[Smt-talk] Harmonic and Melodic Scales

Marcel de Velde marcel at justintonation.com
Wed Dec 4 17:14:51 PST 2013

Dear Martin,

Thank you for your reply.
I find this a very interesting topic and an important area for music 
theory as this is so fundamental to so many things.

As I said earlier, I am not an expert in neurology.
But I have for the past 7+ years done full time research into this exact 
topic and have thought up and performed on myself many many tests to 
probe how we quantize the pitch space in musical context.
This may be of less scientific value though as I have not performed 
large blind studies nor measured the brain in any way. But my results 
are replicable and clearly show pitch classes beyond 12-tones.
I do not have much examples online at the moment but the one I posted 
before does show a difference between an Ab and a G# and to me this 
difference in pitch class survives also in 12-tone equal temperament 
where they are the same key:
The high voice goes G-Ab-G#-A.

I'll further answer your email in parts.

> Dear Marcel,
> The differences between C# and Db or between C-F and C-E# exist 
> exclusively in imagination. They have no physical basis.

If with physical basis you mean sound outside the brain then I agree 
they do not have to have a physical basis. In for instance a 12-tone 
equal tempered piano the E# and F share the same key and produce exactly 
the same physical sound.
But I suspect you mean it in a different way. Then I'm not sure which 
physical basis you're referring to exactly.

> If you read scores, major parts of your brain that are involved are 
> the visual brain, possibly the linguistic brain, and the auditory 
> brain. Now, in which of these parts of the brain varies the processing 
> between C# and Db?
> Answer: It varies in the visual and linguistic parts, but not in the 
> auditory parts.
> The emotional associations may also vary, but it is important to note 
> that such variations are not part of the music but part of the ideas 
> about music.

To this I can not agree. My research has shown me a very different thing.
That for instance a diminished fourth is a different interval than a 
major third.
It does not matter if one reads a score or not. If one listens to the 
music of for instance Beethoven and he plays a diminished fourth then to 
all people this is clearly not a major third.
A diminished fourth has a very unique tension unlike the major third and 
it is a dissonance it must resolve, and it naturally resolves 
differently than the major third. Also on the 12-tone equal tempered 
piano where they are tuned exactly the same.

> The small deviations from equal temperament that we see in historical 
> tunings are relevant for consonance issues. But they are not relevant 
> for pitch class. The interval C-F on the violin varies much more 
> according to musical context than it can vary on the cembalo according 
> to different tuning systems. The origin of this pitch class tolerance 
> lies in the auditory brain, where pitch class is not represented in 
> narrow lines, like on a ruler, but in wide strips, like on the back of 
> a zebra.

Yes I agree with that small deviations in historical tunings are 
relevant for consonance and not for pitch class.
But this does not counter what I stated earlier. Both these things can 
exist at the same time.
As for narrow lines vs wide stripes. I found this depends on the 
context. I will explain this too.

Here 2 examples.
If we play the simple progressions I-IV-V-I and start by playing it with 
Pythagorean major thirds, then play it with ever lower tuned major 
thirds until at some point the major thirds switch pitch class to minor 
Indeed, we did not find any different pitch class in between the major 
third and the minor third. We did not hear it switch to a diminished 
fourth, nor an augmented second.
We only heard different "colorings" of the major and minor thirds, some 
of them quite a bit less consonant sounding but still they expressed 
either a major or a minor third. Not even a "neutral third" like often 
described for some Arab/Turkish/Persian music.
So these are very wide stripes indeed. I could even show you examples 
where depending on context these wide stripes even overlap, in other 
words we can perceive an interval closer in tuning to a minor third as a 
major third in the right context.

