# The Relationship Between Timbre & Consonance / Dissonance



## BrahmsWasAGreatMelodist

A layman's explanation of the hypothesis of this video (I'm unsure how compelling the evidence presented is, but the theory behind it is incredibly interesting and ripe for discussion nonetheless):

Instruments in a Gamelan ensemble have a very inharmonic, metallic timbre to them, yet they have to play in sync with each other and with the human voice (a largely harmonic instrument). Gamelan music has 2 scales: pelog, a 7-tone scale which I think you can (keyword: vaguely) relate to our diatonic scale in the west, and slendro, which is almost 5-TET (interestingly enough, these scales are slightly varied at different "octaves", and octaves themselves are slightly wide).

And if you sweep a harmonic spectrum (like the voice) across this "timbre" and plot the sum of resulting Fletcher Munson curves at each harmonic, you get troughs that represent consonance... what's interesting is that these troughs roughly line up with the notes of the Gamelan scales! (slendro moreso than pelog, the latter of which is a much more recent, post-colonial adoption).

So, TLDR, although the scales of Indonesian gamelan music bear very little relation to Western tonality, they are likely following the same basic principles of consonance and dissonance that we are - but fitted to their inharmonic instruments!

Video:


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## BrahmsWasAGreatMelodist

There is a lot of discussion in microtonal circles about the qualia, or essence, of different intervals, scales, chords, etc. or about frameworks for the characterization thereof. But I rarely ever hear it discussed in the context of the timbre of an instrument, which seems to go hand-in-hand with it. Without this context, you can really only make general statements about certain classes of instruments.


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## Nate Miller

I dont think you hear about that regarding the timbre of the instrument because if the instrument didn't have a defined overtone series then it would be noise and not a musical pitch. The timbre comes from different overtones being stronger or weaker, but they are still there. I think that is why the timbre of an instrument isn't part of the discussion of the intervals created by playing the instrument.


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## BrahmsWasAGreatMelodist

That's not necessarily the case. There are instruments (e.g. metallic instruments) with inharmonic spectra, noise generators, etc. Besides that, even within the realm of harmonic instruments, you have different overtones which may be more or less pronounced & different formants, thereby creating a different hierarchy of consonance & dissonance in intervals.

It seems like microtonal theory generally assumes an instrument with relatively strong peaks on many prime overtones. To be fair, most traditional acoustic instruments seem to match this description. But if you are going to go out of your way to design a new scale, why not also suggest a family of timbres to use it?


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## tortkis

That is an interesting point. I only know that John Pierce designed a spectrum for 8-ET instrument on which tritone is consonance. It has peaks at the following frequencies:

1
2^(10/8)
2^(16/8)
2^(20/8)
2^(22/8)
2^(24/8)

Its dissonance curve has dips at minor third, tritone, major sixth, and an octave. Pierce composed Eight-Tone Canon for computer using the tones with these overtones.
John Pierce - Eight-Tone Canon (1966)

I don't know if it is possible to design and make an acoustic instrument with this kind of artificial spectrum.


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## BrahmsWasAGreatMelodist

@tortkis Yes it is possible, though rather difficult, to design acoustic instruments with carefully chosen inharmonic spectra. An example is William Sethares's hyperpiano .

I came across to this  podcast a while where he describes the process of modelling the physical strings of instruments like the hyperpiano.


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## tortkis

That "this podcast" is a link to Hiromi's Pathetique... Is it related? 😄 (enjoying it now)

The hyperpiano sounds like prepared piano or gamelan. Inserting objects between strings may cause similar effects, but I think Cage's intention was not to explore new consonance. Is there any compositions for the hyperpiano focusing on the different consonance? (not as just another percussion instrument)


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## BrahmsWasAGreatMelodist

Sorry that was unintentional. Glad you enjoy it though, I do as well.

This podcast


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## tortkis

I found this instrument recently. "Polygonola is a two-dimensional, suspended plate musical instrument. The name Polygonola is coined by uniting two words, “polygon” in Greek and “ola” in Spanish. “Ola” in Spanish meaning “wave”. There are several shapes of Polygonola, triangle, square, pentagon, hexagon and circle. A polygon with an infinite number of sides would come close to being a circle. When you hit the Polygonola, it emits sound waves rich in non-integer overtones."





English | 二次元楽器 ポリゴノーラ







www.oto-circle.jp





The circle plate has two vibration modes: pizza mode (node lines going through the center) and doughnut mode (concentric circle nodes). The actual spectrum depends on where on the plate is hit. For example, if you strike a point 1/4 of the radius from the center, the overtones are 2.254, 3.823, 5.706, ... The scale set of circle Polygonola consists of 13 notes, of which the 11th note is 2.25 times the first note (octave). A CD/download of compositions for Polygonola can be listened and purchased on the site.

Music of the Polygonola (a polygonal flat musical instrument)
A unique tone, timber and scale
Keiji Haino, Tonika Ichinose, Makiko Sakurai


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## BrahmsWasAGreatMelodist

tortkis said:


> For example, if you strike a point 1/4 of the radius from the center, the overtones are 2.254, 3.823, 5.706, ...


Why those numbers? I'm curious, I'm not very knowledgeable about the actual mechanics behind this stuff; I'd like to know more.


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## tortkis

Here is a detailed explanation of the vibration modes and the overtone spectrum of a circle object, but there seem no English pages. You may use Google translate tool. In the article, there is a link to A.W. Leissa's paper "Vibration of Plates".

The set of possible vibration modes (the combination of node numbers of pizza/doughnut modes) depends on the hitting point, because vibration nodes cannot exist at the point where the plate is hit. According to the article, the vibration frequency of each mode can be calculated. (I have not read the paper and will not be able to fully understand it.) In the following figure, the second from the top is the spectrum in case a point 1/4 of the radius from the center is hit. The base note is 248Hz in this case, generated from the mode of doughnut=0 & pizza=1, and different modes generate 559Hz (x2.254 of the base note), 948Hz (x3.823), 1415Hz (x5.706), and so on.


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