# Topic: OT - Musicians are just plain NUTS!

1. ## OT - Musicians are just plain NUTS!

OK, So I'm a scientifically trained engineer with heavy math background and I just started learning a bit of music theory. It takes you 4 steps to get from C to E, so what do they call that? A third. Of course getting from C to Eb, which is actually three steps is also called a third. But here's where it gets really weird: the 7 steps from C to G is called a fifth and the 10 steps from C up to Bb is called a dominant seventh.

So, either musicians are nuts or mathematicians are nuts. Let's assume for the moment that it's the mathematicians who are misguided. What can they learn from the musicians' example. First lesson: You don't need all those number words like "seven", "five", and "nine". There's a more straightforward way to count with, for example, only four number words, "unit", "quad" and "dozen":

1. unit
2. semi-quad
3. diminished quad
4. quad
5. augmented quad
6. semi-dozen
7. inverted augmented quad
8. inverted quad
9. inverted diminished quad
10. inverted semi-quad
11. inverted unit
12 dozen

With intermediate values such as:

3.25 diminished minor quad
3.5 minor quad
3.75 augmented minor quad

Second lesson: Number words don't have to actually be what they say they are. For example, just as 10 steps is a "seventh" in music, so we might say that "dozen" really means "nine" (or, more properly, "inverted diminished quad").

How much more sensible it would be to teach students that pi is "diminished quad point unit quad unit augmented quad inverted diminished quad", or to talk in history class about the "inverted augmented quad wonders of the ancient world."

On second thought, maybe it's the musicians who need revamp their descriptive terminology for intervals. Hmmm. Some how I don't think it's ever going to happen, but wouldn't it be nice if an interval of 10 steps was called an 10th instead of being called a "dominant seventh" so that actual mathematical analysis could be done on intervals?

Ah, but so it goes. I may some day understand music theory, if I can get my head around its irrational nomenclature.

--gary shannon

2. ## Re: OT - Musicians are just plain NUTS!

Hee hee, musicians are nuts, you're right.

BUT, there is a very good explanation for why a 3rd is called a 3rd and why a 5th is called a 5th. Counting semitones is not the best way to visualize an interval. I'm assuming you know this and your post is all in fun -- but let us know if you genuinely want an explanation.

chris.

3. ## Re: OT - Musicians are just plain NUTS!

Yes, it's a bit wacky, but perhaps not the most wacky thing about music.

The humor is not lost on me, but in case you actually are wondering about some of this...

The interval numbers are based on diatonic counting, not chromatic, and are inclusive of first and last notes. This is easiest to understand by using only the notes of a C major scale, or the white keys on a piano. The interval between every adjacent set of notes is a 2nd. The quality of the interval is determined by how many half-steps are between the two notes.

Since a unison is 0 steps, you can think of it as (interval_name -1) steps. And by the way, the seventh between C and Bb is called a "minor 7th." "Dominant 7th" refers to the quality of a certain four-note chord, which is not necessarily built on the dominant (fifth scale degree) of the scale! Confused yet?

But if music is putting the hurt on math, math is certainly having its revenge. Consider our very imperfect "circle-of-fifths." The interval of a perfect fifth has the ratio of 2:3 ~0.6667 between two frequencies. However, since we slice up our octaves into 12 even sized steps, the ratio of a perfect fifth as we use it is actually 1: (7 * 12th root of 2) ~ 0.6675, which is close, but not quite there. A major third is even worse... compare the simple ratio of 4:5 = 0.8 to the ballpark figure of 1: (4 * 12th root of 2) ~0.7937.

How do we get around these inconsistencies? Easy... we cheat!

4. ## Re: OT - Musicians are just plain NUTS!

Originally Posted by fiziwig
OK, So I'm a scientifically trained engineer with heavy math background and I just started learning a bit of music theory. It takes you 4 steps to get from C to E, so what do they call that? A third. Of course getting from C to Eb, which is actually three steps is also called a third. But here's where it gets really weird: the 7 steps from C to G is called a fifth and the 10 steps from C up to Bb is called a dominant seventh.

So, either musicians are nuts or mathematicians are nuts. Let's assume for the moment that it's the mathematicians who are misguided. What can they learn from the musicians' example. First lesson: You don't need all those number words like "seven", "five", and "nine". There's a more straightforward way to count with, for example, only four number words, "unit", "quad" and "dozen":

1. unit
2. semi-quad
3. diminished quad
4. quad
5. augmented quad
6. semi-dozen
7. inverted augmented quad
8. inverted quad
9. inverted diminished quad
10. inverted semi-quad
11. inverted unit
12 dozen

With intermediate values such as:

3.25 diminished minor quad
3.5 minor quad
3.75 augmented minor quad

Second lesson: Number words don't have to actually be what they say they are. For example, just as 10 steps is a "seventh" in music, so we might say that "dozen" really means "nine" (or, more properly, "inverted diminished quad").

