Musical Theory question

[Enola Straight]

Given: a simple chord...C-E-G...yields perfect harmony.

What if, instead of a chord of 3 notes, we have a harmony of harmonies?

A harmony of C-E-G each in the keys of C, E, and G?

[Biffy the Elephant Shrew]

You seem to be talking about a polychord consisting of the C major triad (C-E-G), the E major triad (E-G#-B), and the G major (G-B-D) triad combined. That would give you a chord of the notes:

C-E-G-B-D-G#

(Duplicated notes don't matter.)

Spell the G-sharp as A-flat and you could parse that as a C major 9 (add flat 13). Basically what you've got there is a jazz chord. :)

[antonio107]

You seem to be talking about a polychord consisting of the C major triad (C-E-G), the E major triad (E-G#-B), and the G major (G-B-D) triad combined. That would give you a chord of the notes:

C-E-G-B-D-G#

(Duplicated notes don't matter.)

Spell the G-sharp as A-flat and you could parse that as a C major 9 (add flat 13). Basically what you've got there is a jazz chord. :)

I'm not clear on the question, but yes, that sounds like what it is. Don't voice them too closely together!

[outlierrn]

One thing you won't have, is a perfect harmony.

[Pitchmeister]

The reason you wouldn't have perfect harmony are actually the harmonics. Even that C major chord isn't perfect in the technical sense. (The most perfect interval is unison, of course, followed by the octave and the fifth.)

If you look at the harmonic scale (http://en.wikipedia.org/wiki/File:Harmonic_series_intervals.png) you will see that the major third is the 5th harmonic, still pretty low in the series, but the three other notes, D, G# and B are numbers 9, 13 and 15. (They're also odd harmonics, which makes them more dissonant than even ones, generally speaking.)
The harmonics of all those notes played together just create a mess of different frequencies that follows no mathematical principle, which is the basis for consonance.

[Chronos]

It should also be mentioned (I almost said noted) that in most modern tunings, you won't actually get any perfect harmonics, other than the octave. Modern tuning (called equal-tempering) is generally a compromise designed so that all of the harmonics (in any scale) are all very close to perfect, but none of them quite exactly so. So you get a ratio of 1.25992... where you should get 5/4, 1.33484... where you should get 4/3, and 1.49831... where you should get 3/2.

[BigT]

It should also be mentioned (I almost said noted) that in most modern tunings, you won't actually get any perfect harmonics, other than the octave. Modern tuning (called equal-tempering) is generally a compromise designed so that all of the harmonics (in any scale) are all very close to perfect, but none of them quite exactly so. So you get a ratio of 1.25992... where you should get 5/4, 1.33484... where you should get 4/3, and 1.49831... where you should get 3/2.

On a piano, not even the octaves are exact. The harmonics of the strings are not perfect octaves. A perfect octave could only be produced with a perfect string.

If you tune a piano with perfect octaves, it will sound sharp, as the harmonics of the strings are slightly flat. Any interval over an octave will sound off, and piano players rarely stay within one octave (unless it's someone just learning to play Twinkle Twinkle Little Star or something.)

[CookingWithGas]

...you could parse that as a C major 9 (add flat 13). Basically what you've got there is a jazz chord. :)And the reason that such chords are thought of as jazz chords is that the dissonance creates tension. A big part of jazz is managing tension. Two notes a half-step apart, or an octave plus a half step--G and G#--create a huge amount of tension, and are even often called "avoid notes".

ETA: If you make that an E minor triad, then you have built a C major 9, which has lower tension than with the b13. And that is the start of a discussion of how chords are built and extended from the major scale.

[Chronos]

On a piano, not even the octaves are exact. The harmonics of the strings are not perfect octaves. A perfect octave could only be produced with a perfect string.

If you tune a piano with perfect octaves, it will sound sharp, as the harmonics of the strings are slightly flat. Any interval over an octave will sound off, and piano players rarely stay within one octave (unless it's someone just learning to play Twinkle Twinkle Little Star or something.) Interesting, I did not know that. But that sounds like a technological limitation, not a mathematical one. Even with an ideal waveform generator that got both the fundamentals and the harmonics exact, you still can't construct a tuning that gives you exact resonances on all of the chords in all scales.

[AnalogSignal]

Even with an ideal waveform generator that got both the fundamentals and the harmonics exact, you still can't construct a tuning that gives you exact resonances on all of the chords in all scales.

Many synthesizers have workarounds that allow you to re-tune on the fly so all chords sound good. Here is one example from an Emu XL-7 manual:


The Just C Tuning Tables

Well Tempered and Just were standard keyboard tunings up until the 20th century when the current “equal tempered” scale became prevalent. In an equal tempered scale, the octave is equally divided into 12 parts. In Just or Well Tempered scales, the 12 notes are separately tuned to produce pure chords. However, in Just tunings you are limited to playing certain chords and if you play the wrong chord it may sound very BAD!

XL-7 allows you to modulate between keys by providing you 12 user tuning tables. Tuning tables can be changed as you play using a program change (create several presets with the same sound and different tuning tables), by MIDI SysEx command (using a programmable MIDI footswitch or other device), or using a continuous controller (link 2 presets and crossfade between them using a controller).

[GorillaMan]

Many synthesizers have workarounds that allow you to re-tune on the fly so all chords sound good.
It doesn't mean that you can make all the chords sound good with one single setup, though. What these systems allow you to do is tune the instrument so that it's tuned to mean-tone on C, for example, which means that C, F and G chords will sound good, but distant ones such as B or A flat will sound rather unpleasant.

Fine, if the music you're playing doesn't wander far from C major but sticks around closely-related keys, as baroque music generally does. Then, if the next piece you're playing is in F# minor, you can switch the tuning to mean-tone on F#, but then C major will sound terrible.

And changing within a piece will certainly not sound right, not in the context of classical music, anyway!

[AnalogSignal]

It doesn't mean that you can make all the chords sound good with one single setup, though.

You can retune as many times as desired within a piece of music as described in the quote.

And changing within a piece will certainly not sound right, not in the context of classical music, anyway!

What sounds right is a matter of opinion. I think the current fashion for the extreme use of autotune on pop vocals sounds wrong but that doesn't stop songs with extreme autotune from being number one hits.

[GorillaMan]

What I mean by 'in the context of classical music' is using an electronic instrument to simulate the sound of an acoustic one, and that changing temperament in the middle of a Bach fugue would sound as much like a real keyboard instrument as the autotuning in Cher's Believe sounds like live singing.

[AnalogSignal]

What I mean by 'in the context of classical music' is using an electronic instrument to simulate the sound of an acoustic one, and that changing temperament in the middle of a Bach fugue would sound as much like a real keyboard instrument as the autotuning in Cher's Believe sounds like live singing.

If you are not a traditionalist, anything is possible. Here are notes from Wendy Carlos' Switched-On Bach 2000 (http://www.wendycarlos.com/+sob2k.html#tunings):

The smooth sounds you may notice in this recording are to a large degree the result of the tunings used. None of them is our standard equal temperament, a compromise that allows modulation into all keys, and requires only twelve notes in an octave. But along the way a lot was sacrificed. Musicians have remained rather timid about trying out the alternatives, probably believing the myths that anything microtonal sounds weird and out-of-tune. The few pioneers who do venture into these waters get treated with disdain by a majority who exhibit surprisingly little tolerance or curiosity in this area. What is everyone so afraid of?