What is the innate sense that tells us the last note of do-re-mi-fa-so-la-ti-.. ? Yet depending on how the brain hears the introduction, we may or may not be satisfied with the tuning of a the last do. It is a understood problem in music that you can't tune a piano perfectly. To understand why takes a bit of mathematics and physics. The Math Explained: Let us begin by explaining the way a scale is constructed. To avoid black keys on the piano, I'll use the key of C Major. So- called middleC represents a particular frequency, (440). I will utilize the mathematical trick of defining my units so that my middle C has a frequency of, the easy to work with the value of 1. Going up one octave doubles the frequency. Thus, the C one octave up from middle C has a frequency of2 Quadrupling, i.e. 4 × would be the third C , up the scale, defining the second Octave, (between the Second and Third C ) of which the fifth, G is defined by: Tripling the first frequency moves to the perfect fifth, G in the second octave. In our case, this means that the G in the next octave has a frequency of 3. And by Halving that we get theG in the First Octave. This Process Continues, that is times 13. By inverting the rule that says that the same note one octave above another must have double the frequency, we can fill-in the perfect fifth in the first octave, since it should have half the frequency of the G in the second octave.The Intervals Defined: Circle of Fifths: C G D A E B F# C# G# D# A# F C | The Tonic
as1 | The Fifth
3 /2 the Tonic |
|---|
| C 1 | G 1.5 | | G 1.5 | D 1.125 | | D 1.125 | A 1.16875 | | A | E 1.25625 | | E | B 1.8984375 | | B | F# 1.423828125 | | F# | C# 1.06787109 | | C# | G# 1.60180664 | | G# | D# 1.2013549 | | D# | A# 1.80203247 | | A# | F 1.351524 | | F | C 2.027286 |
| Here is the Arithmetic and define the other keys, note the beautiful simplicity. 1/2 of 3/2 or 1/4 of 3/2 or 3/8 to get the results in the first octave ranging between C measured as 1 and 2 Apply Repeatedly to define all the notes, (mathematics is always so simple) The perfect fifth in the key of G is D. Thus we have, by tripling and then halving, then halving again, (same as quartering) Repeated again: the perfect fifth in the key of D is A. and so on, and so on |
| | Tonic | W | Third | Fourth | Fifth | Sixth | H | Tonic |
|---|
| C | D | E | F | G | A | B | C | | C# | D# | F | F# | G# | A# | C | C# | | D | E | F# | G | A | B | C# | D | | D# | F | G | G# | A# | C | D | D# | | E | F# | G# | A | B | C# | D# | E | | F | G | A | A# | C | D | E | F | | F# | G# | A# | B | C# | D# | F | F# | | G | A | B | C | D | E | F# | G | | G# | A# | C | C# | D# | F | G | G# | | A | B | C# | D | E | F# | G# | A | | A# | C | D | D# | F | G | A | A# | | B | C# | D# | E | F# | G# | A# | B | | C | D | E | F | G | A | B | C |
| - Here we see the steps in theMajor scales of the differing keys. Note the orderly progression, this gives the art of music it's amazing mathematical properties. The rearrangement of the half steps give rise to the minor scales and various other interesting properties that the scales have.
- Pythagorean Hammers given 2600 years ago are approximations. Yes it is the same Mathematician that gave us the theorem that predicts the unknown side of the Right Triangle.
- The frequency of the perfect fifth is 3/2 that of the Tonic.
- The frequency of the tonic at the end of the octave is twice that of the original tonic.
- The frequency of the perfect fourth is 4/3 that of the Tonic.
- The frequency of the major third is 5/4 that of the Tonic.
- The frequency of the minor third is 6/5 that of the Tonic.
- Using these strict mathematical rules and combining them as efficiently as possible,(to lessen the induced error) one arrives at the following list of frequencies for the notes in the C major scale, Thusly:
| Pythagorean Comma's |
|---|
| Note | Math | Reduced |
|---|
| C | 1 | 1 | | G | 3/2 | 1.5 | | D | {(3/2) 2 }*(1/2) = 9/8 | 1.125 | | A | {(3/2)3 }*(1/2) = 27/16 | 1.6875 | | E | {(3/2)4 }*(1/2)2 = 81/64 | 1.25625 | | B | {(3/2)5 }*(1/2)2 = 243/128 | 1.8984375 | | F# | {(3/2) 6 }*(1/2)3 = 729/512 | 1.423828125 | | C# | {(3/2)7 }*(1/2)4 = 2187/2048 | 1.06787109 | | G# | {(3/2)8 }*(1/2)5 = 6561/4096 | 1.60180664 | | D# | {(3/2)9 }*(1/2)5 = 19683/16384 | 1.2013549 | | A# | {(3/2) 10 )*(1/2)5 = 65536/32768 | 1.80203247 | | E# | {(3/2)11 ) *(1/2)6 = 177147/131072 | 1.351524353 | | C | {(3/2)12 )*(1/2)6 = 531441/262144 | 2.027286529 |
The Comma is the Error.
Notice that some of these fractions are not equal! In particular, the final C in the scale ought to have frequency twice the basic C. Instead, if we go way up by fifths, then back down again by octaves, we have this fraction, whose decimal approximation is: 2.027286530. If we took half of this to return to our starting point, we'd have: 531441/524288=1.013643265 This discrepancy is known as the Pythagorean Comma. We can't mix a function based on tripling (for the fifths) with a function based on doubling (for octaves).The method that western music has adopted is to use the system of temperament also known as even temperament, whereby the ratio of the frequencies of any two adjacent notes known as a Semitone is constant, with the only interval that is acoustically correct being the octave. Also seeSavart Equal Temperament: spreads the error around in two ways. The errors in any particular key are more or less evenly spread about. There is no key that are better off than any other.Just Temperament: Here the Instrument is specifically tune to the Key that it will be played in.- The Whole Step is Flatten a Semitone (1/16)
- The Half Step is Sharpen the same
- This difference is so small that most people cannot hear it.
- All String Musicians can hear such differences, however.
- The sum of the corrections 5 whole steps and 2 half steps5/16-2/16=3/16
of a whole step,(.125) or- .0234
Here are other concerns and observations related to these commas, That Relate to both the Just and Equal Temperament of the Piano and the Phenomena Known to every Blues Guitarist that slurs his notes. They all have an Ear for it. |
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