12-step Harmonics

I was just listening to a Bette Midler song, and a particular chord sequence really sounded good; it practically sent me into ecstasy (okay, not exactly, but I don’t know how to describe it). My questions are:

  1. Why do certain chords sound so great to us, while others sound awful (such as a minor 2nd or a major 7th)? Is this a cultural preference, or is it universal?

  2. Why do we divide harmonics into 12 half-steps? Are there any instruments that use smaller steps, and does, say, 14 half-step music sound better than 12-step?

“It is better to follow even the shadow of the best than to remain content with the worst.”
–Henry van Dyke

This is a great question and the following explanation is oversimplified. Nit pick at your leisure.

Western music theory teaches us some intervals are considered consonant (good) and some are dissonant (bad). Inversions are usually considered to have identical qualities (the m2 and M7 in your example.)

These are both cultural preferences and very subject to personal tastes. What a Korn fan considers an acceptable interval may vary slightly from what a Celene Dione fan will accept.

Even not all teachers/books agree on the “stability” of certain intervals. Perfect 4th comes to mind .
[I had an argument about this with my theory teacher. At the time it amazed me he wouldn’t even consider calling a P4 consonant. Later I realized he just wasn’t into Megadeth.]

Keep in mind these are just intervals of two notes, fuller chords are much more dependent on the progression, especially the preceding and following chords. This make sense if you consider a chord as multiple intervals.

When physically represented, each note has a little wave form, a pattern of peaks and troughs. If the peaks and troughs correspond it is consonant, the more the are “off”, the more dissonant the interval.

We divide the octave into 12 half-steps. There are many instruments that have both more and less divisions within the octave.

Sitar (India) and Samishin (Japan) both have more. I have played both. Both sound like ass. This is two-fold. One, my ears are not used to the “quarter-tones”, so it sounds out-of-tune. Two, I’m really not that good in the first place.

Since we’re on it, it may interest and confuse you to know even within our octave division scheme there can be differences. In another words a C# is not always equivalent to a Db. If this offends you, as it does me, read up on temperment.

Finally, harmonics are (roughly) overtones of the fundamental note. For example, when a string is plucked it vibrates along many different nodes (it’s entire length, 2 x half its length, etc.). Each instrument has a different mix of harmonics. Among other things, the mix of which harmonics are stronger than others gives that instrument its timbre.

Of course a guitarist (even one as untalented as I) has a different take on harmonics. Natural, artifical, pinch, and harp harmonics still all rely on the above, they just emphasize a particular overtone by (partially) muting the others.

It’s not entirely cultural. The most common and (to some ears) most pleasing intervals are also the most simple mathematically. The octave for example, represents a doubling of the fundamental frequency, or a 2:1 relationship. Dissonant harmonies tend to be more distantly related.

I don’t remember enough music theory to give examples other than the octave, and I’d rather not guess. But some of you musically trained folks out there must know more. Any help?

“non sunt multiplicanda entia praeter necessitatem”

Re-reading Astroglide’s post a little more carefully, I see he mentions lining up the peaks and troughs of the waves. This is equivalent to forming simple whole number ratios between the frequencies, which is what I was saying.

“non sunt multiplicanda entia praeter necessitatem”

Perfect fifth = 3/2
Perfect fourth = 4/3
Major third = 5/3
Astroglide stated it well. Temperment problem occurs when you try to cycle through the 12 notes of the chromatic scale using one of these perfect intervals. Using perfect fifths, for example: C, G, D, A, E… You will come back to C, but it will not be “in tune” with your original C; it will not be in a 2:1 ratio (or multiple of 2:1). You can also see this in the harmonic series when you get top the 8th and 9th overtones. 8:9 (C-D) is a major second. So is 9:10 (D-E); they are not the same size, though.

I always thought the perfect fifth involved Jack Daniels…

Chaim Mattis Keller

“Sherlock Holmes once said that once you have eliminated the
impossible, whatever remains, however improbable, must be
the answer. I, however, do not like to eliminate the impossible.
The impossible often has a kind of integrity to it that the merely improbable lacks.”
– Douglas Adams’s Dirk Gently, Holistic Detective

WTG, guys! Y’all stated the facts clearly and concisely, which is more than a lot of theory teachers can manage.

I would like to add a tiny bit more . . .

We’re also affected by how we hear and feel the tensions as music is made – how the chord progression resolves. Music that does not resolve easily (or as we’re accustomed to, even) is often physically and even emotionally uncomfortable to listen to.

