Author Topic: Millions on General Music Theory  (Read 1573 times)

millions

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Millions on General Music Theory
« on: January 04, 2013, 08:19:27 AM »

Harmonic Function 

What I mean by tonal music being "visceral" or "of the senses" is because all its functions, even the "cerebral" ones (root movement, tension/resolution) are traced back to the ear/brain perception of vertical, instantaneous consonance/dissonance.

Sorry, but here's the long-winded explanation:
________________________________________

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.

The interval ratios in the chart above, to the right, 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.

In the key of C, a simple 1-3-5 G triad is not identical to the simple 1-3-5 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.

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.

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.

Hearing harmonic (vertical) dissonance/consonance happens instantaneously, as the result of the ear/brain perceiving a harmonic relationship, or ratio (a ratio is not a fixed quantity, it is a relationship between two things).

But chord function (horizontal time-line) takes time to establish. This is cognitive, although it is based on visceral harmonic (vertical) "instant" recognition of the ear/brain to consonance/dissonance.

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.

All chord 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 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.]

So, how does the "time line" figure into the cognition of "function" of chords? Cognition of harmonic function always involves hearing a sequence of events.
"In Spring! In the creation of art, it must be as it is in Spring!" -Arnold Schoenberg
"The trouble with New Age music is that there's no evil in it."-Brian Eno

millions

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Re: Millions on General Music Theory
« Reply #1 on: January 04, 2013, 08:21:24 AM »
Key Signatures 

We'll start on C, project a minor third from that, yielding an Eb, then another minor third from there, giving us Gb. From Gb, up a minor 3rd is Bbb, or enharmonically, an A, giving us the familiar 'diminished seventh' chord. As you can see by the 'Bbb/A,' the glitch in our diatonic 7-letter-name scale system is revealed by this. A scale must consist of seven different letter-names. This is a good time to discuss this in more detail.
On a keyboard, Gb and F# are the same note, physically.

If one starts building fifths from a starting point of C, then going "forward" or clockwise around the "circle of fifths" would yield C-G-D-A-E-B-F#(C#-no need for D#).

If, on the other hand, you go in reverse (counter-clockwise), you travel the "circle of fourths", which yields C-F-Bb-Eb-Ab-Db-Gb (Cb).

As you can see, there are three keys which "overlap" under two different names: B (Cb), F# (Gb), and C# (Db). The reason it goes no further has to do with the physical layout of the keyboard itself (there are two semitone steps in the letter sequence), and the subsequent "letter-naming" of notes which results. To be a diatonic scale, you must have seven different letter names.

For example, there is no key of "Fb" because this is E, a sharp key; but if we named it anyway, we would get Fb-Gb-Ab-Bbb (you can't repeat A - there must be seven different letter names with no repeats), Cb-Db-Eb-Fb. This "repeating letter or double-flat" dilemma does not arise on the three "repeat" keys of B (Cb), F# (Gb), and C# (Db), because this is the "seven-letter limit".

In equal tempered tuning, both end points (F# and Gb) are identical, because all the "fifths" have been adjusted flat by 2 cents, to keep from "overshooting" the mark. Otherwise, instead of a closed circle which repeats from octave to octave, we would have an endless spiral, and an infinite number of different notes.

In other tuning systems, which I am just now beginning to understand & study, the physical layout of the keyboard must remain as 12 notes (7 white and 5 black), regardless of what tuning we use.
We don't want to have a separate "F#" and "Gb" black key, although this has been tried.

In either mean-tone or Bach's tuning, what remains consistent in a "key signature" is the RELATIONSHIPS or intervals produced in that octave or key, all in relation to the "key" note.

For instance, in mean-tone tuning, starting from C and building our fifths, we have C-G, G-D, D-A, and A-E. The fifths are adjusted in mean-tone, made smaller, in order to create a good-sounding major third of C-E, which without adjustment would have been too sharp. This was a limited tuning, since going in fifths clockwise yields C-G-D-A-E-B-F#-G#(Ab), or counter-clockwise yields C-F-Bb-Eb. This sequence produces in G#- Eb (or Ab-Eb, or G#-D#) a "wolf" fifth. So there is the limit of mean-tone tuning.

