm not sure I'm hearing it 
as a Gbbb or as a very low "coloured" Fb.
The other music that has trouble fitting Pythagorean logic is Indonesian 
gamelan music. Their tuning is all over the place and differs from 
village to village. No doubt this has to do with the bell like 
inharmonic overtones of their instruments and percussive way of playing. 
I'm not too worried about this one.

As for blues. There are a lot of augmented seconds etc in use in blues 
as well.
So while, as you say one song isn't much sampling material, it's quite 
possible that the interval you're describing is expressing an augmented 
second instead of a minor third.
The just tuning for the augmented second is 19683/16384 which is ~318 
cents. A 6/5 is ~316 cents, an almost imperceptible difference.
Also don't trust the spelling in the blues scores or theory books. There 
are many spelling errors there. We can't apply some of the "rules" we've 
found for classical music to blues as it often "functions" quite 
differently.
Take for instance the "Hendrix chord" / "purple haze chord" which is 
often described as an dominant 7 #9. It is not, that's most definitely a 
minor tenth instead of a augmented ninth, etc.


Sorry for not only the late but also the long reply.
I can't help it with this subject :) Though I think the subject deserves 
it..

Kind regards,

Marcel de Velde
marcel at justintonation.com
Zwolle, Netherlands

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