The frequency relationship can also be written in term of logarithms

Transcription

1 Pitch ad Frequecy by Mark Frech Departmet of Mechaical Egieerig Techology Purdue Uiversity There are two differet kids of scales importat to a guitar player, eve tempered ad diatoic. The diatoic scale defies the relatioships betwee the frequecies of differet otes. The space betwee two otes is called a iterval. Diatoic itervals have very simple frequecy ratios like 3/2 ad 5/4. While they make the math easy, these simple ratios make it impossible to have a regular fret spacig o the guitar eck. A eve tempered scale allows a regular fret spacig at the cost of the resultig otes ot always beig exactly the right frequecy. Havig all the frets spaced the same way makes the guitar icredibly versatile. How may differet ways ca you play a A major chord? O a piao, there is oe key for every ote ad, thus, exactly oe way to play a particular ote. The up side of the piao is that it ca be tued to diatoic frequecies. The dow side is that you have to deal with 88 separate keys. A eve tempered scale uses a fixed proportio to defie the pitch chage betwee otes. If some ote has a frequecy, f 0, ad the proportioal chage i pitch is r, the frequecy of successive otes is f = r f0 so that, the frequecies as the pitch icreases are f 0, rf 0, r 2 f 0 ad so o. The twelfth ote i the series is a full octave above the origial oe. Sice icreasig pitch by a octave doubles the frequecy, it is easy to figure out r. f = r f 2 f = r f 0 2 = r 0 r = The frequecy relatioship ca also be writte i term of logarithms f log = log f log f0 = log r f 0 Pictures are a lot more helpful tha equatios. Here s a plot of the frequecy ratios over two octaves 0

2 4 3.5 Frequecy Ratio r i i Semi-toe Sice the fuctio defiig frequecy ratios is a power series, the curve is straight whe it s plotted o a semi-log scale 10 Frequecy Ratio r i i Semi-toe

3 This all pretty abstract. To make it a little more real, here is a table of a complete octave usig middle A (440 Hz) as a startig poit Notes i Major Scale Iterval (Semi-toes / Note) Eve Tempered Frequecy Ratio Eve Tempered Frequecy (Hz) Diatoic Frequecy Ratio Diatoic Frequecy (Hz) 1 0 / A / A# / B / / C / C# / / D / / D# / E / / F / F# / / G / G# / A The frequecies calculated usig the eve tempered scale are a little bit differet from those usig a diatoic scale (which is what a guitar is attemptig to reproduce). The differece is less tha 1%, though. The guitar player is basically tradig the ability to use diatoic tuig for the ability to use a regularly spaced fret patter to play the same scales all up ad dow the eck. While purists might ot like it, this trade makes the guitar a icredibly versatile istrumet. It helps to have startig poit. For a guitar, the startig poit is the frequecies to which to the strigs should be tued. The ope strig frequecies for a guitar are Strig Note Frequecy (Hz) 1 E B G D A E 82.4 Sice these are ope strigs (o frets ivolved yet), these are diatoic frequecies. These frequecies correspod to those you d fid o a correctly tued piao. So what happes whe you make a chord? A chord is just a collectio of otes from a scale which are played together. Each differet kid of chord uses a differet group of otes. For istace, major chords use otes 1,3 ad 5, mior chords use otes 1, b3 ad 5, ad so o. Cosider a major chord, made usig the 1,3 ad 5 otes i a scale usig A

4 as the startig poit. The frequecy ratios formed usig a guitar are ot exactly diatoic, but they are very close. Note Diatoic Frequecy Ratios Eve Tempered Frequecy Ratios Percetage Differece A 1/ % C# 5/ % E 3/ % Strictly speakig, chords are really defied by itervals. A iterval is just the distace betwee two otes. I a major scale, the iterval betwee the root ote (the ote that starts the scale) ad each diatoic ote has a ame. These ames alog with the frequecy ratios are Iterval Name Jump Semi-toes Frequecy Ratio Major Secod r 2 =1.246 Major Third r 4 = Perfect Fourth r 5 = Perfect Fifth r 7 = Major Sixth r 9 = Major Seveth r 11 = Octave 1-8 r = The frets o a guitar eck are arraged so that the space betwee two frets o the same strig is always a semi-toe. To figure out the fret spacig, we eed to kow the relatioship betwee the frequecy of a vibratig strig ad its legth ω = π L T ρ where T is the tesio i the strig ad ρ is the mass desity of the strig. The first frequecy, ω 1, is the fudametal frequecy ad the frequecies ω 2, ω 3 ad so o are called overtoes or higher harmoics. The frequecy with which a strig vibrates is iversely proportioal to the legth of the strig as log as othig else chages. for istace, doublig the legth of a strig would lower the frequecy by half or oe octave. Kowig this, it is ot hard to write dow the expressio for the positio of the frets o the eck. If S is the scale legth (distace from ut to bridge), the distace, d, from the bridge to the th fret is d S = r

5 The distace from the ut to the th fret is d S 1 = S = S 1 r r The two most commo scale legths are 25.5 (Feder) ad (Gibso). Here are the spacigs for those two scale legths Fret Feder (25.5 ) From Bridge Feder (25.5 ) From Nut Gibso (24.75 ) From Bridge Gibso (24.75 ) From Nut 0 (ut) The ext chart shows the frequecies o a guitar eck. Remember that icreasig the pitch of ay ote by a octave doubles its frequecy ad reducig the pitch by a octave cuts the frequecy i half. The lowest ote o the eck is E o the 6th strig Hz. Doublig that frequecy gives Hz. Doublig agai gives Hz, so there are two octaves betwee strig 1 ad strig 6. If your guitar has 24 frets, you could raise strig 1 by aother two octaves. Thus, the istrumet has a 4 octave rage.

6 Frequecies O A Guitar Neck Frequecies i Hz Fret Strig 6 Strig 5 Strig 4 Strig 3 Strig 2 Strig 1 ut Oe way to make the frequecies o the eck more real is to draw a picture. Each box represets a ote o the eck of the guitar, ad the height of each box is the frequecy for that ote. The result is a 3-D picture of frequecies o the eck.

7 Fr Boxes with the same height are the same frequecy ad, thus, the same ote.