# The Horizontal and the Vertical



## millionrainbows

Music moves through time; that's its "horizontal" dimension. It also consists of simultaneously-sounding elements; that's its "vertical" dimension. How do we perceive tonality? Do we perceive it "instantly," as in the vertical, or do we have to wait for a "horizontal" string of chord changes? Well...it depends on how much information the ear is given at any one moment.

Most dissonant intervals to most consonant intervals, within one octave:
1. minor seventh (C-Bb) 9:16
2. major seventh (C-B) 8:15
3. major second (C-D) 8:9
4. minor sixth (C-Ab) 5:8
5. minor third (C-Eb) 5:6
6. major third (C-E) 4:5
7. major sixth (C-A) 3:5
8. perfect fourth (C-F) 3:4
9. perfect fifth (C-G) 2:3
10. octave (C-C') 1:2
11. unison (C-C) 1:1

The steps of our scale, and the "functions" of the chords built thereon, are the direct result of interval ratios, all in relation to a "keynote" or unity of 1; the intervals not only have a dissonant/consonant quality determined by their ratio, but also are given a specific scale degree (function) and place in relation to "1" or the Tonic. This is where all "linear function" originated, and is still manifest as ratios (intervals), which are at the same time, physical harmonic phenomena.

"One (1:1) is the ultimate consonance. In the beginning was ONE. From this, sprang forth the universe. 
"All musical understanding can be reduced to the understanding of one note."

The interval ratios are just a way of expressing the relationship of two notes. For example, 2:1 is the octave, or doubling of frequency; conversely, 1:2 halves it.

A G triad is not identical to a C triad, because of its position (functioning as V) in relation to the root. The D, resolving down, now becomes root, as well as being the top of a fourth G-C, which is heard as root on top.

We still have not explained how tonic is established to begin with...

Why, universally/acoustically speaking isn't the G triad identical to the C triad to anyone without absolute pitch? They are major triads and are equally dissonant. The functional difference is only apparent once tonic has been established. Tonic is established correctly once the listener has heard and connected (COGNITIVELY) the series of intervals that constitute the diatonic scale.

"G" must be seen in relation to its home key, whether that be "G" or "C", not in isolation. No chord exists in isolation, but all exist in relation to "1", unity, or tonic.

Implicit in any harmonic interval, whether it be 2:3 or 3:4, is an implicit relation, and specific note-position in the heirarchy, in relation to "1" or tonic, as well as its being more dissonant or more consonant in relation to "1" or the root. This holds true for intervals, too.

The functional difference is only apparent once tonic has been established. Tonic is established correctly once the listener has heard and connected (COGNITIVELY) the series of intervals that constitute the diatonic scale."

Or does this happen simultaneously, as the result of a perceived relationship, or ratio? A ratio is not a fixed quantity, it is a relationship between two things. 

In the case of simple pop song progressions, using static chord exchanges of say, C-F, it might be ambiguous whether "C" is I and "F" is IV, or if "C" is V and "F" is I; in fact, many pop songs play on this ambiguity.

In this age of equal-temperament, these relations are harder to identify from a mean-tone tuning in which each scale degree has a more colored, distinct relation to I.

So what I am saying, is that all functions relate to the Tonic, or unity. This is exactly the way interval ratios work, also.

From Harry Partch, "Genesis of a New Music:" 

[A ratio represents a tone and an interval at one and the same time; in its capacity as the symbol of a tone it is the over number that is nominally representative (in the upward manner), but since the over number exists only in relation to the under number, the ratio acquits its second function, as representative of an interval; 

...conventional musical example: 3/2 represents "D" in the "key of G" - upward from "G"; it is thus simultaneously a representative of a tone and an implicit relationship to a "keynote" - or unity.]

Thus, it is seen that the steps of our scale, and the "functions" of the chords built thereon, are the direct result of interval ratios, all in relation to a "keynote" or unity of 1; the intervals not only have a dissonant/consonant quality determined by its ratio, but also is given a place in relation to "1" or the Tonic. This is where all "linear function" originated, and is still manifest as...ratios.

[...the identities of a tonality (tonal polarity around a 1 identity) can of course be deliberately placed to simulate the series of partials, but in this sense they are not partials.]

[The scale of musical intervals begins with absolute consonance (1 to 1), and gradually progresses into an infinitude of dissonance, the consonance of the intervals decreasing as the odd numbers of their ratios increase.]"...Functional harmony is goal-directed. It persuades us that the music is proceeding somewhere."

(end quote) ...So although functional harmony is "vertical," it is a series of relations with a time-oriented goal and travels horizontally through time; it creates in us, through tension & resolution, expectations for dissonance to resolve, and to reach the goal of rest.

