Wagner: "Expanded" tonality?

[bwv 1080]

where does the 'geometry' come from?  Did you just make it up, or is there some justification for the term?

[millionrainbows]

where does the 'geometry' come from?  Did you just make it up, or is there some justification for the term?

Diatonic tonality came out of harmonic, sensual concerns: the triad, the fifth, the third, and the hierarchy created by building triads on the scale steps, all in relation to "1" (the tonic). Viewed in these terms, the octave is divided at the fifth.

When all the chromatic notes began to be used, the "geometry" of 12 notes began to emerge.

Characteristics of these chromatic geometries are:

the octave divided at the tritone, 6 semitones (6+6=12)
diminished chords, minor thirds, 3 semitones (3+3+3+3=12)
whole tone scale, augmented, stacked major thirds (2 semitones) (2+2+2+2+2=12), (4+4+4=12)
the chromatic scale, 1 semitone (1+1+1+1+1+1+1+1+1+1+1+1=12)

Unlike fourths (5 semitones, 5x12=60) and fifths (7 semitones, 7x12=84), not divisible into 12, which require larger numbers 'outside' the 12 (octave), these smaller intervals (1, 2, 3, 4, 6) are recursive, meaning that they stack or repeat and coincide back into the octave instead of traveling outside (to 60 or 84).

This makes them 'inward-going' instead of tonality's 'outward going.'

Chomaticism applied to root movement allows modulation to distant areas without having to follow the circle of fifths, by using these in-octave stations via diminished sevenths and whole-tone augmented mechanisms.

Bach and Mozart were well-aware of these "chromatic/fifths' connections.

In this link, Bernstein discusses how Mozart uses diatonic tonality and chromaticism:

https://www.youtube.com/watch?v=DCP58BigEfw

Notice how he distinguishes the two methods, seeing them as separate methods, yet working together in the service of making diatonic tonality more interesting and alive.

This is the perfect example of the "dual nature" of tonality: its diatonic origins and the use of chromaticism, in the form of diminished seventh chords. Chromaticism serves tonality, yet it creates movement away from tonic and creates vagueness. It is a destabilizing influence.

[bwv 1080]

Diatonic tonality came out of harmonic, sensual concerns: the triad, the fifth, the third, and the hierarchy created by building triads on the scale steps, all in relation to "1" (the tonic). Viewed in these terms, the octave is divided at the fifth.

When all the chromatic notes began to be used, the "geometry" of 12 notes began to emerge.

Characteristics of these chromatic geometries are:

the octave divided at the tritone, 6 semitones (6+6=12)
diminished chords, minor thirds, 3 semitones (3+3+3+3=12)
whole tone scale, augmented, stacked major thirds (2 semitones) (2+2+2+2+2=12), (4+4+4=12)
the chromatic scale, 1 semitone (1+1+1+1+1+1+1+1+1+1+1+1=12)

Unlike fourths (5 semitones, 5x12=60) and fifths (7 semitones, 7x12=84), not divisible into 12, which require larger numbers 'outside' the 12 (octave), these smaller intervals (1, 2, 3, 4, 6) are recursive, meaning that they stack or repeat and coincide back into the octave instead of traveling outside (to 60 or 84).

This makes them 'inward-going' instead of tonality's 'outward going.'

Chomaticism applied to root movement allows modulation to distant areas without having to follow the circle of fifths, by using these in-octave stations via diminished sevenths and whole-tone augmented mechanisms.

ISTM thats modular Arithmetic not geometry

https://en.m.wikipedia.org/wiki/Modular_arithmetic

[Karl Henning]

Unlike fourths (5 semitones, 5x12=60) and fifths (7 semitones, 7x12=84), not divisible into 12, which require larger numbers 'outside' the 12 (octave), these smaller intervals (1, 2, 3, 4, 6) are recursive, meaning that they stack or repeat and coincide back into the octave instead of traveling outside (to 60 or 84).

This makes them 'inward-going' instead of tonality's 'outward going.'

Next door to gibberish.

[BasilValentine]

Same goes for any other kind of music that we talk about as being tonal. In the end it's just all sound that we like to play around with. It's all make believe. We made this up by experimenting with the physical properties of sound and some of us in some location at some time liked some things we heard and that became the music of that culture....then we added words to describe it...and words are stuff we made up as well. Meaningless sound until we agree that 'chromaticism' means a certain thing. We just believe anything to be so and it is so. It's all a kind of fictional reality we are dealing with where you can say what you just said even if you do look like a bit a loony by saying it.

Because I think most people understand that Wagner's use of tonality is the result of his chromaticism anyway. The way he uses chromaticism to create mediant relationships in Tristan und Isolde that result in localised tonal centres, the way he moves between subdominant functioning and dominant functioning chords and the way he even does provide resolutions to these progressions can all be described as being functionally  t o n a l  as anything else written in the Common Practice Era.

