I ran across this web site during an informal scan for interesting things...

www.hermode.de

This fellow obviously has a great deal of knowledge about tuning systems and he has developed something quite interesting that can now be implemented thanks to all the computing power inside of our synthesizers. His web site is a highly educational experience, for people like myself, who are not experts in tuning systems.

So I sat and thought about how I could implement something akin to his Hermode tuning system for Kyma. It is clear that in general he tries to bend the frequencies of the major triads to produce pure harmonic 3rds and 5ths, but in general he also bends the tonic, or base key, of the triad so as to "minimize" or "equalize" the amount of bending needed for the complete triad.

He also states that there are some conditions under which he will sacrifice pure harmonic intervals, although I have no idea when that happens.

The attached Sound file implements two Sounds. The first is simply an example Sound developed from the Kyma Tunings tool for Just intonation. But the second Sound called "C Major Scale.kym" implements my attempt at something like Hermode tuning for triads in the C-Major scale.

The idea in that sound is to allow the base or tonic key (am I even using that term correctly??) to be what it normally is in the equal tempered scale, then apply intervals of 4 and 7 for the 3rd and 5th respectively. Those three note numbers are then mapped back to the C Major scale, and then the intervals of the resulting bent 3rd and 5th are computed against the bent tonic note. Those intervals are then mapped into a Just tuning scale relative to the tonic.

My aim was to have pure harmonic 3rds and 5ths in those mappings where we really do produce 3rds and 5ths in the C Major scale. But due to the mappings from intervals (0, 4, 7) back to C-Major we can have actual intervals of [3,4,5] for the "3rd" and [6,7,8] for the "5th". In other words, holding the triad to the C-Major scale sometimes produces minor chords.

[Sorry for being such a rank amateur about this stuff... I have only a rudimentary music theory background... maybe I should sign up with the Online Berklee College courses?]

At any rate, this is one attempt at something like Hermode tuning.

-- DavidMcClain - 17 Oct 2004

A perfect 3rd is about 14 cents short of an equal tempered 3rd, while a perfect 5th is about 2 cents stretched. This sound didn't do this, but the Hermode Tuning site recommended adding an offset to all three notes of a major triad in order to minimize the maximum deviation from equal temperament.

In order to do this, simply add 4 cents to each note -- when you actually have a major triad. That puts, e.g., the C at +4 cents, the E at -10 cents, and the G at +6 cents. In this case the maximum deviation would be the E at -10 cents, while the sum of the C and G offsets is also +10 cents.

As a physicist there are two points that I would make:

1. The most important thing here is to align the fundamentals of these three notes so that their higher partials coincide or else form difference beat notes that coincide with other partials or behave as subharmonics. For example: the triad beginning at A440 would have D660. Their difference is A220 exactly one octave below the tonic.

2. Minimizing the sum of the cents deviations on either side of the equal temperament scale is an odd way to "fix" things back to the world of equal temperament. I would have attempted to minimize the maximum deviation of all notes in the triad. So instead of adding 4 cents to each note, I would have sought to add 6 cents, making the deviations +6, -8, and +8 cents in the major triad.

Hermode tuning also generally provides a fractional amount of correction, where 100% is full Hermode tuning correction. My Access Virus has this Hermode tuning implemented in the latest OS. They often recommend using about 50% correction, called "Natural", (100% is denoted as "Pure") to allow some beating and motion among the partials. They like to demonstrate this effect by using sine wave oscillators sent through harsh electric-guitar-like distortion, which as far as I can tell, is simply strong clipping in order to generate a raft of strong harmonics. A major 3rd played in this way with equal temperament sounds very harsh indeed, but with Natural or Pure tuning you can actually get away with this in "power chords".

-- DavidMcClain - 17 Oct 2004

[Sure would be nice to be able to exchange pictures and charts...]

I just did a simple math analysis showing how the partials from each component of a pure triad align out to the 10th harmonics. In retrospect, the system is easy enough to do without graphical aids.

With a perfect 5th tuned exactly 1.5 times the tonic frequency (ratio 3:2), all harmonics of the 5th alternately align perfectly with higher harmonics of the tonic, or else they fall exactly halfway between the harmonics of the tonic. Hence all difference frequency beat notes are exactly one octave below the tonic, and all sum frequencies either align with higher harmonics of the tonic, or else they fall precisely halfway between them.

The perfect 3rd tuned exactly 1.25 times the tonic frequency (ratio 5:4) has harmonics that align every 4th partial of the third with a tonic partial, and every 5th partial aligns exactly with a partial from the perfect fifth. Difference frequencies are exactly 1 and 2 octaves below the tonic fundamental, and sum frequencies fall halfway between the tree of fifth partials and the tonic partials.

This all happens nicely because the ratios are so simple 3:2 and 5:4. The result is that there ought to be a strong sense of a sub-octave partial which is exactly what we hear. The alignment remains "perfect" in some sense for all higher partials of all three notes in the major triad.

