In
several places in the articles on various organs, I have made, or will
make reference to the equally tempered scale. Exactly what is this?
There is much on the Internet about the many different musical temperaments that have been in use for centuries and of their advantages and shortcomings. A temperament is a tuning system by which the pitches or notes of the scale are tuned so that polyphonic music is possible in more than one key. You see, if you tune all the notes of the scale absolutely accurately so that you can play a song perfectly in the key of C, for example, it would not work if you want to play the same song in F. You'd need to create another scale and tune it perfectly for F. Obviously, if you have a keyboard instrument with a separate scale for each key, it would become pretty complicated and unwieldy. So, even as early as many centuries ago, musicians and instrument makers began to experiment with various ways of tuning so that things would sound good in several different keys and yet the instruments would not get too complicated.
The ultimate result of this is our modern equally-tempered musical scale. It takes the twelve tones of a complete musical scale and adds a slight amount of deliberate error to each pitch so that the error is spread out evenly across all twelve notes and no one error is too large. Then you can play a song in any key you choose and it will still sound good.
Modern pianos and organs would not be possible unless some type of temperament for tuning was available. The equally-tempered scale is not the only temperament available, but it's the best for keyboard instruments. Other temperaments were in use earlier which attempted to distribute tuning errors differently, making several keys sound really good and leaving certain intervals in other keys that were way off and were, interestingly enough, referred to as "wolves" probably because the glaring out-of-tune sounds that resulted from playing these so-called wolf intervals suggested the howling of a wolf.
The equally tempered scale takes any particular note's frequency and multiplies it by the twelfth root of two, which then gives the frequency for the next higher note on the scale. In modern music, It is essentially agreed that the pitch of Middle A will be 440 Hz. I understand that this was largely brought about by the J.C. Deagan company in the early 20th century as they were the leading manufacturer in the USA of tuned steel bars and tuning forks and other related items. Regardless of exactly who was responsible, we need to have a pitch standard if we're going to have more than one instrument playing simultaneously. A band or for that matter a symphony orchestra would not be possible unless everybody agreed on a standard pitch and everybody's As and Cs and F#s were all the same. Anyhow, if we take the note Middle A which by standard convention has a frequency of 440 Hz and we multiply that by the 12th root of two which is 1.05946, we will get a frequency of 466.16376151. That's many more decimal places than we really need and is of academic interest only. For all practical purposes we can round that off to two decimal places and say that the frequency is 466.16 Hz. So, if Middle A is 440, then Middle A# (or Bb) will be 466.16 Hz. Even if we rounded it off to the nearest whole number or 466 Hz, it would still be sufficiently accurate to sound right. Modern electronic tuning devices are usually accurate only to two decimal places, and theoretically perfectly accurate tuning is a physical impossibility anyhow. If in the mid-range of an instrument, the tuning is within ± 0.5 or a half cycle per second, it will be very good indeed.
With the exception of octaves, there are no other exact pitch intervals in the equally tempered scale. Octavely related pitches have a 1:2 frequency ratio. True fifth-related pitches have a 1:1.5 ratio. If we were to play Middle A and the next higher E, for example, that musical interval is known as a fifth. It should have an exact 1:1.5 frequency ratio, so that if A is 440 Hz, then E should be (1.5)(440) = 660 Hz.
If, however, we are working with an equally tempered scale, A will of course be 440Hz, but E will be 659.26 Hz. The difference in pitch between the true E, and the equally tempered scale E in this case is only 0.74 Hz which is actually a very small error and is not really noticeable. But that slight discrepancy still lets that E work well with A, and it also works with any other key that we might wish to sound it with; thus the equally tempered scale lets us play in any key and still sound good. We trade off absolute perfection in any particular key for acceptable results in all keys and we also keep our keyboard instruments manageable, easy to play and affordable.
If we consider the normal musical scale and count the black as well as the white keys, we have 12 different tones or pitches in an octave, such as C, C#, D, D#, E, F, F#, G, G#, A, A#, and B. The next note of course would be the next C which has exactly twice the frequency of the first C. Therefore, the way to divide things evenly so that each successive pitch of the twelve is higher by the same ratio is to take the 12th root of two and use that number as the constant factor by which to multiply the frequency of any particular note or tone to get the next higher one.
Today, we have modern crystal-controlled electronic tuning devices which sound a very accurate series of pitches which we can use as standards or references to tune an instrument. Prior to the development of these devices, however, tuners would develop an equally tempered scale by ear, by sounding notes in pairs and listening for the slight amount of detuning necessary to get the desired result. Although this sounds quite formidable, instrument technicians could learn, in some cases fairly easily, what the various intervals should sound like and once having established an accurate C or an accurate A, could then easily get the remaining eleven notes. Even today many tuners still prefer to work by ear. This is, by the way, especially useful in piano tuning, because the upper harmonics of piano strings are not always exact, whole number integral multiples of the fundamental pitch, and an accurate "by ear" tuning can actually sound better then a tuning based on an electronic standard with perfect 1:2 octaves.
The actual creation of an equally tempered scale is done by sounding notes in progressive fifths and fourths. In the equally tempered scale, all fifths are slightly narrowed so that they are all off from a true 1:1.5 ratio by 0.112%. Likewise the fourths need to be widened so that they are off from a true 3:4 ratio by .114%. A very typical procedure used by many instrument tuners calls for working in the middle octave of a piano, for example and first accurately tuning Middle C as closely as possible to a standard. If A = 440 Hz, then Middle C, to be correctly tunied must have a frequency of 261.63 Hz.