How much more sensible it would be to teach students that pi is "diminished quad point unit quad unit augmented quad inverted diminished quad", or to talk in history class about the "inverted augmented quad wonders of the ancient world."

On second thought, maybe it's the musicians who need revamp their descriptive terminology for intervals. Hmmm. Some how I don't think it's ever going to happen, but wouldn't it be nice if an interval of 10 steps was called an 10th instead of being called a "dominant seventh" so that actual mathematical analysis could be done on intervals?

Ah, but so it goes. I may some day understand music theory, if I can get my head around its irrational nomenclature.

--gary shannon
Actually, it takes 2 1/2 steps to get from C to E. From C natural to C Sharp is only a half step. From C natural to D natural is a whole step. So from C natural to E natural is two whole steps and one half step. From C natural to C sharp is a minor second; From C natural to D natural is a major second; From C natural to D shapr is an augmented second; and from C natural to E natural is a major third; but from C natural to E flat is a minor third. From C natural to F natural is a perfect fourth. From C natural to G natural is a perfect fifth; and from C natural to G flat is a minor fifth; and from C natural to G double flat is a diminished fifth. C natural to G sharp is an augmented fifth; C natural to A natural is a minor sixth; and from C naturl to B natural is a (I may falter here) dimished seventh.

C is the tonic (I Major); d is the supertonic (ii minor); E is the Mediant (iii minor); F is the subdominant (IV Major); G is the Dominant (V Major); a is the Submediant (vi minor); b is the Leading Tone (vii diminished) [there should be a small circle next to the vii]; and then back to the Tonic as your octave.

fiziwig I do hope that your are completely confused now. Just kidding.

5. ## Re: OT - Musicians are just plain NUTS!

... musicians are nuts?
I eat them. Am I accused of cannibalism now?

Raymond

6. ## Re: OT - Musicians are just plain NUTS!

Well, I'm gradually getting it figured out. But the whole idea of a "semi-tone" or "half step" doesn't make any sense either. Either you take a step or you don't. A half step is where you raise your foot, but never quite put it back down again. You can take a long step or a short step, but never a half step. If you move from one note to another you have stepped, you haven't half-stepped, or almost stepped. You have stepped.

I understand why it's called what it's called, but it still doesn't make any sense. Even calling it a "third" doesn't properly express it since it's only TWO whole steps in size. Starting from C, step once to D, then step once more to E. Two steps, not three.

Anyway, my point is simply that viewed from the inside by someone who has become thoroughly familiar with the system, it's inherent irrationality has become invisible. Viewed by a mathematically-minded outsider, it's all shear madness that makes no logical sense whatsoever.

Math teacher to class: "Children, today we learn about four. Now four can mean either three, or four or five, and I won't always tell you which one it means. But if I really mean four then I might say perfect four, or I might take it for granted that if I don't say augmented four or diminished four that I mean perfect four. Now how much is four plus one?"

That's my semi-quad cents worth.

--gary

7. ## Re: OT - Musicians are just plain NUTS!

Originally Posted by fiziwig
Well, I'm gradually getting it figured out. But the whole idea of a "semi-tone" or "half step" doesn't make any sense either. Either you take a step or you don't. A half step is where you raise your foot, but never quite put it back down again. You can take a long step or a short step, but never a half step. If you move from one note to another you have stepped, you haven't half-stepped, or almost stepped. You have stepped.
I was planning to answer this message. But after having read my comments about half-steps, whole steps, almost steps, shorter steps, longer steps, I just looked at myself and thought that given my body size I better can walk the 12-tone system. Smaller steps.

Raymond

8. ## Re: OT - Musicians are just plain NUTS!

Originally Posted by Samantha Penigar
...
C is the tonic (I Major); d is the supertonic (ii minor); E is the Mediant (iii minor); F is the subdominant (IV Major); G is the Dominant (V Major); a is the Submediant (vi minor); b is the Leading Tone (vii diminished) [there should be a small circle next to the vii]; and then back to the Tonic as your octave.

fiziwig I do hope that your are completely confused now. Just kidding.
Yes, completely confused. I just don't know why you guys have to have seventeen different words that all mean "3".

