Making sense of tonal, "non-functional" music (Dmitri Tymoczko et. al.)

[John Ladasky]

I haven't posted here in years, but a few "old-timers" from the 1990's chimed in recently, and I thought I would contribute something.

The early 20th century is my favorite period in orchestral music: Debussy, Ravel, Stravinsky, and their peers. My ears immediately "understand" that this lush and complex music has resonance, and deep tonal implications. My music theory courses barely scratched the surface as to why. Most people agree that the theories that were developed to explain common-practice functional harmony are inadequate for the task. I own Hindemith's _Craft_of_Musical_Composition_, and Persichetti's _Twentieth_Century_Harmony_, but even these books don't go very far in explaining what my ears already seem to know.

Recently, I have been experimenting with musical scales (in 12edo) which are off the beaten path, for my own compositions. My subjective impression from auditioning various scales is that I did not like the sound of two consecutive semitones in a scale. I Googled "consecutive semitone avoidance", and I found the article "Scale Networks and Debussy" by Dmitri Tymoczko:

http://dmitri.mycpanel.princeton.edu/debussy.pdf

Soon after, I also found "The Consecutive Semitone Constraint on Scalar Structure: A Link Between Impressionism and Jazz":

http://dmitri.mycpanel.princeton.edu/files/publications/consecutivesemitone.pdf

These articles have a lot to say, but the main observation that I took away from them was that the collection of scales which are used in tonal Western music, from the Renaissance to post-bop jazz, are exactly the scales which have the following two properties:

1) They have no consecutive semitones.
2) Consecutive intervals are either semitones or major seconds.

With those constraints, you obtain all the "church" modes (Ionian through Locrian), the whole-tone scale, the "diminished" scale (alternating m2 and M2), and all the "jazz" modes (ascending melodic minor, etc.). I found that to be highly interesting.

Some additional Google searches have led me to discover a school of thought called "Neo-Riemannian" music theory.

https://en.wikipedia.org/wiki/Neo-Riemannian_theory

I have barely begun to read about this subject, but I think that what the Neo-Riemannians are trying to develop is an efficient framework which would do for chords and chord progressions what Tymoczko's two rules above does for scales. According to the Wikipedia article I referenced, Tymoczko apparently published a criticism of some Neo-Riemanninan ideas, but it seems to me that his criticisms are about some details of the theory and not the general goal.

I never learned to enjoy serial music. The constraints that serialists impose on themselves with 12-tone rows are too tight to allow much in the way of melodies that sound connected, harmonies that sound connected. I love dissonance, cacophony not so much.

If anyone has any thoughts on Tymoczko, Neo-Riemanninans, or any other systems which encompass post-functional harmony, please share. Thanks!

[J.B. Wood]

Post by John Ladasky
I haven't posted here in years, but a few "old-timers" from the 1990's chimed in recently, and I thought I would contribute something.
The early 20th century is my favorite period in orchestral music: Debussy, Ravel, Stravinsky, and their peers. My ears immediately "understand" that this lush and complex music has resonance, and deep tonal implications. My music theory courses barely scratched the surface as to why. Most people agree that the theories that were developed to explain common-practice functional harmony are inadequate for the task. I own Hindemith's _Craft_of_Musical_Composition_, and Persichetti's _Twentieth_Century_Harmony_, but even these books don't go very far in explaining what my ears already seem to know.

<snip>

Hello, and I wish Margo was still around to provide erudite comment.
I've collected a lot of material on tunings, temperament and musical
acoustics over the last 20+ years and have come to the conclusion that
for Western music (to include notation, acoustic (non-electronic)
instrument construction and tempo/performance) when all factors are
weighed, there simply is no substitute for 12-TET (or 12 EDO). IOW, the
"bad" (if that's a good term) is spread out amongst these things. One
reason 12-TET works IMO is because the human ear/brain (for the majority
of folks) allows deviations from the ideal. While an ET M3 (400 cents)
sounded out of a musical context might sound harsh to some when compared
to a just M3 (386 cents), it doesn't really amount to much in the grand
scheme. The analogy here is to the colors of the visible light
spectrum. Each color has a band of wavelengths (frequencies) associated
with it. So we have to ask how far we can deviate in wavelength from
the color red, for example, before we would say it was red-orange or
orange-red or just plain orange. You might say red-orange or orange-red
would be equivalent to a musical semi-tone.

