Prime Factors

I just found this post lurking unpublished. I like it so let's let it out into the world.

So I wondered what would happen if instead of using base 10, each place in a number represented a prime factor. The "units" is instead the number of times each number is multiplied by 1 (this is always 1 for every number equal or greater than 1). The "tens" becomes the number of times 2 features as a prime factor in a number. So 11 in this prime factor number system is 2 in decimal. 21 is 4 in decimal as 4 = 2^2 * 1. The "hundreds" is instead the number of prime factor 3s. Is there a name for this system and is it used for anything? I can't be the first person to have thought of this.

Decimal Number	Prime Factor Number	Name
1	1	Root
2	11	Octave
3	101	Fifth
4	21	Octave
5	1001	Third
6	111	Fifth
7	10001	septimal (not seventh)
8	31	Octave
9	201	Second
10	1011	Third
11	100001	Tritone?)
12	121	Fifth
13	1000001	(Minor Sixth?)
14	10011	septimal
15	1101	Major Seventh
16	41	Octave

Phenomenology of Rhythm

Introduction

So recently I started studying drum rudiments. Despite being an accomplished drummer I had never looked at rudiments because I thought they would be artificial and not apply to playing drumset as well as they did to marching bands. Through studying the standard 40 rudiments I have discovered that:

1. I could already play most of the rudiments, I just didn't know the names. Oftentimes I understood there to be something "essential" about certain patterns which turned out to be either rudiments themselves or very close to established rudiments (i.e. my feel and the maths were almost, but not quite in line).
2. Rudiments are artificial and don't apply to playing drumset as well as to playing in a marching band. The original list of Swiss rudiments was an essential piece of theory for marching bands sending messages to troops via their playing. The 40 strong list of rudiments we have today is a fudge between this and some weird attempt to fall short of being definitive.
3. The world lacks a sophisticated phenomenology of rhythm.

Phenomenology of Rhythm

By "phenomenology of rhythm" I mean a list of things that can be played on a drum kit. I think we need to build one, and I think I am uniquely well placed to do it. I am an accomplished drum kit player with a background in the feel and practice of playing drum kit rather than in the theory of it. I know enough about music theory to allow me to undertake this task, but not so much that my approach will be clouded by it. I have already explored harmony (in this blog) in a way that has set me up to use a similar technique to explore rhythm. Finally, and most importantly, I think we are at a point with rock music where we can look back, in a similar way to how one might look back at the classical period, and categorize and analyse the practice of the time. Or, in other words, there is a corpus or a canon to study.

Limitation of Combinatorics

There are a number of different approaches that could be taken to this. You could sit down and use combinatorics to list all the possible combinations of rhythms. The standard rudiments hold that they are informed by a laissez-faire version of this technique. But in practice musicians don't do this. There are a more limited number of rhythms that are employed. The situation is analogous to phonemes; there are comprehensive tables of sounds or letters that can be created in the human mouth and throat. However, no language uses them all. I am interested in the sounds that have been produced by drummers, not the rhythms that could have been played.

Canon

I suppose I better define a canon. What I am interested in is rock music including a drum kit. So I'm not at this point concerned with drum machines, or latin percussion or bodhrans or anything else. I want music played by a human drum set player with limbs.

To define a drum kit I will arbitrarily say it is a thing consisting of multiple percussion instruments, including, as a minimum, a bass drum operated by a pedal and a snare drum. A drum kit is played with a stick held in each hand (although there are very few exceptions). I do not consider a rhythm played with alternating finger tips to be drum set playing. I would usually expect the kit to include a modern hihat, and also ride and crash cymbals and a set of tom-toms. However, so long as the pedal-operated bass drum and snare are present, I think that is a drum kit. Once the core of this work is complete, we should dismantle these definitions and broaden the scope of the phenomenology. To summarise:
1. A drum kit has (at least one) pedal-operated bass drum.
2. A drum kit has a snare drum (or a functional equivalent taking its place such as a timbale).
3. A drum kit is played with a stick held in each hand (very few exceptions), and with the feet operating pedals.