For those of you that wonder why so many tunes are built around the same pattern of progressions, well, here’s why; it works.

your humble TubaDiva
dancing the whole and the half steps

Correcting my own typo - 5/4 = major third, nor 5/3.

Hello group. My name is george, and I am a…

Oh, 12 step harmonics. 'Scuse me.

Work like you don’t need the money…
Love like you’ve never been hurt…
Dance like nobody’s watching! …(Paraphrased)

BTW, our current ideas about scales and notes originate with the ancient Greeks. No less than Aristotle, Plato, and Ptolemy wrote about them.

There are instruments in use that a allow quarter tones (or other semi-tones). They are called “trombones”. With the use of the slide, a trombone can play every possible pitch in its range easily. There have been a few contemporary pieces written that take advantage of this ability.

Of course, many instruments also have this ability. Fretless string instruments (violins, for example), electronic keyboards with pitch-bending controls, and many electric guitars with a “whammy” bar. Also, most experienced brass and woodwind players can exert enough control over their instruments to bend pitches.

Carpe hoc!

One thing I thought I might add is that what is considered the nastiest interval in music, the tri-tone or Devil’s Interval (six half-steps), has some interesting math facts. Whereas (as above posts have mentioned) pleasant intervals are ratios of small whole numbers, the tri-tone’s ratio is the square root of two to one - an irrational number. Not only is not a ratio of SMALL whole numbers, it’s not a ratio of whole number AT ALL. Never can be, no matter how large the denominator.

The even-tempered scale is a geometric set, and the tri-tone is right in the middle of that set. The geometric mean of one and two is 1.414… (irrational). The arithmetic mean, 1.500, is very consonant and occurs at the (pure) perfect fifth. Of course I’m mixing just intonation with even temperment, but you get the point.

I don’t know if I’ve pointed something out which is common knowledge, or if I’ve made a neat little discovery, or if my math is wrong. But I thought y’all might be interested.

First of all, this is a very complex subject. You can get PhD’s in it, OK?

Now, that being said, the main reason for the 12-tone scale seems to be that it is one of the best solutions to the circle-of-fifths problem. After the octave (2/1), the next most obvious interval is the fifth (3/2). If you chain fifths together, the first few cases that come near to a power of two (and thus the octave) are:

3^5/2^7 = 243/128 = 1.90
3^7/2^11 = 2187/2048 = 1.07
3^12/2^19 = 531441/524288 = 1.01

Note that 5 is the number of notes in the pentatonic scale (black keys of the keyboard), common in many folk musics, and 7 is the number of notes in the diatonic scale (white keys of the keyboard), and, yes, the two scales generated by making a circle of fifths with 5 and 7 notes are indeed the pentatonic and diatonic scales, just as the 12-tone circle-of-fifths scale is the chromatic scale.

I suspect that the failure of quarter-tones as such (i.e., exact 24-note scales, as opposed to various other scales with more than 12 notes) is that the 24-note circle-of-fifths scale isn’t as good as 12:
3^24/2^38 = 282429536481/274877906944 = 1.03
The first circle-of-fifths scale that improves on 12 has 53 tones, which is probably too many.

John W. Kennedy
“Compact is becoming contract; man only earns and pays.”
– Charles Williams

Found this while reading in bed last night –

– Music, the Brain and Ecstasy, Robert Jourdain

He goes on to say that mid-range tones that are only one or two half-steps apart are in each others’ critical bands, but bass tones have to be more widely separated to avoid this phenomenon.

So there is some physiological basis for dissonance, too. It’s not entirely a matter of acculturation.

“non sunt multiplicanda entia praeter necessitatem”

John W. Kennedy, Member, posted 08-24-1999, 11:44 AM

As can one in throwing frisbees.

A coupleof years ago , on NPR, I heard a piece about a professor of music, who composed some music for different tonal scales, as has been described. Some of this sounded really good, others really weird-I wonder if anyone knows where this CD can be obtained?

Guitarists don’t need a whammy bar to play notes that fall between the traditional twelve-tone scale. All you have to do is bend the strings with the fingers playing the notes. Such notes (assuming you don’t bend up a half- or a whole-step) sound like the fretted note played slightly sharp, giving them a slight emphasis.

This also shows part of the problem with trying to come up with a “new” scale that includes more notes. A scale is an established standard that is recognized by the listener. Notes that fall slightly outside the scale are perceived not as new notes but as variations on established notes.

FYI, you can also forcibly bend (slightly!) the guitar neck to modify the pitch for the whole chord. It’s physically hard (and it can’t be good for the instrument!) but I’ve seen it done, even in concert.

“non sunt multiplicanda entia praeter necessitatem”