Getting back to semitones, I think it's the ability of most folks to
easily distinguish between a C and C#, for example, whereas something
between (quarter tones) these pitches might prove difficult.
"Chromatic" means "color" and those 5 extra pitches added to the 7-tone
diatonic scale are valuable both for variety and the ability to perform
in any one of 12 keys (another benefit of 12-TET). So while I can
appreciate some of the advantages provided by just intonation (very
restrictive with only 12 tones), meantone or unequal (dubbed "well" by
some) temperaments, none of these can provide the advantages of 12-TET,
which is where I believe these historic tunings for Western music were
evolving. These 12 pitches are the paints on an artist's palette and
mean little in and of themselves. The end result demands the creativity
of the artist. We take in the whole picture, not merely focusing on
adjacent colors. My two "cents" worth. Sincerely,

[John Ladasky]

Post by J.B. Wood

Post by John Ladasky
I haven't posted here in years, but a few "old-timers" from the 1990's chimed in recently, and I thought I would contribute something.
The early 20th century is my favorite period in orchestral music: Debussy, Ravel, Stravinsky, and their peers. My ears immediately "understand" that this lush and complex music has resonance, and deep tonal implications. My music theory courses barely scratched the surface as to why. Most people agree that the theories that were developed to explain common-practice functional harmony are inadequate for the task. I own Hindemith's _Craft_of_Musical_Composition_, and Persichetti's _Twentieth_Century_Harmony_, but even these books don't go very far in explaining what my ears already seem to know.

<snip>
Hello, and I wish Margo was still around to provide erudite comment.
I've collected a lot of material on tunings, temperament and musical
acoustics over the last 20+ years and have come to the conclusion that
for Western music (to include notation, acoustic (non-electronic)
instrument construction and tempo/performance) when all factors are
weighed, there simply is no substitute for 12-TET (or 12 EDO).

The issues of tuning and temperament are tangential points to my original thoughts, but I'll take them up for a moment. I'm going to agree with you with some elaborations and caveats.

I can hear and appreciate non-equal temperaments and microtonal tunings. Many of my old posts here in rec.music.theory concerned Arabic music, which is an interest of mine, and which decidedly does not use 12-EDO. I won't get into the issue of whether 24-EDO is the "correct" way to think about Arabic music today.

That being said: the fact remains that 12-EDO covers a lot of territory. It provides many frequency combinations which are close enough to small-integer ratios to sound "harmonious." 12-EDO is the underpinning of nearly all modern tonal music, including a large body of recent work for which we are still seeking logical theories. The musical oeuvre to which I am referring (the so-called Impressionists, Stravinsky, and jazz) is definitely written by composers who were thinking in 12-EDO, for instruments which are designed to play in 12-EDO.

As for me personally, I am a composer in the Western tradition who likes three things which basically demand 12-EDO: 1) keyboard instruments (not exclusively, but often), 2) thick harmonies which "mesh" (to use Tore Lund's word), and 3) the ability to modulate all over the place.

I would love to branch out into other compositional styles. I certainly listen to some music which does not sound remotely like post-functional Western tonal music. But, you only get one lifetime!

[Anon Anon]

Post by John Ladasky
I haven't posted here in years, but a few "old-timers" from the 1990's chimed in recently, and I thought I would contribute something.
The early 20th century is my favorite period in orchestral music: Debussy, Ravel, Stravinsky, and their peers. My ears immediately "understand" that this lush and complex music has resonance, and deep tonal implications. My music theory courses barely scratched the surface as to why. Most people agree that the theories that were developed to explain common-practice functional harmony are inadequate for the task. I own Hindemith's _Craft_of_Musical_Composition_, and Persichetti's _Twentieth_Century_Harmony_, but even these books don't go very far in explaining what my ears already seem to know.
http://dmitri.mycpanel.princeton.edu/debussy.pdf
http://dmitri.mycpanel.princeton.edu/files/publications/consecutivesemitone.pdf
1) They have no consecutive semitones.
2) Consecutive intervals are either semitones or major seconds.
With those constraints, you obtain all the "church" modes (Ionian through Locrian), the whole-tone scale, the "diminished" scale (alternating m2 and M2), and all the "jazz" modes (ascending melodic minor, etc.). I found that to be highly interesting.

I haven't read this or the referenced articles thoroughly at all but....

You/he seem to be leaving harmonic minor and harmonic major out of your
scalar palette and they both have an important role in jazz as well as
in post-tonal classical music.
I understand why scales with consecutive semitones are problematic for
extracting harmonies.
But an augmented 2nd interval in a scale doesn't usually cause those
types of problems.
Even the harmonic major scale yields recognizable tertian triads and 7th
chords.
C E G B
D F Ab C
E G B D
F Ab C E
G B D F
Ab C E G
B D F Ab

And the 5th mode of harmonic minor is employed regularly on V7b9 chords
in minor keys.
E.g. C harm min on G7b9 in C minor. (Aka G Mixolydian b2b6.)