From my mathematical perspective the major scale on any root tone 0 is "0 2 4 5 7 9 11 12 (= 0 mod 12)". Each note has ONE name. Each interval has ONE name. Each chord has ONE name. "0 4 7" is the same chord regardless of what key it's played in. "0 3 7" is minor, and I don't call it "0 4 7 minor", it's important enough to have its own name, "0 3 7". Relationships become more obvious when you can do the math. e.g. 7-4 = 3 and 4-0 = 4 while 7-3 = 4 and 3-0 = 3 showing the interval relationship between major and minor as "step(4,3)" for major vs. "step(3,4)" for minor. You get from major to minor by inverting the step order. Simple math. Everything that's so confusing in music theory becomes crystal clear when translated to modular integer functions on integer steps (mod 12).

--gary shannon

9. ## Re: OT - Musicians are just plain NUTS!

Originally Posted by Raymond62
I was planning to answer this message. But after having read my comments about half-steps, whole steps, almost steps, shorter steps, longer steps, I just looked at myself and thought that given my body size I better can walk the 12-tone system. Smaller steps.

Raymond
Maybe as we get older we should use some middle eastern system that has quarter tone intervals. Smaller steps still.

--gary shannon

10. ## Re: OT - Musicians are just plain NUTS!

Heheh, I understand your concern. I teach AP Music Theory to high schoolers who have the same questions. Think of it this way. Numbers mean different things depending on the context
• Three can mean Thirty-Six, if you're talking about feet (Thirty-Six inches) or dozens (Thirty-Six delicious donuts).
• Three can mean 15 if you're talking about hands (5 fingers per hand).
• Three Families can mean "6 people" or "107 people" depending on the sizes of the families.
• When you are in Third Grade, you might be in your 4th or 5th year of school, depending on whether you attended Pre-school, Pre-K (I don't know the equivalents in other countries).
• A Martial Arts expert who has reached 3rd Dan has been through many belts or rankings.
Look at this set of numbers:

12345678

3 is obviously the THIRD number, even though it is TWO steps away from the number 1. Same thing with a musical scale:

CDEFGABC

Sure, E may be two steps away from C, but it is obviously the THIRD note in the series. This is why it's called the 3rd.

The confusing part, of course is that the major scale is made up of both whole steps AND half-steps. So a scale step can be either, depending on where you are in the scale. The distance between C and D is not the same as the distance between E and F.

In a major scale you have:

1x2x34x5x6x71

The numbers are the scale degrees. The xs are notes between the scale degrees. If you're talking about intervals and scale degrees, just ignore the x's and count numbers. 5 is the FIFTH number. 7 is the SEVENTH number. 2 is the SECOND number. If you're talking about semitones, include the x's (which would make 5 "seven half-steps" from 1).

Of course a mathemetician would read this paragraph and think it's crazy! Why not just have 12 notes? ABCDEFGHIJKL? Because music came from our ears long before it was written down and analyzed. And because for hundreds and hundreds of years, most western music has used the major and minor scales, and modal variants of them. Music is written in terms of scale degrees, because that is the way we hear it, play it, and write it. So there are 7 scale degrees, which can be lowered and raised in certain contexts. If we called them ABCDEFGHIJKL, the A major scale would then be:

A C E F H J L A

Ok Math Boy, does that make more sense to you?

If you're interested in mathematical approaches to music, you might be interested in looking into Shenkerian Analysis and Set Theory (Pitch-Class Sets, Pitch-Class Theory, etc.). And of course Schoenberg's 12-tone system and other methods of "serial composing." A lot of composers since the early-mid 20th century (but wait, why is it 20th when it's the 1900s? those calendar makers are nuts) have gotten very mathematical with their music.
• Boulez and Babbit are two who come to mind. Many composers of that generation wrote "math-music," where the music expressed very complex ideas--it's hard to listen to if you're new to this style, but the analysis can be fascinating if that's how your mind works. I think either Berg or Webern (or maybe it was Xenakis) was a mathematician and wrote music based on mathematical principles.
• Several composers have used the fibonacci series and "golden ratio" in their music, including Bartok, Debussy, and Mozart.
• Dufay (I think) wrote a very famous mass which somehow mirrored the ratios of the architecture in the cathedral in which it was performed.
• There's a group of composers who write "Spectral Music" where intervals are tuned to match the overtone series (there's much more to it than that though)--check out the music of Tristan Murail. IRCAM in France puts out a lot of mathematical music, including "algorhithmic music," where music is composed based on complex algorithms (sometimes by a computer program, but sometimes by a person following a forumula).
• Mozart had music "game" where he composed melodies by rolling dice (I don't know the details). A few hundred years later, John Cage did a lot of "chance" music based on statistics and rolling of dice and other random events.
• Pythagoras himself was a musician and he applied his mathematics to an understanding of acoustics, sound, musical scales, etc.
I'm sure there's a lot more out there -- google "music and math" or variations on that theme and I'm sure you'll find a lot of fun stuff.

chris.

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