Chromaticism is something Wagner used to extend his harmonic vocabulary whilst still staying within the limits of tonal harmonic progressions. 'Chromaticism' isn't something separate from tonality in this sense. It's a part of it.

Yes(!) to the conclusion and "Huh?" to your first paragraph.

The Tristan Chord-disintegration of tonality story is my candidate for the most irritating and silly trope in the history of music theory. If one interprets the Tristan chord as a Fr+6 chord with an appoggiatura, as I do, then what we have is a powerful tendency chord that goes exactly where it is supposed to, to the dominant (of one key or another), over and over. That the dominants don't resolve is irrelevant. We feel them as unresolved dominants, that is, in accordance with their tonal function, every time. It is hard to imagine a stronger affirmation of tonality using chromatic language. Moreover, the whole point of the frustrated resolution is thematic, musically and with respect to the story, in a freaking obvious way! Its success is entirely dependent on feeling the tonal function of the harmonies — and Wagner picked the strongest sequence of tendency chords he could. Whoever invented this trope should be disinterred and kicked in his bony ass.

[bwv 1080]

Yes and to take it one step further, there are wrong notes in the first movement of Schumann's Gesang der Frühe more striking than Tristan.  It does not destroy the tonality, they just sound like wrong notes because the piece is tonal.  Then you get triads in Debussy or Messiaen but because they are within pieces based on symmetrical scales there is no real tonality there

[millionrainbows]

Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. The "division" of the octave at the fifth is a matter of ratio, not quantity.
Dividing the octave at the tritone is conversely more a matter of equal distance, which makes it a geometric consideration.

Unlike fourths (5 semitones, 5x12=60) and fifths (7 semitones, 7x12=84), not divisible into 12, which require larger numbers 'outside' the 12 (octave), these smaller intervals (1, 2, 3, 4, 6) are recursive, meaning that they stack or repeat and coincide back into the octave instead of traveling outside (to 60 or 84).

This makes them 'inward-going' instead of tonality's 'outward going.'

Barok, with his in-octave stations of tone-centricity, is inward-looking, introspective, concerned with the inner world.

[millionrainbows]

Same goes for any other kind of music that we talk about as being tonal. In the end it's just all sound that we like to play around with. It's all make believe. We made this up by experimenting with the physical properties of sound and some of us in some location at some time liked some things we heard and that became the music of that culture....then we added words to describe it...and words are stuff we made up as well. Meaningless sound until we agree that 'chromaticism' means a certain thing. We just believe anything to be so and it is so. It's all a kind of fictional reality we are dealing with where you can say what you just said even if you do look like a bit a loony by saying it.

Generally, tonality is a universal phenomenon, based on the harmonic model, and adapted to different scales.

Western CP tonality is a special case, beginning with the use of 12 notes. In this sense, Western tonality has always been intimately connected to chromaticism.

Because I think most people understand that Wagner's use of tonality is the result of his chromaticism anyway. The way he uses chromaticism to create mediant relationships in Tristan und Isolde that result in localised tonal centres, the way he moves between subdominant functioning and dominant functioning chords and the way he even does provide resolutions to these progressions can all be described as being functionally  t o n a l  as anything else written in the Common Practice Era.

Of course; it's been that way since Mozart's G minor, at least.

Chromaticism is something Wagner used to extend his harmonic vocabulary whilst still staying within the limits of tonal harmonic progressions. 'Chromaticism' isn't something separate from tonality in this sense. It's a part of it.

That's just one way of using chromaticism, which was around for a while before Wagner. Chromaticism is its own, and does not depend on diatonic tonality for its existence or autonomy.

[bwv 1080]

Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. The "division" of the octave at the fifth is a matter of ratio, not quantity.
Dividing the octave at the tritone is conversely more a matter of equal distance, which makes it a geometric consideration.

Unlike fourths (5 semitones, 5x12=60) and fifths (7 semitones, 7x12=84), not divisible into 12, which require larger numbers 'outside' the 12 (octave), these smaller intervals (1, 2, 3, 4, 6) are recursive, meaning that they stack or repeat and coincide back into the octave instead of traveling outside (to 60 or 84).

Not conversely - 12/7 and 12/6 are both ratios, one simplifies to an integer the other doesn't.  This is arithmetic.  It's not geometry that tells me there are 4 quarters or 20 nickels in a dollar

[millionrainbows]

Not conversely - 12/7 and 12/6 are both ratios, one simplifies to an integer the other doesn't.  This is arithmetic.  It's not geometry that tells me there are 4 quarters or 20 nickels in a dollar

Ratios are relationships, not quantities. Tonality is based on these ratios, such as 3/2.

The division of the 12 note octave at '6' is based on quantity: the number of semitones. But this is not merely arithmetic, in that it contains symmetries.
Symmetry is an aspect of geometry which applies to music. The division of the octave at the tritone creates symmetries.

Don't look at music only in terms of arithmetic; look at symmetry as well. A strictly arithmetical approach seems too literal. The use of circles and lines  have been used for years to demonstrate these sorts of things.