The other modes of both harm maj and harm min have a more limited role
compared to the diatonic modes, but depending on the artist they might
be employed more often.
E.g. I'm pretty fond of the 3rd mode of harm maj on dom7#5#9 chords.
I.e. Ab harm maj on C7#5#9. (Aka "C Phrygian b4", lol.)
Etc.

Post by John Ladasky
Some additional Google searches have led me to discover a school of thought called "Neo-Riemannian" music theory.
https://en.wikipedia.org/wiki/Neo-Riemannian_theory
I have barely begun to read about this subject, but I think that what the Neo-Riemannians are trying to develop is an efficient framework which would do for chords and chord progressions what Tymoczko's two rules above does for scales. According to the Wikipedia article I referenced, Tymoczko apparently published a criticism of some Neo-Riemanninan ideas, but it seems to me that his criticisms are about some details of the theory and not the general goal.
I never learned to enjoy serial music. The constraints that serialists impose on themselves with 12-tone rows are too tight to allow much in the way of melodies that sound connected, harmonies that sound connected. I love dissonance, cacophony not so much.
If anyone has any thoughts on Tymoczko, Neo-Riemanninans, or any other systems which encompass post-functional harmony, please share. Thanks!

[e7m]

One has to keep in mind that tonality was first defined by melody and not by harmony and the harmony was then a result of the combination of multiple melodic lines and a few rather simple use rules derived by what sounded natural to the composer.

The pattern of the I IV V7 I and other cadence formulas came later as composers began to look for ways to add to their musical pallet and the early composers used the harmonic series as a guide. So in order to find tonality don't forget the importance or the melodic lines

LJSE7M

[John Ladasky]

Post by e7m
One has to keep in mind that tonality was first defined by melody and not
by harmony

Historically, of course, that is correct. I would argue that vertical relationships eventually gained equal standing in the minds of listeners and composers, some time in the late 19th Century.

Post by e7m
and the harmony was then a result of the combination of multiple melodic
lines and a few rather simple use rules derived by what sounded natural to
the composer.
The pattern of the I IV V7 I and other cadence formulas came later as
composers began to look for ways to add to their musical pallet and the early
composers used the harmonic series as a guide. So in order to find tonality
don't forget the importance or the melodic lines

Agreed. To take one example that I've already discussed, I think that the tendency to avoid consecutive semitones in scales stems from both harmonic and melodic challenges that arise when they are used.

[e7m]

True enough. I believe that the melody is still the most universal determative of tonality if for no other reason than we all pretty much listen to music and remember it by the melody. There are,of course some exceptions. These exceptions seem to be "learned" rather than innate.

Unless one has learned what a chaconne is, one would not hear it as a progression but would more likely know it better by the melody above it. In pop music, non trained musicians, would hear the melody over the I vi ii V pattern that was so often used in so many golden oldies. Then also, if you have a somewhat tonal melody using selected tones in 12 tone serial music, you will lose the sense of atonality. Thus I have to think that the melody is the determining factor not only in tonal music but especially in atonal music as even in strict 12 tone sedil, a tonal melodic phrase will quickly give a sense of tonality very easily irregardless of the atonal harmonic structures.

LJSe7m

[John Ladasky]

Post by Anon Anon

Post by John Ladasky
I haven't posted here in years, but a few "old-timers" from the 1990's chimed in recently, and I thought I would contribute something.
The early 20th century is my favorite period in orchestral music: Debussy, Ravel, Stravinsky, and their peers. My ears immediately "understand" that this lush and complex music has resonance, and deep tonal implications. My music theory courses barely scratched the surface as to why. Most people agree that the theories that were developed to explain common-practice functional harmony are inadequate for the task. I own Hindemith's _Craft_of_Musical_Composition_, and Persichetti's _Twentieth_Century_Harmony_, but even these books don't go very far in explaining what my ears already seem to know.
http://dmitri.mycpanel.princeton.edu/debussy.pdf
http://dmitri.mycpanel.princeton.edu/files/publications/consecutivesemitone.pdf
1) They have no consecutive semitones.
2) Consecutive intervals are either semitones or major seconds.
With those constraints, you obtain all the "church" modes (Ionian through Locrian), the whole-tone scale, the "diminished" scale (alternating m2 and M2), and all the "jazz" modes (ascending melodic minor, etc.). I found that to be highly interesting.