By rock music I mean the style of music characterized by the heavy use of a backbeat that may have begun in the 1950s with artists such as Little Richard, although I am reluctant to draw any line. Led Zeppelin played rock music. The Sex Pistols played rock music. The Beatles played rock music. Nirvana played rock music. I am also wary of defining this canon in reference to the backbeat because any phenomenology that results will inevitably beg the question. As an arbitary cut-off point I'll draw the other end of the line for now at the year 2000.

Goals

I hope that in studying a phenomenology of rhythm, some better understanding of drum set playing will be arrived at. I hope this may become a tool for students, distilled into some form appropriate for that task. Finally, and more loftily, I hope that some unifying idea can be arrived at that links rhythm and harmony. I believe that there is some emotional similarity between the interval of a fourth, which is created by two tones in a relationship of 4/3, and of a polyrhythm of four beats contrasted to three beats. Each example repeats over a period of twelve units; with the drums we are talking about beats in a bar, and with the piano we are talking about wavelengths. I am hoping to demonstrate that rhythms are nothing but slow sophisticated harmonies.

Method

To begin the project I will start experimentally by collecting examples of rhythms from the canon and categorizing them. Once a corpus of data has been built up then I hope the road ahead will become clear. I expect to find something like 95% of rhythms to be related to the backbeat, that is, with an accented snare on the third beat (in four time). I am interested in patterns moreso than tempo, and hope to work from the usual to the unusual.

Thanks for reading. Let's begin.

Minor Seventh Equal Temperament

Many scales have been defined by propagating a certain interval up or down until an multiple-octave interval (or an approximation thereof) from the root is arrived at. Each of the intermediate notes is then transposed through all of the octaves to fill in all of the notes of the scale. Equal Temperament is defined like this, using an interval of a "fifth" where seven octaves equal twelve "fifths". Meantone Temperaments tried to even out the error between in tune fifths and in tune thirds by propagating with flattened fifths. There are many interesting just (and non-just) intervals that can be used to create scales in this manner.

I'm particularly fascinated by the minor seventh at the minute. I came to a greater understanding of the thirds by creating scales propagated on those intervals, so perhaps I will get somewhere propagating a scale using the minor seventh? The problem is that the equally tempered minor seventh is powerful and fascinating because the brain reads it as both 7/4 and the complement of 9/8. You cannot have a just version of the equally tempered minor seventh, because the minor seventh is a product of equal temperament. (An approximation would be possible, but what would be the point in abstracting it one further step?)

So let's define a scale by propagating the equally tempered minor seventh. What happens? You get 12-TET normal everyday standard equal temperament. This is true for any equally tempered interval. In fact, you could define notes intervals in any given equal temperament as those notes which when propagated produce each other. This is because the temperament is equal. That, at least, is interesting in itself.

Notes on a Modulating Just Scale

I have now had the chance to play a keyboard tuned to the scale I proposed here.

The difference between the Fourth of 4/3 and the Super Minor Third of 27/20 was perceivable but too small to be of any value. I would suggest that a piece composed with many II chords and few IV chords would do well to adopt 27/20 rather than 4/3, but that otherwise 4/3 is quite suitable. For example, much Godspeed You! Black Emperor would benefit from this adjustment. I think the powerful identity of 4/3 (it is after all the complement of a very low numbered partial) is such that one tends to hear 27/20 as an out of tune 4/3. This relative identity is an interesting concept that should be explored further.

Two Bbs were used, 7/4 and 9/5. The 7/4 was indeed useful and appropriate, although it did not ring as clearly in the I7 as I would have expected. In some ways it had more of the quality of a 7th chord, and in another way it did not. I think by denying the minor seventh an identity as the complement of the wholetone, part of the leading feeling of the 7th chord is lost, but that by isolating and perfecting the identity of the minor seventh as the seventh partial, another aspect to the character of the 7th chord is distilled and amplified. This is a tool to use with discretion. The two tones were readily identifiable and it was entirely appropriate to have both on the same instrument. I would consider placing the 7/4 on the B and the 9/5 on the Bb and doing away with the major seventh altogether.