Ok, from the above we can see that major triads are ambiguous, when it comes to establishing a sense of tonality. Let's look at all seven diatonic chords of the major scale:
I major
ii minor
iii minor
IV major
V major
vi minor
vii diminished

All the major & minor chords are ambiguous; the vii is the only one that is not.
Does adding a note, and creating seventh chords reduce the ambiguity?

I: 1-3-5-7 major seventh
ii:1-3-5-7 minor seventh
iii: minor seventh
IV: major seventh
V: dominant seventh
vi: minor seventh
vii: diminished seventh

Adding a note did help. The only chord affected by this is V, which is the only b7 chord. "I" could still be mistaken for IV, and vi for iii, etc.

How about ninth chords?

I: 1-3-5-7-9 major ninth
ii: minor ninth
iii: minor seventh, with b9...unusable?
IV: major ninth
V: dominant ninth
vi: minor ninth
vii: minor seven b5

There is still the I/IV ambiguity; still a ii/vi ambiguity; and the vii, now a min 7 b5, has actually been degraded, now seeming to favor the minor vi key area, with major III as a new V chord. 

It seems that the I/IV ambiguity cannot be resolved, even if 8 notes are used. In the key of C, we get a giant chord C-E-G-B-D-F-A-C, and on F we get F-A-C-E-G-B-D-F, which is really the same chord in permutation, BUT...

Actually, the "F" root establishes a root feeling; the major seventh (F-E) establishes a root on F. 
Our "C" chord, however, also contains a fourth above that "root" (C-F-B) and this spoils it...

The other set of notes gives us G-B-D-F-A-C-E-G, which actually works as a chord.

So the whole mess seems to resolve into three giant chords: a "C" with major seventh and "fourthy" characteristics, an "F" with a major seventh and NO fourth, and the "G" chord, actually a G 13. Our major scale, then seems to be ambiguous unto itself. Maybe George Russell was right...

Let's apply the same procedure to George Russell's lydian scale, and see if it yields a greater sense of tonal function.
The lydian is a mode of the major scale we just discussed, so all we're doing is changing the root of the tonality to F rather than C.

Well, for starters, the lydian scale, F to F, has a good stable sense of tonality; a fifth, and a "leading tone" to that fifth (B-C). Gone is the fourth, which does nothing to reinforce our tonality.

We are still faced with the ambiguity of F/C, which both yield root-reinforcing major sevenths when notes are added. The "IV" chord B-D-F can sound like an incomplete dominant G7. This is a II7, used normally for modulation (V of V). With this reinforcement of C with its dominant (the assumed G7), the key most reinforced seems to be C, not F. Puzzling...

But wait: let's consider the fourth again. When we hear a fourth, such as C-F, we tend to hear the TOP note as root. The opposite is true with a fifth, such as C-G; we tend to hear the BOTTOM note as root.

In the lydian scale, there is no perfect fourth from F, our root. Therefore, F is still heard as root, even if we hear F-B; B will be heard as a flatted fifth, not an augmented fourth, because the closer tone is C (a half-step up), not A (further away, a whole step down). 

The V in lydian, C-E-G, is really not heard as a "V", but as a continuation of the F scale, because of the "leading tone" B which led us to C.

Therefore, the lydian scale reinforces its own tonality more than the C major scale does. The lydian has no proper "IV", and its "V" is not heard as V, but as a continuation of F, aided by the "leading" tone b5, B-C.


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## Razumovskymas

I'll keep that in mind when I listen to Beethovens "Heiliger Dankgesang" of opus 132


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## Woodduck

Razumovskymas said:


> I'll keep that in mind when I listen to Beethovens "Heiliger Dankgesang" of opus 132


Beethoven was thankful at reaching the end of an illness. Yeah, I can see sort of a parallel...


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## Guest

millionrainbows said:


> Music moves through time; that's its "horizontal" dimension. It also consists of simultaneously-sounding elements; that's its "vertical" dimension. How do we perceive tonality? Do we perceive it "instantly," as in the vertical, or do we have to wait for a "horizontal" string of chord changes? Well...*it depends on how much information the ear is given at any one moment.*


...and on how long one is listening for. If I only hear the first note of Beethoven's 5th, that's not much to go on - anything I 'perceive' is of little value. Unless we believe that moment-by-moment perception is the right way to analyse what we're listening to?


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## Vox Gabrieli

I feel as if I missed a conclusion here. As if I were engaged in a conversation with somebody and they just get up and leave while talking. You wouldn't do that in normal social circumstances, so why do it here?

