[Harp-L] Evolution of Temperaments

[icemanle@xxxxxxx]

will this be on the quiz?





 





 



-----Original Message-----

From: Vern Smith <jevern@xxxxxxx>

To: harp-l@xxxxxxxxxx; icemanle@xxxxxxx

Sent: Fri, Oct 30, 2009 4:00 pm

Subject: Re: [Harp-L] Evolution of Temperaments












Thank you, Iceman, for that clear, articulate explanation of 
the need for equal temperament and how it arose. 

 


Now, for the mathematically inclined, (if any), here are 
some formulas that will relate frequency to haltone notes 
and cents.  They will work in Xcel, Basic, or on any 
scientific pocket calculator. 

 


Definitions: 


N = musical interval in number of halftones, a + or - 
number 


C = musical interval in number of cents, a + or - number 


Fo = known starting frequency/pitch in Hertz (cycles per 
second) 


f = unknown frequency/ pitch in Hz at the other end of an 
interval. 


^ indicates exponentiation 


* indicates multiply 


/ indicates divide 


+ and - indicate add and subtract 


log indicates the base-ten logarithm 

 


To find the frequency (f) at the end of any interval as a 
function of the starting frequency Fo and N halftones. 


f = Fo * 2 ^ ( N/12) 


Note that when N =12 halftones, then N/12 =1 and f = Fo * 2, 
an octave. 

 


If you assign Fo the value of 440 Hz, then f is the 
frequency of any musical note where N is the number of 
halftones counting from the note A4, (A above middle-C) 


Example: N = 12 at A5, 14 at B5, etc. 

 


To find the number of halftones N in an interval from Fo to 
f: 


N = 12 / log(2) * ( log(f) - log(Fo) ) 

 


A cent is perceived by the ear as 1/100th of a halftone. 
Small deviations from musical notes in discussions of tuning 
are usually quantified in cents. 

 


To find the frequency at the end any interval as a function 
of the starting frequency Fo and the mumber of cents: 


f = Fo * 2 ^ ( C / 1200) 

 


Example: 24 cents of pitch deviation has been mentioned in 
Iceman's email. 


f = 440 * 2 ^ (24 / 1200)  =  478.16 Hz. or 38.16 Hz out of 
tune. 


At an octave up Fo = 880, then f = 956.32 Hz or 76.32 Hz out 
of tune. 

 


To find the number of cents C in the interval Fo to f: 


C = 1200 / log(2) * ( log(f) - log(Fo) ) 

 


Now that's probably a lot more than anyone wanted to know 
about equal temperament.  That is what the "DELETE" key is 
for.   ;o) 

 


Vern