But, now here a different example with a different context.
Play a C E G chord on the first and third beat with a melody on top 
either C - Db - E or C - C# - E where the Db or C# falls on the second 
beat. (it is a nonchord tone / passing tone in this example)
Tune it to Pythagorean with the Db 90 cents from C and the C# 114 cents 
from C.
And voila, we hear 2 different intervals. That Db is in this context 
expressing a different interval than the C#.
And this with a difference of only 24 cents. (it is even more clearly if 
we widen the difference a little, this is often done in maqam music).
So while this is not the same as narrow stripes on a rules with nothing 
in between, we do have narrower stripes than in the previous example.
This is because of context.
Btw, I'll be happy to render this example for you with piano and sine 
wave and put it online.

Musical context is of course a complex thing, but what I can say is that 
we tend to automatically go for the simplest interpretation which is 
often the shortest chain of fifths span.
In the first example there was such strong context for a major third or 
minor third and very little for a diminished fourth or augmented second 
so tuning did not matter there.
In the second example there is almost equal context for either a minor 
second or augmented prime and the small tuning difference is enough to 
make clear which is which.

So I cannot match this to a 12-tone chroma pitch quantization.
But I do think that one would have to build tests with this knowledge 
beforehand to show it.
If one does for instance random tone probe tests I can understand if the 
outcome will be simply the simplest intervals and give something similar 
to a 12-tone chroma pitch space.
But this will not have excluded my second example and can therefore not 
be considered good proof.
If you disagree though I'd very much like to see the papers that proof 
the 12-tone chroma space. There must be something wrong there then.

> More instructive than musical mathematics and philosophy is what the 
> musicians and the instrument makers do. In Indian classical music we 
> have the success of the 12-key hand-pumped harmonium. In European 
> music we have the introduction of 12-tone equal temperament in the 
> fret spacing of the lute. The latter predates Baroque music and 
> excessive modulations. It is interesting that 12-tone equal 
> temperament in the fret spacing of the lute was used, even though 
> there was no compelling musical reason to do so, and uneven fret 
> spacing was also used and is quite popular today. The fact that even 
> fret spacing was used 500 years ago, for music that was not modulated 
> much, demonstrates the affinity of such a technique to the chroma map 
> of the auditory brain.

I thought the majority of Indian instruments are and were based on the 
shruti system.
This system is most commonly described either in Pythagorean as a 
continues chain of 21 fifths giving the 22 shrutis, or as 2 Pythagorean 
chains a Syntonic comma apart (which gives about the same result in this 
case as the fully Pythagorean description with a max difference of about 
2 cents).

Furthermore, there is Arab/Turkish/Persian maqam music.
Their music has the small and large "neutral" intervals, like the small 
neutral second and the large neutral second.
Their tuning varies per region and performer and can be anything from 
very roughly 115 to 140 or so cents for the smaller neutral second, and 
roughly 160 to 180 or so cents for the larger neutral second. (on top of 
that there's a lot of mislabeling going on, some may play even a minor 
second where a larger neutral second is indicated by theory)
While I personally hear a portion of these neutral seconds as simply 
colored minor and major seconds, at other times they do express a 
different interval as a minor or major second.
And for a long time people have tried to harmonize these unique 
intervals without success. The thing is that when one plays something 
that truly expresses something different from a minor or major second 
and then harmonizes it as if its a major or minor second then it will 
simply become interpreted as a colored minor or major second because of 
the created context of the harmony.
I have found that when we interpret these intervals as augmented primes 
and diminished thirds and harmonize them accordingly that the unique 
expression is preserved. Though this is easier said than done, it is not 
that easy to harmonize a diminished third naturally in the context of 
specific chromatic melodies. Truly new theory is needed here for 
composition in order to avoid mistakes in resulting interpretation.
In any case, before going too far off topic, the thing is that 
traditionally they indicate this difference between a minor second and 
small neutral second and large neutral second and major second with 
tuning in monophonic music. And this is truly "functional" / pitch class 
difference not mere "color" difference.

I really hope my perspective is in some way useful to you.

Kind regards,

Marcel de Velde
Zwolle, Netherlands
marcel at justintonation.com

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