The harmonic series and its implication on music

So, I have always been taught that you can find the diatonic scale in the harmonic series, just look at this picture:

Harmonics 8-16 (omiting the 14th harmonic) give us the diatonic scale, right? This, I used to think was the natural basis of the diatonic scale. But the truth is, the 8th-16th harmonics sound nothing like the diatonic scale, actually it sounds like a hybrid between an Arabic Maquam, with all the neutral intervals and such, and the diatonic scale. This is because the above picture is actually just the approximation of intervals in 12-TET... Some of the intervals are as much as 50 cents off of 12-TET!

We still can however, find the diatonic scale in the harmonic series, just not in a neat scalar order. You, for example can find the diatonic scale by using these harmonic ratios:

1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

Take note that the diatonic scale can be tuned many more ways in just intonation than these ratios.

So, my conclusion. The diatonic scale itself isn't natural, but the intervals that it uses (or in the case of equal, or many other temperments, approximates) are.

Although, if you stack a series of 6 perfect fifths, you will make a diatonic scale (lydian to be percise), possibly how people discovered the scale. Also, you can get the major pentatonic scale, VERY popular in various musical cultures, from a stack of 4.

Another fun fact, the diatonic scale has the most major/minor triads than any other 7-note scale.

I may be rambling on here, but another intersting point of discussion (which I know little about) would be the undertone series.

Read this book:

Lydian Chromatic Concept of Tonal Organization.

Read this book:

Lydian Chromatic Concept of Tonal Organization.

Way too expensive... And they don't have it in my library. I know the basics (E.A, what little information on it there is online) though.

I have been to the website before, and one thing that confused me (in the FAQ) was this:

6. What is the fundamental difference between the Lydian and Major Scale?

As described in the answer to the previous question, the Lydian Scale has one single tonic, otherwise known as the “DO” of the scale. The Major Scale is known as a diatonic (meaning: two tonic) scale. Therefore, the essential difference between these two scales is that the Lydian (a single tonic scale) is in a state of unity with itself, and the Major Scale, with its two tonics, is in a state of resolving.

Last time I checked, the Major scale had one tonic, Both the major scale, AND the Lydian scale are diatonic, And the etoymology of "diatonic" actually means "Progressing through tones"

The harmonic series can be used only to determine the functions of harmonies, not the melodic functions of any scale. That is, each relationship between partials in the harmonic series has a unique harmonic (chordal) function and connotation in music, but we cannot infer melodic functions from any of these just intonation ratios. For instance, the ratio 5/4 is the ratio between the fifth and fourth harmonic, and it has the very familiar function of the bottom interval in the major triad, but one can not instantaneously infer the function of the "major third" in the diatonic scale from that ratio.

The melodic function of notes in a scale can be determined by their locations in a matrix of notes created by stacking up an interval (called a generator) on top of itself multiple times and repeating each new note's location within another larger interval (called a period). Two notes' locations and closeness in this stack of generators determines the tonal relatedness of those two notes melodically in a scale created by the continuously stacked generator.

The harmonic function of the notes in a scale created by a continuous stack of a generator reduced into one period can then be determined by defining the generator and period as tempered just ratios, and then defining what combinations of these two intervals represent other prime just ratios.

Example:

We play our music with scales created by a 700 cent generator repeated in each 1200 cent period (Cent=unit for measuring musical intervals where there are 1200 cents per octave.

We use scales created by stacks of 700 cents. They are the pentatonic, diatonic, enharmonic scales.)

We traditionally define these two intervals as the just ratios 3/2 and 2/1. (Defining them as just ratios allows us to make the tie in of scales and melody with harmony and the harmonic series.)

We then define other prime ratios as a combination of 3/2's and 2/1's. We already have the primes 3 and 2, so the next is 5/1. We reduce 5/1 to 5/4 to put it in one octave, and we define the ratio 5/4 as equivalent to up four 3/2's and down two 2/1's (or up five "fifths" and down to "octaves"). Mathematically it looks like this:

5/4 ≈ [(3/2)^4] / [(2/1)^2]

This tells us that in this matrix of notes created by 3/2's and 2/1's, we have defined the harmonic function of the interval 5/4 as up 4 fifths and down 2 octaves from the original note. Though this mathematical statement is not mathematically true, the point is that we are *representing* a 5/4 with the combination of four fifths minus two octaves. This applies across all values that could represent the interval 3/2 other than 700 cents which lead to completely different tunings, all of which will still function similarly however due to the creation of their scales through the same approximated just interval and also due to the same definition of prime limits.