In contrast, the equal temperament scale shows various misalignments among the higher partials of these three tones, ranging from subtle at the 5th harmonic of the tonic, to very large and incommensurate misalignments elsewhere. Hence there would develop a whole forrest of subharmonics leading to the sense of harshness that we hear but frequently disregard in our western minds.

I suspect that well trained musicians on winds, strings, and other instruments that permit subtle tone bending (human voices?) will subconsciously bend their harmonies to achieve more nearly perfect tuning with each other. The amounts of bending are so slight as to be almost imperceptible without some reference tone to judge against.

-- DavidMcClain - 17 Oct 2004

Some considerable time ago it was suggested that if you add together two oscillators, such as the fundamental tonic and its 5th, you actually see the subharmonic pattern in the amplitude display. This is true, yet (and I was stumped by this explanation at the time...) there really is no such subharmonic energy in the combined output signal. That visual appearance has nothing to do with detecting energy at an octave subharmonic. We do in fact hear that beat note suboctave sound. But that is due entirely to the nonlearity of our human hearing loudness response.

Plant a narrow filter tuned to the suboctave and nothing will exit that filter. That's because our filters are linear systems. But do something to the signal ahead of that filter, such as rectification (Absolute value) or squaring or anything nonlinear (which means anything apart from simple gain or attenuation) and then you will see some energy pouring out of that narrow suboctave filter.

Now back to the visual representation of the signal envelope... perhaps our minds are sufficiently nonlinear that we detect the subharmonic pattern in our visual cortex - a prescient indicator of what we will hear. But no energy actually exists at that lower pseudo-tone. Only our eyes and our ears detect it - due entirely to the nonlinear response of our senses...

[I'm reminded of another pseudo-phenomenon from physics... Imagine a very large pair of scissors in the sky. So large that it extends as far as you can imagine. Now start to close those scissors very slowly. At some point far enough away, it will appear that the intersection of the scissors blades will be closing and advancing at speeds that equal or exceed the speed of light... But no energy is contained in that intersection point and so no violation of causality will occur. Kind of like the difference between phase velocity and group velocity of signal propagation...

In fact, in my ancient studies of why galaxies form spiral arms, we once had a theory known as "Spiral Density Wave Theory". A density wave is nothing more than an illusion -- imagine flying in a traffic helicopter over busy rush hour traffic. You will notice that cars appear in bunches near traffic lights and then spread apart to lower densities between these lights. That is a density wave. The group velocity of the wave describes the motion of individual cars -- all of them progressing. But under some conditions (and true for Galaxy spiral arms as well) it will appear that the density "wave" is actually propagating backwards, and at other times forward. That's phase velocity and it carries no energy of meaning... cars really do reach their destinations eventually...

In a galaxy, the stars and gas and dust are in orbit about the center of the galaxy. Yet their spiral patterns appear where incoming and outgoing spiral density waves constructively interfere. Gas, dust, and stars bunch up in gravitational wells. Yet they continue forward in their orbits. However, the phase velocity of that spiral pattern will often appear to be moving in the opposite direction of the mass motion in orbit.]

-- DavidMcClain - 17 Oct 2004

I wonder if you could sonify this concept using phase modulated (aka FM) oscillators. In phase modulation, you can sometimes read backwards through the wavetable and sometimes forward. Could you create a sonic representation of the process that forms the spiral arms of a galaxy?

-- CarlaScaletti - 18 Oct 2004

Hi David

looking for the sub harmonic is one way to test if a musical cord is in harmony, and many have used this as conveniant way of testing cords, and then jumped to the conclusion that thats what we humans do. They may be right, but you can move your narrow filter up into the harmonic aria of the sound and see this beating effect while still keeping it totaly linear. I suspect that our sence of tuning may be continuesly being recalibrated as so many sounds that we hear are single notes which have harmonics that are perfectly in tune. Just imagine if we could get a volenter to wear a pair of headphones which was fed the outside sound except that it was first fed through a single side band ring modulator that put all the harmonics out of tune with each other. If he wore them for two weeks, would he then recalibrate his hearing such that taking the headphones off would make every thing sound out of tune, similar to the upside down glasses experiment. My guess is that he would and that our sence of harmony is not related to beat frequencies but more related to familiarity. This is purely gut feeling on my part, but is based on the thought that an orchestra haveing many instruments that are far from being absolutly in tune, with strings that are vibratoing at all different phases, can still sound harmonious in spite of the total destruction of any regular sub harmonic which may be present in the sound.

-- PeteJohnston - 19 Oct 2004

Hi Pete!

You certainly have a twisted imagination! (That is intended as a compliment!!) But it does seem that we humans, playing violins, might naturally bend the finger positions to create a deeper sense of harmony... on the other hand, the box being so close to the ears also makes the sound appear flatter than it really is. Most of us have been just fine with equal temperament, so growing up with that sound may well condition us to expect what we hear as pleasant enough.

I was playing around with a sampler in Reason last night with some horn samples and found that despite the lack of anything like Hermode tuning capabilities, those horns made near perfect sounding chords. But since I have no way of switching on Just tuning, perhaps my mind is hearing what it wants to hear?

Cheers,

-- DavidMcClain - 21 Oct 2004