Bach's tuning was not "equal", but it was "well" tempered, meaning that, unlike mean-tone tuning, he could get a decent sound in all twelve keys.

In mean-tone tuning, there IS no "Gb", only F#.

In the Bach tuning, the difference in F# and Gb would show up as the OTHER keys those notes are in; for example, in the key of D (a sharp key), the F# is the major third. This would be a different-sounding major third than the C-E in the key of C, it might be a wider or narrower interval span.

Similarly, F# could be the fifth in the key of B (also a sharp key). This B-F# fifth is unique to this key.

Gb could be the fifth of the key of Cb, because a fifth below Gb (keeping our letter rule) would be Cb. This would be a unique fifth for that key, maybe more "perfect" and restful, or "sharper" and restless.

Gb can't be the major third in any key; there is no key of "Ebb". This is really D, in which case we must call it "F#", not Gb.

In Bach tuning, the keys of F# and Gb would also be identical.
"In Spring! In the creation of art, it must be as it is in Spring!" -Arnold Schoenberg
"The trouble with New Age music is that there's no evil in it."-Brian Eno

millions

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Re: Millions on General Music Theory
« Reply #2 on: January 04, 2013, 08:23:14 AM »
Root movement 

There are only 6 possible intervals of root movement (as in tonal functon/stations).

This way of looking at root movement makes more sense if you look at the chromatic scale as circular, and notes as identities, without regard to intervallic distance.
On a circle, all intervals can be easily seen as invertible. This reveals the fact that triad construction and root progression are both contingent upon identity, i.e. note names, as in a 'station' on the circle, rather than a quantity or static intervallic distance from a given note. Thus tonality, as we know it, is dependent on direction around an hierarchical circle, with note-names as identities, more so than static "distances" or quantities from a given note. This is because of the harmonics of a note, going in a "direction" of importance from a fundamental to its harmonics.
If tonality is seen as "gravity", then this way of separating pitch identity from interval distance may become more valuable when looking at a floating, global, gravity-based view of tonality as a localized "sphere of influence" (as in Bartok).

Schoenberg speaks in his text "Structural Functions of Harmony" of traditional root progression, and he classifies the root progressions into three categories:

1. Strong, or Ascending: (a) A fourth up, identical with a fifth down; (b) A third down
2. Descending: (a) A fourth down, identical with a fifth up; (b) A third up
3. Superstrong: (a) One step up; (b) One step down.

That makes 5 ways to move a root, if we remember that a step (seconds) and thirds can be minor or major. That is, (2) thirds + Fourth/fifth + (2) steps (minor or major second), with the tritone as "odd man out" or 6.

FURTHER THOUGHTS: To remember these root progressions & their relative strengths, it will help to think of them simultaneously as being steps from one root note to another, traveling forward through time horizontally, and also as harmonic intervals, sounding simultaneously at once, vertically:

1. Strong, or Ascending: (a) A fourth up, identical with a fifth down;

...this is because when we hear fourths as a simultaneous sounding of two notes, we always hear fourths with the top note as root; in root function terms, we have moved UP TO this note (higher in pitch, within the upper octave above the root); this makes it 'strong' or 'ascending' TO the root on top.

A fifth down (lower in pitch below the root, in the octave below the root) is the inversion; when we hear both notes at once, we always hear fifths with the bottom note as root, as we do with all fifths. Every head-banger knows this. Spread out horizontally, a fifth down means the second (bottom) note is root.

(b) A third down: we hear the second note as root, or resting point. Makes sense. Listen to Cream play "Spoonful."


2. Descending: (a) A fourth down, identical with a fifth up; we hear the top of the fourth as root, still, but we have moved away from it to a weaker note; thus, it is "descending." If we hear a fifth, again, the bottom note is root; but we have moved a fifth up to the weaker note; thus, it is "descending," or getting weaker. By 'weak' or 'strong' we mean simply reinforcing a root or key center, or 'weakening' the root or key, and moving away from it, to perhaps another key.


