Absolute Pitch Education

[aruffo]

The idea of "saturation" as a dimension of pitch continues to appeal to me. On the one hand, it's an easy explanation for the fact that middle octaves, being more saturated, are better identified; on the other hand, it describes the desaturated sound of a fuzz guitar. What I'm still not sure about is how to move around the harmonic lattice (which I've attributed to Mathieu, but I've since learned the lattice has been around for centuries).

The idea of a "scale" that moves not through semitones but through increasingly complex ratios, starting with fifths (1:3) and thirds (1:5), is sensible and appealing-- but how do you move through it? It's easy enough to slide from one semitone to another by mathematically increasing or decreasing the vibratory frequency, but how do you gradually change 1:2 to 1:3 without passing through a host of ugly ratios in between?

[paul-donnelly]

Is moving around the scale by simple ratios directly related to saturation? If not, I don't really see an issue. If you're sliding from semitone to semitone you pass through an infinite number of ugly ratios, while also starting and ending on ugly ratios. Why is it then a problem to find ugly ratios between two purely tuned notes as well?

The big difficulty I'm having with your tonal cylinder is how you actually go about increasing the saturation of a pitch.

Saturation is a difficult concept for me because there are so many ways to make the fundamental more difficult to perceive. You can take power away from all the lower harmonics, leaving only closely spaced high ones. You can distribute power unevenly to imply a different fundamental (e.g. adding power to the second and fourth harmonics will sound like the octave relationship normally only obvious at the root). You can increase the number of non-harmonic tones to add confusion. And each one of these strategies has an infinite amount of variability in its implementation.

While it's fairly easy to come up with a way to calculate a tone's saturation, going the other way looks like a huge problem to me.

By way of contrast, color is very easy. As we have only three color receptors, any way you slice it gives you a three-dimensional field (ignoring the overlap between the red, green, and blue receptors in our eyes, anyway). You can take a brightness for each of the three colors and combine them into a single luminance value, have a hue value for your position in the spectrum, and then add in the mysterious saturation. In visual media, saturation is an easy process-of-elimination concept.

If the color you're working with excites the red and green receptors, adding some more to blue receptor desaturates. From there, you can shift your red and green levels a little to keep the hue in the right place, and Bob's your uncle. Although it's possible to create a color that requires you to increase the level of two receptors in order to desaturate it, it's still easy to do. Essentially, you add a "harmonic" (more light at a new wavelength) right in between those two receptors, and hit them both. In either case, you're adding only one "harmonic." For the situation in which only the middle-spectrum green receptor is excited, you do have to add two new "harmonics", but this is likewise obvious. Adding only one would change the hue but not the saturation.

Again, I'm mostly ignoring the physiology of the eye, as I don't think it's relevant. The important part is that saturation in color theory follows directly from the fact that we only detect three color ranges. Adding more light frequencies is irrelevant, because our eye will simply sum them (simplified! Overlap between receptors does enhance our color discrimination). If we could see ultra-violet light our color model would gain a dimension (probably another hue, but who knows? Ask a tetrachromat how she thinks of it).

But how can we translate that to pitch? Rather than having three receptors, we have a huge number of them. Our whole auditory bandwidth from 15 or 20 Hz up to 20 to 30 kHz, divided up in intervals of 300 cents or so. So 80ish (!Warning: here be back-of-the-envelope math!). How do we decide which of these 80 bands to put energy into when we want to desaturate a pitch?

EDIT:
tl;dr -- While desaturing a tone and a color are similar and similarly hairy in theory -- both pitches and colors come with infinite spectra of companions -- color is vastly simplified by the simplicity of our color perceptors. We can ignore the infinite number of wavelength combinations possible, and focus on how the affect only three types of cone cell. This boils down to a three-axis concept of color, which can easily be transformed into the HSV system. Note, it is impossible to take an HSV or RGB color and procure the exact combination of wavelengths that produced it! The best we can do is create a color that will achieve that effect.

Sound however is made immensely more hairy by having 70 or so more receptors! This suggests to me that any direct translation of the RGB or HSV system would need as many axes. I'm not prepared to say that there is no meaningful three axis system that we can use to rotate tones, but I am prepared to say that a direct translation will have quite a few axes. Probably one for overall SPL, one for saturation, and many, many, many for "hue" or chroma, as you might call it.