Another cool thing about defining things this way is that you determine which harmonic functions become redundant, or are "tempered out" in this matrix due to the mathematical definitions of your prime limits.

Example:

Given that 5/4 ≈ [(3/2)^4] / [(2/1)^2], it becomes true that 80/64=81/64, and therefor that 81/80 =1.

That means that any intervals whose difference is 81/80 become functionally the same. The interval 81/80 has been "Tempered out".

Example:

The ratios 10/9 and 9/8 are both referred to as just intonation minor seconds. They are each equivalent to 80/72 and 81/72 respectively. Because the interval 81/80 is "tempered out" in our previously defined matrix, and the difference between the two intervals is that ratio, then the two intervals are defined functionally as the same in our matrix.

Finally, once you've gone from generators, periods, and a matrix to just intervals, limits, and tempering, you can then reapply everything back to the harmonic series. That is, you can then apply all your new approximations of just intervals and use them to define the new intervals between partials for a timbre that is perfectly matched for consonance in the scales created by your matrix.

The harmonic series can be used only to determine the functions of harmonies, not the melodic functions of any scale. That is, each relationship between partials in the harmonic series has a unique harmonic (chordal) function and connotation in music, but we cannot infer melodic functions from any of these just intonation ratios. For instance, the ratio 5/4 is the ratio between the fifth and fourth harmonic, and it has the very familiar function of the bottom interval in the major triad, but one can not instantaneously infer the function of the "major third" in the diatonic scale from that ratio.

The melodic function of notes in a scale can be determined by their locations in a matrix of notes created by stacking up an interval (called a generator) on top of itself multiple times and repeating each new note's location within another larger interval (called a period). Two notes' locations and closeness in this stack of generators determines the tonal relatedness of those two notes melodically in a scale created by the continuously stacked generator.

The harmonic function of the notes in a scale created by a continuous stack of a generator reduced into one period can then be determined by defining the generator and period as tempered just ratios, and then defining what combinations of these two intervals represent other prime just ratios.

Example:

We play our music with scales created by a 700 cent generator repeated in each 1200 cent period (Cent=unit for measuring musical intervals where there are 1200 cents per octave.

We use scales created by stacks of 700 cents. They are the pentatonic, diatonic, enharmonic scales.)

We traditionally define these two intervals as the just ratios 3/2 and 2/1. (Defining them as just ratios allows us to make the tie in of scales and melody with harmony and the harmonic series.)

We then define other prime ratios as a combination of 3/2's and 2/1's. We already have the primes 3 and 2, so the next is 5/1. We reduce 5/1 to 5/4 to put it in one octave, and we define the ratio 5/4 as equivalent to up four 3/2's and down two 2/1's (or up five "fifths" and down to "octaves"). Mathematically it looks like this:

5/4 ≈ [(3/2)^4] / [(2/1)^2]

This tells us that in this matrix of notes created by 3/2's and 2/1's, we have defined the harmonic function of the interval 5/4 as up 4 fifths and down 2 octaves from the original note. Though this mathematical statement is not mathematically true, the point is that we are *representing* a 5/4 with the combination of four fifths minus two octaves. This applies across all values that could represent the interval 3/2 other than 700 cents which lead to completely different tunings, all of which will still function similarly however due to the creation of their scales through the same approximated just interval and also due to the same definition of prime limits.

Another cool thing about defining things this way is that you determine which harmonic functions become redundant, or are "tempered out" in this matrix due to the mathematical definitions of your prime limits.

Example:

Given that 5/4 ≈ [(3/2)^4] / [(2/1)^2], it becomes true that 80/64=81/64, and therefor that 81/80 =1.

That means that any intervals whose difference is 81/80 become functionally the same. The interval 81/80 has been "Tempered out".

Example:

The ratios 10/9 and 9/8 are both referred to as just intonation minor seconds. They are each equivalent to 80/72 and 81/72 respectively. Because the interval 81/80 is "tempered out" in our previously defined matrix, and the difference between the two intervals is that ratio, then the two intervals are defined functionally as the same in our matrix.