I would like to make more observations of the two As in the scale.

Overall, the whole thing held together and rang beautifully. The modulations worked just as I had intended, and it was very pleasant as a melodic scale. This proves that the ideas I have about abstract numbers on paper translate properly into real world musical situations. I know what I am doing.

Phenomenology of Scales

I want to start work on a phenomenology of scales.

Some attributes that I think might be useful are as follows:
1. "Limit": this is the highest partial involved in the scales. For example, Partch's 43-tone scale is one flavour of an 11-limit scale. Western music is largely an approximation of 5-limit. Organum is 3-limit.
2. Symmetry: what happens beneath the tonic? Is a scale mirrored (complementary) or rotationally symmetrical or non-symmetrical? The Dorian and Ionian modes are unique in being rotationally symmetrical about the tonic (rather than the logarithmic mid-point). Partch's 43-tone scale is mirrored due to an error in understanding the nature of utonality. Organum is both mirrored and rotationally symmetrical. Most western modes and western "minor" are non-symmetrical (I think this is a result of being 5-limit; 3-limit and 7-limit reflect more easily. In general, if the limit is one less than a power of 2 the scale may be easily designed to exhibit good symmetry. Is this related to methods for searching for prime numbers?).
3. Utonal limit: which partials are considered suitable for transposing tones into the home octave? In Pythagorean tuning, only the Octave and Fifth are suitable (it has a Utonal 3-limit). In Partch's 43-tone scale it is an 11-limit. Organum is also a 3-limit. (In these latter two examples the Utonal limit is the same as the Otonal limit.) When I started this, it was because I wanted to create scales with a Utonality of exactly 19. I now see that this is nonsense, but scales could be constructed using this method. Scales with a high Otonal limit and a low Utonal limit will be close to the Harmonic series (the Harmonic series is a scale with an infinite Otonal limit and a Utonal limit of 1).
4. Dissonance-limited: rather than limiting scales by partials, they could be limited less crudely by measuring the dissonance of their intervals.
5. Respect for the Octave: does a scale repeat every 2/1? A pentave, repeating on the 3/2 could be used instead, or even other intervals.

I am sure there are more things that would be useful that will present themselves as time goes on.

Beautiful Numbers

So I've been converting a lot of values between cents and hertz, between musical tone and physical frequency. The way the numbers work out is absolutely beautiful.

I've been working with 440Hz as my baseline because modern practice is to tune A=440Hz. It seems that this was a very sensible choice given the biases of modern music. I find the following pattern:
440Hz Tonic
550Hz Major third
660Hz Fifth
(770Hz Septimal)
880Hz Octave
That's really quite beautiful. Is this a product of choosing 440 as the root?

If we pick 330Hz for the tonic, we get:
330Hz Tonic
440Hz Fourth
550Hz Fourth + Major Third
660Hz Octave
So there's different patterns for different starting values. What we are doing here is logarithmically modifying our selected utonal rules.

Or, with 100Hz as the tonic:
100Hz Tonic
110Hz ?
120Hz Minor Third
130Hz ?
140Hz (0.63Hz out from the Super Major Third)
150Hz Fifth
160Hz Fourth + Minor Third
170Hz ?
180Hz Fifth + Minor Third
190Hz ?
200Hz Octave

I could go on picking out different patterns forever. I realise that I am using the same ratio method I have been using but viewed from the opposite angle. Nonetheless, this is a quite different way to think about things, an alternate cognition of tone. Perhaps it will be fruitful.