I haven't read this or the referenced articles thoroughly at all but....
You/he seem to be leaving harmonic minor and harmonic major out of your
scalar palette and they both have an important role in jazz as well as
in post-tonal classical music.

Obviously, I can't speak for Tymoczko, or the thoroughness of his survey of music.

But would you say that you can easily find persistent use of harmonic minor or harmonic major in compositions, in long passages, the same way that you find the other scales he listed? My musical experience says they're used much less commonly, in passing. Of course, if anyone wants to compose with harmonic minor and harmonic major, go right ahead! But if these scales are less common (I think they are), there might be a reason. That's what music theory is about, attempting to find a concise explanation for musical decisions that composers have made using their own ears.

Post by Anon Anon
I understand why scales with consecutive semitones are problematic for
extracting harmonies. But an augmented 2nd interval in a scale doesn't
usually cause those types of problems. Even the harmonic major scale
yields recognizable tertian triads and 7th chords.

I agree, that's an interesting harmonic observation. I know that I always found the harmonic minor scale to sound a bit wonky. I haven't tried composing harmonies with it. I've never even tried harmonic major.

Let me make a suggestion then: composers may have found the augmented 2nd / minor 3rd scale step to sound out of place in a heptatonic SCALE, though perhaps it causes no great difficulty with harmonies.

So what is a desirable property for scales, whose job it is to provide horizontal connections, rather than vertical? Some people say that "evenness" is acoustically important. If we're dividing 12 equal semitones into seven steps, the average step will be 12/7 steps, about 1.71. A "maximally-even" collection of seven steps will have two semitone steps and 5 whole tone steps.

If we require even one minor 3rd in a heptatonic scale, we're forced to change a M2 to a m2 to compensate. We have to push two intervals away from the mean, making one smaller and one larger.

The arrangement of those intervals matters too. If we put two semitones together, it somehow weakens the scale-like properties to many listeners, myself included (see above).

I won't say that I whole-heartedly support every reference that I post, but here's a link to an article, "Circular Distributions and Spectra Variations in Music: How Even Is Even?" It discusses a mathematical definition of evenness which might possibly be used to explain the observation "church modes first, jazz modes next, and (apropos to you) harmonic minor only occasionally, and other heptatonic combinations even more rarely":

http://archive.bridgesmathart.org/2005/bridges2005-255.pdf

I will propose the following: for a pitch collection to have versatility and wide use in tonal composition, in general, it has to have desirable harmonic AND melodic properties, because the composers of tonal music are generally trying to make simultaneous use of both melody and harmony. That doesn't mean that you can't take something a little off-kilter like harmonic minor, make it work as the centerpiece of a tonal composition, and make it sound great -- it just means you'll have to work harder, because the resources are more limited, and there are more "corner cases" and "avoid notes."

Think about the modern "Berklee School" chord-scale system for jazz: it seeks to map every chord to (generally, familiar) scales which are supersets of its chord tones. The system doesn't get you to everything that sounds harmonic and tonal -- but it gets you quite a lot of it.

https://en.wikipedia.org/wiki/Chord-scale_system

I don't know this chord-scale system as well as a professional jazz player. But I take the idea behind the system, that "every chord fits into a scale" as implicit support for my proposal that tonal composers settled on the scales in common use because of their simultaneous utility as scales and as chord material. It wasn't completely arbitrary.

All that being said: you have inadvertently touched on an issue that I might eventually discuss in more depth. In searching for new material, I have stumbled across two observations: one is an eight-note chord, and the other a six-note chord. Both "break the rules" in interesting ways, and yet they're still strongly tonal to my ears. But this post has gone on long enough.

[Joey Goldstein]

Post by John Ladasky

Post by Anon Anon

Post by John Ladasky
I haven't posted here in years, but a few "old-timers" from the 1990's chimed in recently, and I thought I would contribute something.
The early 20th century is my favorite period in orchestral music: Debussy, Ravel, Stravinsky, and their peers. My ears immediately "understand" that this lush and complex music has resonance, and deep tonal implications. My music theory courses barely scratched the surface as to why. Most people agree that the theories that were developed to explain common-practice functional harmony are inadequate for the task. I own Hindemith's _Craft_of_Musical_Composition_, and Persichetti's _Twentieth_Century_Harmony_, but even these books don't go very far in explaining what my ears already seem to know.
http://dmitri.mycpanel.princeton.edu/debussy.pdf
http://dmitri.mycpanel.princeton.edu/files/publications/consecutivesemitone.pdf
1) They have no consecutive semitones.
2) Consecutive intervals are either semitones or major seconds.
With those constraints, you obtain all the "church" modes (Ionian through Locrian), the whole-tone scale, the "diminished" scale (alternating m2 and M2), and all the "jazz" modes (ascending melodic minor, etc.). I found that to be highly interesting.