However, I do take heart in the fact that we can ignore some of the variety of pitch and focus on under 100 classes of pitch, corresponding to critical bands, and that no one has to work directly with their extreme variability. Making a computer work out a way to increase a tone's saturation in 80 variables whilst maintaining its character in most of them (preserving chroma and timbre) doesn't strike me as anything like a show-stopper. The main issue, as with color, is picking which frequencies exactly to use. In the visual realm we've settled on three colors to use for computer monitors, but they are far from producing true-to-life colors (some, such as cyan and emerald, are missing entirely). It's possible that similar imprecision (which I predict will be heard as "roughness" of tone) will be tolerable in audio.

[paul-donnelly]

So it's less a question of how to come up with a less saturated color than it's a question of which one we should use.

Not how to rotate a tone, but which way to turn it. I'm not quite clear on whether new axes change the hue axis into a hue space, or whether they create more concepts like "hue" and "value."


Assuming I'm right, you might want to glance at some stuff on 4 dimensional space. It can make the murky water of greater-than-3 dimensional rotation a little less murky. in 4D space, you get two (I believe) additional axes of rotation, which translate to some slightly incomprehensible motions when you move at right angles to the plane's we're used to.

[TS]

The ratios are due to how the harmonic overtones of single tones fit together.

Here's an article that explains it:
http://eceserv0.ece.wisc.edu/~sethares/consemi.html
And here's the original paper with more details:
http://eceserv0.ece.wisc.edu/~sethares/papers/consance.html

The infinite octave was about moving from 1:1 to 2:1, and that was accomplished by amplifying the harmonics that the 2:1 tone had, and diminishing the harmonics that the 1:1 tone had, so perhaps what is needed to move from 1:1 to 3:2 (perfect fifth) is to amplify the harmonics of the 3:2 tone, and diminish the harmonics of the 1:1 tone. Some harmonics will gradually disappear, some new ones will appear, and some will stay the same because they appear both in the 1:1 tone and the 3:2 tone.

[paul-donnelly]

[aruffo]

[paul-donnelly]

[aruffo]

[paul-donnelly]

[aruffo]

[TS]

[aruffo]

Oh! That makes sense..! I like it especially since it continues to be a manipulation of overtones which means that it continues to be "pitch". I'll have to try that, both to see if I do in fact understand what you're describing and to see if it produces the desired effect.

I suppose I came near what you're describing when on the main page I came up with the tone which was 5% C-power; I just didn't take it all the way and fade out the C completely. In a way it already proves the point, because it's compositionally neither C nor E but could be perceptibly either.

A functional problem that I had had with that, and also with the problem of sliding along the infinite octave, is that where you start from determines the overtones immediately available to you. Once you move from C3 to C4, you can't move from C4 to C5 without introducing the C5's partials. However, once you fade in the C5's partials you can keep right along (which is exactly what I did to achieve the three-octave slide). So there must be something similar which can be done to move around the harmonic lattice-- even though fading from C to G means you don't have the G's partials immediately available to you, what you have is (in my current parlance) a somewhat desaturated G, and once you fade its partials in you can fade out the ones you need to remove to get to the next third or fifth.

The thing I'll be interested to find out, then, once I have a tone that is an acceptable midpoint between (as the best example) G and E-- does this, then, create an interval with the C-root which is ambiguously either a fifth or a third, without creating an interaction that is something entirely different?

If I find I can create a sliding harmonic interval space like this, I know exactly what experiment I want to conduct on it.

[paul-donnelly]

This isn't 100% on-topic, but it follows from what we've been talking about: I just realized that I don't have the faintest idea why you're talking about harmonic lattices. Pardon me if you've written something on the topic already and I've forgotten or missed it (in which case a link would be wonderful), but I've never thought about lattices outside of a just intonation context, and the last time I checked you were primarily interested in the usual 12 tone equal tuning.

[aruffo]

Hm... why am I talking about harmonic lattices? It's taken on somewhat a life of its own... I should check why I originally started paying attention to em and compare that with what I'm thinking about em now.

Oh yes. Once I started thinking that harmonic relative pitch was a more musically appropriate measure (and skill) than "distance" between intervals, I was pleased to encounter the lattice because it provided the alternative to the standard chromatic scale, describing how intervals could have harmonic rather than scale-step relationships.

The reason I've been wanting to move freely around the lattice space is that I want to find out if harmonic space is a legitimate continuum like the standard arithmetic scale, or if instead it's just a theoretical convenience.

[paul-donnelly]