Modulating Just Scale

I've designed a scale. It's like this:

1/1           C  Tonic
9/8           D  Supertonic / Fifth Fifth
6/5           Eb Minor Third
5/4           E  Major Third
4/3           F  Fourth
54/40 = 27/20 F  Super Minor Third
45/32         F# Super Major Third
3/2           G  Fifth
24/15         G# Fourth Minor Third
20/12 = 10/6  A  Fourth Major Third
27/16         A  Super Fifth
7/4           Bb Septimal 
16/9          Bb Subtonic
18/10 = 9/5   Bb Fifth Minor Third
15/8          B  Fifth Major Third
2/1           C  Octave

In this scale you can modulate between I, II, IV and V chords with both minor and major thirds with absolute just accuracy.

Frankly I don't know what's going on with the minor seventh, so I've included all the options. Obviously to facilitate modulation we need the the 18/10 Fifth Minor Third. There is open (and probably unanswerable debate) as to whether the 12-TET minor seventh is approximating a septimal (i.e. derived from the 7th partial) interval or is the complement to the major second. It's probably both, so I've included both. But I don't think anyone needs three B flats, so there's work to be done here. Perhaps with more experience I would be able to make a firmer choice.

This scale will play modal forms such as C major and D dorian adequately well, although better scales could be designed for that purpose.

So that's basically it. I believe this scale is capable of playing most 20th century popular music and most folk more beautifully than 12-TET (it won't play III, VI or VII chords or 7th chords, except the root seventh). Now I need to build an instrument to test it.

Apologia
Some of these intervals are as small as 21 cents. Some are over two semitones. If you're playing a melody it would make no sense to run up and down the notes, just as it makes little sense to play a 12-TET chromatic melody. It might take a while to feel which notes are right, but it may also be possible to suggest harmonic change just by playing a melody. How effective this is remains to be tested.

Some of these intervals appear to be more dissonant than perhaps would be expected. This is because they are intended to suggest a shift in the root note ("key change"). Accepting this more firmly than is reasonable, the largest dissonance that remains is 9/8.

Mathematically Modelling Progression and Resolution

Partch makes an offhand statement that a move from one of his 43-tone notes to a certain other is a move from dissonance towards resolution. This was the first moment in which I saw the link between consonance and resolution.

Resolution is when a melody or harmony comes cleanly to an end. Typically it means returning to the root note or chord at the end of a phrase. The second most resolved place to finish a melody is on the fifth. Progression is when you move away from the root note, developing a melody or harmony into something interesting. Guitarists often say that progression is when you move forward around the circle of fifths, and resolution is when you move back. In both cases, what you are doing is controlling the amount of dissonance relative to the tonic. More dissonance is progression, less dissonance is resolution. In the case of the circle of fifths, you are both playing with powers of 3/2 and increasing the dissonant error in your tuning when you move forwards.

We can do maths with this. Let's assume a simple single note melody (if you like you can imagine a single note drone underneath).

At any time the dissonance is equal to the product of the ratio of the interval between the current note and the tonic.

Let D = dissonance, o = odentity (the numerator of the ratio), u = udentity (the denominator of the ratio)

D = ou

The progression or resolution of a melody is the rate of change of D by time.

Let m = progression/resolution, t = time

m = ΔD/Δt

As an example, let's assume a melody playing four equal-length notes on C, G, D and C. As the melody plays the second note, G, the dissonance is equal to 2x3 = 6 (D = ou). When it was playing the first note, D = ou = 1x1 = 1. So the change in D is 5, and the change in t is 1 unit. m = 5/1 = 5. Coming to the third note, D = ou = 9*8 = 72. m = (72-5)/1 = 67. Finally, at the last note, m = (1-72)/1 = -71.

time, t | dissonance, D | progression/resolution, m
0       | 0             | 0
1       | 1             | 1
2       | 6             | 5
3       | 72            | 67
4       | 1             | -71

The negative value of m at t = 4 indicates that it is resolution, not progression, that is occuring. It will be noted the m is not zero-sum, just as in musical practice.

I believe that it's not possible to do proper calculus with this because the series involved are discrete and not continuous.