(b) A third up. We hear the bottom note as root, but we jumped up, away from it. It's getting weaker.



3. Superstrong: (a) One step up; (b) One step down. We hear the second note as a new root. I need to think about this one some more.

So my point is this: we need to ponder the differences, similarities, and correspondences between horizontal thinking, which progresses on a time-line (which is what root movement is concerned with), and vertical thinking, which happens all-at-once, and lets the ear decide what emerges from aggregates of vertical notes.
« Last Edit: January 04, 2013, 08:47:40 AM by millions »
"In Spring! In the creation of art, it must be as it is in Spring!" -Arnold Schoenberg
"The trouble with New Age music is that there's no evil in it."-Brian Eno

millions

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Re: Millions on General Music Theory
« Reply #3 on: January 04, 2013, 08:24:37 AM »
The dominant chord rules! 

The only way to understand this is to hear it, so get a beer, go to your Bosendörfer Imperial Grand, and try this:

"Dominant" refers to the function of a chord, it tendencies, and what it "wants" to do. In any major scale, each note is a possible root from which to build a triad. The seven note functions are represented with Roman numerals. Capital numerals denote major; small numerals denote minor (or diminished on vii). The functions are: I (tonic), ii (supertonic), iii (mediant), IV (subdominant), V (dominant), vi (submediant), and vii (leading tone).

These functions change in minor keys & other scales. There are function-names for every chromatic note, not just the seven diatonic (within the key) ones above; for example, flatted submediant, etc.

A "dominant" chord is a "V" chord; any chord with a major third and a flat-seven. This interval is a tritone. It is the symmetrical mid-point of the 12-note chromatic scale. For example, in the octave from C to C, C-F# and F#-C are the tritones. This interval is symmetrically invertible; i.e. when you invert it, it is still a tritone, unlike the other intervals (except the octave). Inverted major thirds become minor sixths, etc.

Here's where "function" comes into play. The root determines the function of the notes in the tritone.

If the tritone is occupying the notes F-B, and the root is G, then F is the b7 and B is the third. The F wants to resolve down to E, and the B wants to resolve up to C, making it a V-I cadence in C. This is the typical V-I progression.

Alternately, if the root is C#, then the functions (3-b7/b7-3) are reversed: F (called E# in the key of C#) is the major third, and B is the b7. The tritone can be resolved to F#: F down to E (b7), and B down to Bb (or A# in the key of F#), the major third.

Getting complicated, isn't it? If we combine these root movements, we see other possibilities emerge: play F-B in your RH. All these will resolve F down to E, and B down to Bb.
The possible root movements are: G-C; G-Gb; C#-F#; and C#-C. So you see, we get two different V-Is, and two different chromatic half-step resolutions, all out of one tritone relation.

After playing with this, you begin to see that a series or cycle of V-Is is similar to a chromatic movement. Both cycles will eventually exhaust all twelve notes; the V-Is do it by fifths, and the chromatics do it in succession.

The be-bop jazz players exploited this characteristic of dominants & tritones, calling it "tritone substitution." Jazz is mostly cycles of V-Is, so instead of G7-C7, they would substitute a new root and go C#7-C7, or G7-F#7.
Try this as an endless cycle or loop: G-F-B to F#-E-Bb; C-E-Bb to B-Eb-A; F-Eb-A to E-D-G#; Bb-D-G# to etc.
Notice the I-Ching hexagram-like transition? The last two notes of each second group are carried over to the next first group: E-Bb/Eb-A/D-G#, etc.

The Second Viennese School used this chord (see the New Grove "Second Viennese School"). The chord is (low to high): Bb-D-E-Ab, which, if shifted down one semitone to A-C#-D#-G, is equivalent to the same chord transposed a perfect fifth down: Bb-D-E-Ab to Eb-G-A-Db, which are the same notes (Eb=D#, G, A, Db=C#). This illustrates what I was talking about above, the "chromatic-fifths connection."
"In Spring! In the creation of art, it must be as it is in Spring!" -Arnold Schoenberg
"The trouble with New Age music is that there's no evil in it."-Brian Eno