I haven't read this or the referenced articles thoroughly at all but....
You/he seem to be leaving harmonic minor and harmonic major out of your
scalar palette and they both have an important role in jazz as well as
in post-tonal classical music.

Obviously, I can't speak for Tymoczko, or the thoroughness of his survey of music.
But would you say that you can easily find persistent use of harmonic minor or harmonic major in compositions, in long passages, the same way that you find the other scales he listed? My musical experience says they're used much less commonly, in passing. Of course, if anyone wants to compose with harmonic minor and harmonic major, go right ahead! But if these scales are less common (I think they are), there might be a reason. That's what music theory is about, attempting to find a concise explanation for musical decisions that composers have made using their own ears.

Post by Anon Anon
I understand why scales with consecutive semitones are problematic for
extracting harmonies. But an augmented 2nd interval in a scale doesn't
usually cause those types of problems. Even the harmonic major scale
yields recognizable tertian triads and 7th chords.

I agree, that's an interesting harmonic observation. I know that I always found the harmonic minor scale to sound a bit wonky. I haven't tried composing harmonies with it. I've never even tried harmonic major.
Let me make a suggestion then: composers may have found the augmented 2nd / minor 3rd scale step to sound out of place in a heptatonic SCALE, though perhaps it causes no great difficulty with harmonies.
So what is a desirable property for scales, whose job it is to provide horizontal connections, rather than vertical? Some people say that "evenness" is acoustically important. If we're dividing 12 equal semitones into seven steps, the average step will be 12/7 steps, about 1.71. A "maximally-even" collection of seven steps will have two semitone steps and 5 whole tone steps.
If we require even one minor 3rd in a heptatonic scale, we're forced to change a M2 to a m2 to compensate. We have to push two intervals away from the mean, making one smaller and one larger.
The arrangement of those intervals matters too. If we put two semitones together, it somehow weakens the scale-like properties to many listeners, myself included (see above).
http://archive.bridgesmathart.org/2005/bridges2005-255.pdf
I will propose the following: for a pitch collection to have versatility and wide use in tonal composition, in general, it has to have desirable harmonic AND melodic properties, because the composers of tonal music are generally trying to make simultaneous use of both melody and harmony. That doesn't mean that you can't take something a little off-kilter like harmonic minor, make it work as the centerpiece of a tonal composition, and make it sound great -- it just means you'll have to work harder, because the resources are more limited, and there are more "corner cases" and "avoid notes."
Think about the modern "Berklee School" chord-scale system for jazz: it seeks to map every chord to (generally, familiar) scales which are supersets of its chord tones. The system doesn't get you to everything that sounds harmonic and tonal -- but it gets you quite a lot of it.
https://en.wikipedia.org/wiki/Chord-scale_system
I don't know this chord-scale system as well as a professional jazz player. But I take the idea behind the system, that "every chord fits into a scale" as implicit support for my proposal that tonal composers settled on the scales in common use because of their simultaneous utility as scales and as chord material. It wasn't completely arbitrary.
All that being said: you have inadvertently touched on an issue that I might eventually discuss in more depth. In searching for new material, I have stumbled across two observations: one is an eight-note chord, and the other a six-note chord. Both "break the rules" in interesting ways, and yet they're still strongly tonal to my ears. But this post has gone on long enough.

I'm not sure I totally understand the points you are trying to make here
and I have not read all you've written or examined the files you posted.
But certainly harmonic minor and harmonic major are not utilized
traditionally in the same ways that the pure diatonic scale has been
utilized traditionally, in that an entire composition can be seen to be
"based upon" them.
These scales are used as chromatic alterations of the major and natural
minor scales and add colour/variety to a maj/min key-based composition.
I brought harmonic major up simply because the OP seemed to have missed it.
And again... Major, mel min (ascending), harm min, and harm maj are the
only heptatonic scales possible, in 12TET, with no consecutive
semitones, so they contitute a unique group.

[J.B. Wood]

Hello, all, and to correct my previous post on the topic, I should have
used "irregular" as a more appropriate descriptor rather than "unequal"
when referring to what some authors dub "well temperaments". Such
temperaments allow for performance in all 12-keys, but with different
sonorities (key color). That is often touted as an advantage but their
use when the aim is transposition (e.g. to accommodate the voice range
of a singer) is questionable. Sincerely,