Finally, once you've gone from generators, periods, and a matrix to just intervals, limits, and tempering, you can then reapply everything back to the harmonic series. That is, you can then apply all your new approximations of just intervals and use them to define the new intervals between partials for a timbre that is perfectly matched for consonance in the scales created by your matrix.

The only thing you said that I didn't know was the thing about melodic functions being identifed by the stacked generators. This sounds like something simliar to the Lydian Chromatic Concept, please elaborate.

It would be intersting to explore this idea of melodic functions being defined by generators and there proximity in one of the Magic Temperments. Maybe even in the less firmilar tuning systyem of slightly sharp minor thirds as a generator. (Kleismic Temperment)

The only thing you said that I didn't know was the thing about melodic functions being identifed by the stacked generators. This sounds like something simliar to the Lydian Chromatic Concept, please elaborate.

The lydian chromatic concept, though I don't know much of it, seems to me to be a product of the fact that, contrary to popular belief, there is a major and minor fourth (and fifth). The "sharp eleven" that seems so unnatural to our eyes but so correct to our ears is simply the use of a *major fourth* in an extended major chord.

Major and minor intervals are defined as so because some interval number exists twice in the diatonic scale. It exists in one small and one large form, so we name the smaller of the two minor and the larger one major.

For example, in the diatonic scale containing the C ionian (major) mode, the third occurs in two sizes. The third occurs in a large size between C-E, G-B, and F-A, and in a small one between D-F, E-G, A-C, and B-D. We call the large one major and the small one minor.

Now look at how many interval sizes span four steps in the diatonic scale. There are *two different ones*, the larger being between F-B, and the small being between C-F, G-C, D-A, B-E, A-D, and E-A. If standard naming conventions were to be followed this larger fourth would be called major and the smaller one minor. Harmonically and melodically the major fourth tends to function very... welll, majorly (as the LCC supports).

(The same is true for the fifth. The small lies between B-F, and the large lies between F-C, C-G, G-D, D-A, A-E, and E-B. The small should be called major, and the large minor.)

Major VS minor can also be determined by the occurrence of that interval in the different diatonic modes as one goes around the circle of fifths. The diatonic scale can be created by a continuous stack of seven fifths centered around D (Re), extending from F (Fa) to B (Ti). As one progresses through this stack of fifths starting from the bottom, each movement up a fifth yields a mode with one more minor interval.

The modes on each end, lydian and locrian, are each completely major and minor, respectively. That is, every interval relation to the tonic is major in lydian, and every interval relation to the tonic is minor in locrian. As you move up one fifth from lydian you get ionian, with one minor intervall, the *the minor fourth*. Up another fifth to mixolydian yields the minor seventh, dorian the minor third, aolean the minor sixth, phrygian the minor second, and locrian the *minor fifth*.

But anywayyyy...

It would be intersting to explore this idea of melodic functions being defined by generators and there proximity in one of the Magic Temperments. Maybe even in the less firmilar tuning systyem of slightly sharp minor thirds as a generator. (Kleismic Temperment)

This definition does arise from rank 2 temperament theory. In the Syntonic (or meantone) temperament, which is created by stacking 3/2's (fifths) and 2/1's (octaves), functions are determined by our Circle of Fifths, regardless of the actual size of the fifth, and therefor tuning. The same patterns apply to pythagorean, quarter comma meantone, and sixth comma meantone.

http://en.wikipedia.org/wiki/Syntonic_temperament

It is therefore true that for the magic temperament, instead of a circle of fifths, there is a circle of major thirds that could be used to define the melodic functions in that temperament. These functions would apply to 19-edo, 16-edo, 13-edo, and every other tuning of the magic temperament, regardless of the actual size of the tempered major third.This pattern continues on into other temperaments like the hanson (or kleismic) temperament where tempered minor thirds generate the tunings.

http://en.wikipedia.org/wiki/Magic_temperament

The functions of temperaments other than the Syntonic, however, have not been thoroughly explored at all, and so their patterns are no well known. Some patterns may stay the same between temperaments, and many things certainly change. Not until these other temperaments and their included tunings and scales are widely explored can we be sure.

I have played around a little in the magic temperament using a synthesizer called the TransFormSynth that allows you to change the tuning, in real time, along a smooth continuum defined by your current temperament. I have one video up if you'd like to hear it:

In I play in many of the magic temperament's tunings, and I use the MOS (moment of symmetry) scales that arise from continuous stacks of thirds.

John Moriarty