The beginning of the contemporary era of music and our gradual shift to post-modernism has had few discrepancies, so I will refer to anything past the 1900s as ' contemporary ' for ease of term.

I would like to start by saying that it is both horizontal* and *vertical, especially with the constant change of tonality that contemporary works set out to prove. If I were to think about pre-1900s, and the thing about the generally accepted structure of our ancients, then we have a slightly more comprehensive answer. I tried adding more with pictures and explanations, but my keyboard is acting strangely. Hopefully, this discussion kicks off.


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## millionrainbows

Ok, I'll explain some of this. The main point of this particular post was to show how the degree of general tonality can be determined by examining vertical elements only. By vertical, I mean listening to the harmonic content of the scales/chords. If this is a difficult concept for anyone, I suggest they do some more reading and watching these videos. The concept of "verticality" is established, and in use.

The main closing argument was that the diatonic C major scale establishes tonality less effectively than the Lydian scale starting on F. Anyone familiar with George Russell's Lydian Chromatic concept will know who he is. I created this post to offer evidence of his claims, which I agree with.

Those astute readers will recognize that "adding notes" to the triads is really a way of stacking thirds, then examining the results on each scale step, to determine which chords provide the strongest reinforcement of the tonality of the scale.



Gabriel Ortiz said:


> I feel as if I missed a conclusion here. As if I were engaged in a conversation with somebody and they just get up and leave while talking. You wouldn't do that in normal social circumstances, so why do it here?


_Patience, patience. Soon all things will be revealed, and then the critics can take their turn.

Ideally, I hoped that readers who are really interested in this would read it carefully enough to see the point, without my having to explain it. It's a detailed post, and you can't just skim over it. You have to carefully read and ponder these ideas, and try them for yourself.
_
_


Gabriel Ortiz said:



I would like to start by saying that it is both horizontal* and *vertical, especially with the constant change of tonality that contemporary works set out to prove.

Click to expand...

That's the way I started the post: 


millionrainbows said:



Music moves through time; that's its "horizontal" dimension. It *also* consists of simultaneously-sounding elements; that's its "vertical" dimension.

Click to expand...

_

This can also be done by "stacking fifths." Anyone with a piano, or preferably an electronic keyboard, can do this for themselves, which I encourage. Hearing is believing, and nothing I say here will make much of a difference unless you out the work in, and try it, away from your computer.

Or, for those who are hopelessly tied to their PC, here is a YouTube demonstration of these same ideas.


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## millionrainbows

Function, and tonal meaning, come first from the instantaneous perception of sound and its inner relations, not from successions of events, which simply elaborate this.

BTW, these three video explanations, especially the third one, confirm and give external creedence (from 2 PHDs) that my ideas on verticality (posted as the "All Music is Vertical…" thread are, indeed, worthy of consideration and pondering.


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## Frasier

millionrainbows said:


> Function, and tonal meaning, come first from the instantaneous perception of sound and its inner relations, not from successions of events, which simply elaborate this.
> 
> BTW, these three video explanations, especially the third one, confirm and give external creedence (from 2 PHDs) that my ideas on verticality (posted as the "All Music is Vertical…" thread are, indeed, worthy of consideration and pondering.


What you say might be true in physiological terms - sounds impinge on the ear. They are perceived. They may attach to a familiar anchor, they may not. But music depends on the horizontal, the progressions that provide tension and release. That's what, in olden times, cadence was about - expectation. Assimilation of harmony or otherwise is instant (and may be vertical) but assimilation of music is horizontal.

Why I said "maybe vertical" is in consideration of monody. Harmony is implied as it progresses.

...


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## Razumovskymas

Why not look at music as only horizontal?

Why is a single note, which we know consists not purely of one frequency (sinus wave), regarded as something different as a chord? Of course the way people make music and music evolved is a result of this distinction. But the end result is nearly always alterations of very complex frequencies spread over time. Maybe there's another way of analysing music. Instead of breaking it down to the building blocks it's made with, looking more at the end result and so to say, leave big chunks of sound "unanalyzed" 

Actually, I'm listening to Prokofievs 5th symphony at the moment and when I listen to it as if it's not a combination of single notes and single instruments combined in harmony but a succession of all these incredibly complex sounds, it gets more organic and rich. Listening to it as if you never heard any music, don't know what an musical instrument is, let alone musical theory.


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## JeffD

"I don't like vertical music because it doesn't seem to go anywhere."

"Well at least you understand it."


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## Vox Gabrieli

I just find it frustrating that this question exists, there are different ways to analyze a piece. Why not let the consensus be both without saying it's both. Seems to me like a very streamlined discussion to be having.


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