Obviously this is an abstraction, in real music you do not get isolated one note sine-wave melodies. We need to take into account:
Polyphony
Moving root notes ("key changes")
Timbre
But none of these problems are insurmountable.

It should be obvious that by reversing this method, it would be possible to design a melody or harmonic sequence in which exactly the desired degree of progression and resolution are manufactured. Perhaps you are trying to illustrate a story or lyrics and you want your song to have a certain shape. No longer do you have to have a memory for trial-and-error and tradition generated chord sequences. You could trivially design a program to create them for you, perfectly. I imagine such a program would require a defined set of notes (a key) and some rhythmic ideas to get going, but you could really come up with some cunning harmonies far more complex than a human can efficiently design.

Intonation 8: What is Minor?

I have argued at length that I think that the western use of a bastardised version of the aeolian mode as "minor" is the result of an authoritarian practice approaching the dorian by trial and error.

Partch thinks that "minor" is relationships within utonality, by which he means the denominators of ratios. What does this mean? It means that all the partials which are brought into the home octave by bringing them down a number of octaves are related. It means that all the partials which are brought into the home or second pentave by bringing them down a number of fifths are related. It means that all the partials which are brought into the first three 3-aves by bringing them down thirds are related. It means very little.

At first I thought he had found a justification for why A minor is relative to C major; I thought he had defined this relationship. But it doesn't line up.

Then I thought he had struck upon some kind of inversion/fourths-based/lydian harmony which he had confused with minor. But I can't make sense of that, it's just wishful thinking on my part wanting things to be simple.

This can't just be nonsense, can it? Perhaps I have misunderstood. I suspect that Partch has tried to adopt a fashionable theoretical apology for the minor mess without questioning it, but really I am in ignorance.

Utonality refers to nothing more than the unit by which the scale repetition is defined. Partch is unique in giving equal gravitas to all available options simultaneously. I think he jumped in too early with the numbers and just plain didn't know what he was doing.

I would like some more advanced guidance in this. I tried drafting emails to music professors, but what could they possibly tell me?

Intonation 7: Consonance and Dissonance

It's self evident that some combinations of notes sing pleasantly and other clash nastily.  In between, there are endless shades of grey.  At what point does consonance become dissonance?

For medieval church music, the answer was that only harmonies of octaves, fifths and fourths were permissible.  For Harry Partch, ratios built out of prime numbers less than 11 are acceptable (he also includes 9, "because it is an odd number".  He mentions 15 in this same context and it is clear to me that he is saying that numbers with two prime factors have a value greater than other numbers but less than primes).  He halted at 11 and did not proceed to 13 simply because he thought that gave him enough to be getting on with.

A simple mathematical explanation of consonance is that it is a measure of how often frequency intervals line up. An octave is extremely consonant because it has the maximum possible agreement between two different frequencies (the wavelength is half or double that of the other).  A fifth is very consonant because three wavelengths of the fifth frequency line up with two wavelengths of its tonic.

So, we can measure consonance by multiplying together a ratio's numerator and denominator (Partch calls these the odentity and udentity, I don't know if this is common practice).  It's important to have the simplest equivalent fraction, for example, 2/1 is correct but 4/2 is not.  This is because what we're actually after is the product of the identities (prime factors) of the note.  Here are some examples:

2/1 (octave) - dissonance 2
3/2 (fifth) - dissonance 6

5/4 (just major third) - dissonance 20

45/32 (a small just tritone) - dissonance 1440 (this is universally agreed to be dissonant)

It makes no sense to try and use this method to calculate the consonance of equally tempered intervals because the ratios involved are irrational.  The values that result only reflect your level of accuracy in calculating them.

I'm no closer to defining the point at which consonance becomes dissonance, but I now have a method for objectively comparing intervals and stating whether one is more or less consonant than another.

I wonder if modern music theory understands this as plainly as I have set it out?  Partch understands the general principle but survives only by rules of thumb.  I need a more advanced textbook.