This "gibberish" all goes back to interval projection, as described by Howard Hanson in his text "Harmonic Materials of modern Music."

[millionrainbows]

Next door to gibberish.

Multiplication (music)

[bwv 1080]

Ratios are relationships, not quantities. Tonality is based on these ratios, such as 3/2.

The division of the 12 note octave at '6' is based on quantity: the number of semitones. But this is not merely arithmetic, in that it contains symmetries.
Symmetry is an aspect of geometry which applies to music. The division of the octave at the tritone creates symmetries.

Don't look at music only in terms of arithmetic; look at symmetry as well. A strictly arithmetical approach seems too literal. The use of circles and lines  have been used for years to demonstrate these sorts of things.

This "gibberish" all goes back to interval projection, as described by Howard Hanson in his text "Harmonic Materials of modern Music."

And all this trivial symmetry is just a product of equal temperament, a mid 18th century tuning convention.  4 just or mean-toned minor thirds does not equally subdivide an octave

[millionrainbows]

And all this trivial symmetry is just a product of equal temperament, a mid 18th century tuning convention.  4 just or mean-toned minor thirds does not equally subdivide an octave

Before you could modulate, none of this mattered. I don't think most people really grok the difference between ratio and quantity in music. Nothing said here seems to contradict that. I mean, Hanson wrote a whole book about this, and it's in Wik, yet the responses are so flip and shallow. You are still worried about the term geometry...oh, well, PBF...

[Parsifal]

Since ratios were brought up, here is a simple example of why self-consistent tuning impossible. 

Lets start with C in low register, call it 32 Hz (32 vibrations per second). Now we want to tune our harpsichord by ear. We can perfectly well hear when an active is true. That is when the frequency of the second note is double the first. We get higher octaves of C as 64 Hz, 128 Hz, 256 Hz, 512 Hz, 1024, 2048 Hz, 4096 Hz, multiplying by 2 each time. That's a very high sounding C.

But that only gives us C's. What about the other notes?

Now, suppose we tune a different way. We can very well hear a perfect fifth. That is when the frequency of the second note is higher than the first by a ratio of 3/2. Now we can tune every note of the scale by going up the "circle of fifths." Starting from C we tune G, which is a ratio of 3/2 above C. So G is 32*3/2 = 48 Hz. From G we tune D. D is a ratio 3/2 above G, so D = 32*(3/2)*(3/2) = 72 Hz. We can keep going until we get to C again.

C  32 Hz
G  32*(3/2) = 48 Hz
D  32*(3/2)*(3/2) = 72 Hz
A  32*(3/2)*(3/2)*(3/2) = 108 Hz
E  32*(3/2)*(3/2)*(3/2)*(3/2) = 162 Hz
B  32*(3/2)*(3/2)*(3/2)*(3/2)*(3/2) = 243 Hz
F sharp (G flat)  32*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2) = 364.5 Hz
C sharp (D flat)  32*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2) = 546.75 Hz
G sharp (A flat)  32*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2) = 820.128 Hz
D sharp (E flat)  32*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2) = 1230.1875 Hz
A sharp (B flat)  32*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2) = 1845.28125 Hz
E sharp (F)  32*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2) = 2767.921875 Hz
B sharp (C)  32*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2)*(3/2) = 4151.8828125 Hz

Notice strictly speaking I tuned B sharp, but in equal temperament B sharp and C are the same note.

Wait!

By tuning from C to C to C, up 7 octaves we got C = 4096 Hz.
By tuning from C to G to D, around to circle of fifths we got C = 4151.9. That's way off!

That is the problem, if we get from one note to another by different paths (up 7 perfect octaves, or up 12 perfect fifths) we don't get to the same note.

Suppose I gave you a ruler 1 foot long and another ruler 1 yard long, but the yard was longer than 3 feet. (Sorry, metric people, suppose I gave you a 10 cm ruler and a 1 m ruler, but the 1 meter ruler was longer than 10 of the smaller rulers)? How would you measure anything?

In any case, seems like numerology is more useful than geometry in thinking about tuning systems.

What amazes me is that people were able to devise rules for tuning keyboard without any way to actually measure the frequency of the notes. They went by "beats," which is the subtle interference heard between the overtone series of two notes.

[Monsieur Croche]

Before you could modulate, none of this mattered.

When exactly was that? -- at least, within classical music history.

Modulation is a word which came, of course, from MODES.

[millionrainbows]

When exactly was that? -- at least, within classical music history.

Modulation is a word which came, of course, from MODES.

Just a figure of speech. If music stays in a single tonic area, without shifting that pitch, as in just intonation, then modulation to different key areas does not matter. You seem to be very concerned with literal academic uses of terms, while ignoring the overall thrust of a statement. This will make my job very tedious.

[millionrainbows]

Yes, Scarpia. The octave cannot be divided by these ratios, since it is a 2:1.