what links mathematics's with playing a guitar?

Answer 1

Tons of things. If you look at the fingerboard you'll see that the fret spacing isn't linear. That's because the intervals between notes isn't linear in the 'well-tempered' scale that we use today. If you break an octave up into twelve equal parts, you get what is known as a 'perfect' scale. That is, one in which all of the intervals are rational. And if you play such an instrument, you find out in a hurry that they actually **do** sound better (less dissonances). But the problem is that you can't transpose because the scale is based on whatever you choose as your 'root' for the scale. But if you make all of the intervals proportional to the 12'th root of 2, then you get what is known as the 'well-tempered' scale and you **can** do transpositions, key changes, etc. with a minimum of potential dissonance. The well-tempered scale was invented by a pretty fair little amateur mathematician and composer named Johann Sebastian Bach ☺

Then there are the dynamics of a vibrating string. The 2 dimensional differential equation that models the mechanics of a vibrating string has an entire, infinite series of solutions which are superposed upon one another to yield a final solution to how the string vibrates. This is what a musician refers to as the 'overtones' of their instrument and they contribute to the 'sonority' of the instrument. Also, the manner in which the string is plucked has a great deal to do with deciding the 'overtone series' that will be created. Then there are the even more subtle effects of tremelo and vibrato caused by the performer moving their hand in front of the resonance chamber (or the mouth) of the instrument, and by moving their left hand backwards and forwards on the neck to increase and decrease the tension on the string.

That's just a quick intro to the mathematics of playing a guitar ☺


Doug

Answer 2

If this doesn't take the romance out of the guitar, I don't know what will:

The diatonic musical scale, which is used to tune the guitar, is exponential.

For the musician, frequencies of the note A are 440*2^n.

For the mathematician, all frequencies of the note C are powers of 2.

The frequency of any note can be defined by the relation f=2^(n+m/12), where m is an integer from 1 through 12 (1=C#, 2=D, 3=D#, 4=E, 5=F, 6=F#, 7=G, 8=G#, 9=A, 10=A#, 11=B, and 12=C), and n is the "octave number", taking on any integer value. A correction multiplier of 1.022 must be used to satisfy the musical purist (produce a 440 A). Middle C is then note 12 of octave 7. The base E-string of the guitar must then vibrate at a frequency of 1.022*2^(7+4/12), or 165 cps.

From physics, the frequency of a taut vibrating string is inversely proportional to the length of the string. That, and the exponential nature of notes, is why the frets are not evenly spaced. I have not studied the relationships that are required to get different strings of equal length to produce different frequencies, but it has to do with the diameter of the string and its tensile properties, and the tension placed on it.

Are you thoroughly bored yet?

Answer 3

Math is about thinking. Math is about problem solving. Math is about working with what you do know to give you a framework and a method of exploring and understanding what you don't know, about seeing relationships and patterns. Mathematics is a mind-set, and an attitude when you face something you do not understand. But there is also a beauty and a wonder about mathematics that only insiders know about. Words like elegant and beautiful are used constantly by mathematicians to describe paths of reasoning and proofs.

Certainly many tasks in the life of a musician fall into this category. Arranging a melody on an instrument and finding fingerings that correspond to certain sequences of notes is definitely a type of math problem. Playing the same melody on different instruments is math, as is playing a stringed instrument and changing the tuning. And when you find the best key to play a certain melody on a guitar, for example, there is a sensation that is known to math insiders as elegance.

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Answer 4

when a string vibrates, it does so at certain wavelengths that are ratios of the length of the whole string. For example if lightly touch one finger on a string at the half way point on the guitar (should be exactly the 12th fret), and pluck it, you will get a harmonic that is exactly one octave higher then if you pluck the string open.

Likewise, if your at shorten the string at other ratios of the whole, you will get other notes that are harmonically related to the open string. Harmony and ratios and highly interconnected, and that's just one aspect of the math involved in playing an instrument.

Answer 5

Listen to Johan Sebastian Bach and you'll HEAR the mathematics in music! Rhythm and meter are but a part of it, as harmony and melody relate directly as well.

Answer 6

The duration of the widdle-tastic 1980's perm-rock guitar solo is inversely proportional to the number of people left in the audience.

Answer 7

Rhythm, Meter.

Answer 8

Staff.