Optimal Overtone Numbering System

Comment

R.I.P. Comment by Jan Heinke on July 30, 2008 at 5:48am

hi peter,
thanks, you've touched an interesting topic:

"I know that in cents step 13 of the overtone scale is a bit closer to aflat - but the ear wants a, when you play it on guitar or piano.."

yes, we all are following our expectation while listening to music or nature. that's why we can hear the harmonics 11 and 13 (and some more prime numbers) as major or minor intervals depending on different contexts.
singing a natural 7th (=7th harmonic=970 ct) to a well-temperated piano (=1000ct) sounds really discord but the dominant 7th chord is in both versions (harmonically and chromatically) one of the most recognizable vertical structures in music.
just sing the ode of joy-melody with overtones on a single constant pitch and you will notice an (almost) perfect fourth despite the 11th harmonic is nearly 550 ct, means exactly the middle between the temperated fourth and tritone. sing 10, 11 and 15 and you will listen the same interval as lydian! #11....
and so on for the 13th harmonic=840ct (use 12th/13th/14th/16th harmonic so it sounds major, with 12th/13th/15th/16th it's a minor sixth)
a selection/combination of limited harmonics or a wellknown melody leads to expectations whose influences our perception. what we are listening depends on actual circumstances.
of course we are able to notice these abberations with a little practise. anyway, it seems to be a precious tool or trick to realise a more complex music.
best, jan

Comment by Filip Rydlo on July 26, 2008 at 7:06pm

Thank Thee for Thy comment, Bayreuther.

Comment by Peter Bayreuther on July 22, 2008 at 8:06pm

I am very much for the "scientific" system numbering the fundamental as 1.
I would call it step 1 of the overtone scale.
In every scale you have two things: STEPS and INTERVALLS
so in the normal C scale the first step is C and the first Intervall is from C to D which is a second...
As a Jazz musician I anyway think in chords and scales - so the overtones on C up to step 12 of the overtone scale would be:
C 7/9/#11 as chord and
c g c e g b cdef#g as scale
lately I like to use this scale on guitar and violin in the exact structure saying internally:
fundamental fith fundamental third fifth seventh fundamental ninth third sharp eleven fifth...

up to step 16 of overtone scale
C7/9/#11/13/j7
as scale
c g c e g b cdef#gabhc
I know that in cents step 13 of the overtone scale is a bit closer to aflat - but the ear wants a, when you play it on guitar or piano..
internally I see c with fifth, then C7 chord, then lydian scale on C with added bflat (minor seven).

So I am very much in favor of saying step 1 ( 2 3 4etc.) of the overtone scale - step 1 certainly being the fundamental C ( or other tone ) like we do it with other scales too.
On the other hand the traditional intervall names should be known, because they give you a feeling how leaps do sound and make clear the harmonc nature of the overtone scale, strongly based on the C7 chord( or other fundamental) , which is used with such joy by intuitive blues musicians...

best greetings

Peter

Comment by Filip Rydlo on July 22, 2008 at 7:11pm

Some people are just having difficulties while learning them: because of messing up ovt.numbers with intervals. That's all.
e.g. third, fourth, fifth, sixth ...

Comment by Bodo Maass on July 22, 2008 at 10:55am

I made a mistake in my last post, too bad this site doesn't let you edit or preview posts. In the alternative numbering starting with 0, the octaves would be on 0,1,3,7 etc.
Here is the perfect fifth again, but this time with fundamental=0, first overtone=1:

As in the previous image, the red lines are the octaves, and the green lines are those that match between the two sliders.

Comment by Bodo Maass on July 22, 2008 at 9:42am

Hi Filip,

I had this discussion with Wolfgang when we started the work on Overtone Analyzer. He convinced me that what you call the 'scientific' system is the best, and I agree.
Here you can see two tones forming a perfect fifth:

As Marco and Jan have pointed out, numbering the fundamental as 1 is just the most intuitive system that I can think of, because every harmonic frequency is a whole multiple of the fundamental.

So, as in the picture:
1 = 220Hz 1th Harmonic, Fundamental
2 = 440Hz 2nd Harmonic, 1th Overtone
3 = 660Hz 3rd Harmonic
etc.

I guess as a musician you wouldn't think about the frequency numbers, but as in Marco's reply, I still value that the octaves are all on powers of 2: 1, 2, 4, 8, etc.
On the other system, they would be on 2,3,5,9 etc, and you can no longer double the number to go to the next octave.
In conclusion, I think the 'scientific system' has a lot of merit.

Could you perhaps explain more about what you find wrong with this system, and why people have difficulty with it?

R.I.P. Comment by Jan Heinke on July 22, 2008 at 6:28am

hi filip and friends,
nice to find this theme. yes, i also prefer to count in harmonics, means the fundamental is 1. (to separate fundamental and overtones seems to be an historical mistake and leads to unexplainable difficulties. imagine any polyhedron, call one of the x angles 0 and try to do something usefully with it.) all the mathematical relations between numbers/geometrical forms resp. frequencies are represented by audible properties. not only for tones but for rhythms to. nobody would call the first beat in a bar 0. polyrhytmical patterns are functionally adequate to musical intervalls (1/2, 2/3, 3/4...). in this point of view we have a mathematical explanation why we prefer these intervalls and rhythms in the music we do. (in other words we have a distinct sense for abstract regularities very closed to our inner feelings.)

our diatonic scale with 7 tones is based on the harmonic scale but on another level of complexity. thats why we have different numbers for the same intervall ( perfect fifth=3th harmonic, major third=5th...) it only seems contradictory. a diatonic second is represented by the proportion 8/9 or 9/10. the same intervall is included in the harmonic scale. (fundamental=1=C, then you have these second between c and d = 8th and 9th harmonic resp. d and e = 9th and 10th harmonic and so on and vice versa. to handle these definitions is little effort in view of the beauty behind the numbers. with a few rules (running on different levels of complexity) is it possible to create remarkable colours and movements. ...to find these rules we need useful definitions. (but it's obviously impossible to CREATE harmonical systems. we allways unearth very old bones.)
best wishes
jan

Comment by Filip Rydlo on July 22, 2008 at 1:48am

Welcome friends!

The picture of the group is actually a small part of Gayatri Mantra displayed in "Rainbow" colormap on Overtone Analyzer. ;)

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