Once the tuner has accurately tuned C, he next sounds the G which is a fifth above. The easiest way to tune this fifth accurately is first to tune it as close to exact as possible, so that it does indeed have a 1:1.5 relationship to C. Then, he proceeds to lower G very slightly. If G is not exactly equal to 1.5 times the pitch of C, or in this case (1.5)(261.63) = 392.45 Hz, then a slow beat will develop between the third harmonic of middle C and the second harmonic of Middle G. With practice, you can learn to hear this beat. As soon as the tuner slightly lowers the pitch of G, he will hear the beat between the aforementioned harmonics. When Middle G is tuned correctly in equal temperament with C, this beat rate will be about 0.9 beats per second. If we take C at 261.63 and multiply it by 1.5, we will get 392.45 Hz. But the tempered scale pitch should be 392 Hz which leaves a difference of 0.45, which we must multiply by two since we're dealing with the second harmonic of G. That's where we get 0.9 beats.
Once he has done that, the tuner then goes down a fourth to Middle D. Here, he first tunes Middle D exactly to the G that he just tuned, and then proceeds to lower D until he once again hears the beat, this time between the fourth harmonic of D and the third harmonic of G. To do this, he first tunes D dead on with G so that it comes in at 294 Hz. This is a precise 3:4 ratio between D and G. Then he lowers the D slightly until it is at 293.67 Hz. This will result in a beat between D's fourth harmonic and G's third harmonic of 1.32 beats per second.
All of the stuff in the above three paragraphs is strictly academic. In the middle octave of a keyboard instrument, it works out that the beat rates are very close to 10 beats in ten seconds for fifths, and 14 beats in ten seconds for fourths. (Actually each fifth and fourth is slightly different but who cares?) Believe it or not, you can secure a very nice-sounding, accurate equally tempered scale by using this ten-fourteen rule in the middle octave. Fifths beat ten beats in ten seconds; fourths beat 14 beats in ten seconds.
On a practical matter, it is not worth the time to tune each fourth and fifth to the exact, calculated beat rates, because a piano or organ tuner has to make a living and cannot afford to spend a whole day setting a theoretically perfect temperament correct to 3 decimal places, when the normal tolerance of the instrument he is tuning is not that close to begin with! With practice, it becomes after a while, very easy to recognize ten beats in ten seconds and fourteen beats in ten seconds, and then a tuner can secure a good temperament in about ten minutes or so. The discrepancy between a theoretically perfect equal temperament and that which you obtain with the ten-fourteen rule amonts to errors around a tenth of a percent, and as I just mentioned, there is no piano or pipe organ which does not over the course of a few minutes drift by more than that amount anyhow, just from normal temperature variations, and in the case of pipe organs, normal slight variations in the air pressure in the instrument and also atmospheric pressure as well as ambient temperature.
If you have access to a piano, pipe organ, or an electronic organ with individually tuneable notes and want to experiment, here's a handy chart of notes to play together and the beat rates between them. Stay entirely in the octave from Middle C to next higher B!
There is much on the Internet about the many different musical temperaments that have been in use for centuries and of their advantages and shortcomings. A temperament is a tuning system by which the pitches or notes of the scale are tuned so that polyphonic music is possible in more than one key. You see, if you tune all the notes of the scale absolutely accurately so that you can play a song perfectly in the key of C, for example, it would not work if you want to play the same song in F. You'd need to create another scale and tune it perfectly for F. Obviously, if you have a keyboard instrument with a separate scale for each key, it would become pretty complicated and unwieldy. So, even as early as many centuries ago, musicians and instrument makers began to experiment with various ways of tuning so that things would sound good in several different keys and yet the instruments would not get too complicated.
The ultimate result of this is our modern equally-tempered musical scale. It takes the twelve tones of a complete musical scale and adds a slight amount of deliberate error to each pitch so that the error is spread out evenly across all twelve notes and no one error is too large. Then you can play a song in any key you choose and it will still sound good.
Modern pianos and organs would not be possible unless some type of temperament for tuning was available. The equally-tempered scale is not the only temperament available, but it's the best for keyboard instruments. Other temperaments were in use earlier which attempted to distribute tuning errors differently, making several keys sound really good and leaving certain intervals in other keys that were way off and were, interestingly enough, referred to as "wolves" probably because the glaring out-of-tune sounds that resulted from playing these so-called wolf intervals suggested the howling of a wolf.
The equally tempered scale takes any particular note's frequency and multiplies it by the twelfth root of two, which then gives the frequency for the next higher note on the scale. In modern music, It is essentially agreed that the pitch of Middle A will be 440 Hz. I understand that this was largely brought about by the J.C. Deagan company in the early 20th century as they were the leading manufacturer in the USA of tuned steel bars and tuning forks and other related items. Regardless of exactly who was responsible, we need to have a pitch standard if we're going to have more than one instrument playing simultaneously. A band or for that matter a symphony orchestra would not be possible unless everybody agreed on a standard pitch and everybody's As and Cs and F#s were all the same. Anyhow, if we take the note Middle A which by standard convention has a frequency of 440 Hz and we multiply that by the 12th root of two which is 1.05946, we will get a frequency of 466.16376151. That's many more decimal places than we really need and is of academic interest only. For all practical purposes we can round that off to two decimal places and say that the frequency is 466.16 Hz. So, if Middle A is 440, then Middle A# (or Bb) will be 466.16 Hz. Even if we rounded it off to the nearest whole number or 466 Hz, it would still be sufficiently accurate to sound right. Modern electronic tuning devices are usually accurate only to two decimal places, and theoretically perfectly accurate tuning is a physical impossibility anyhow. If in the mid-range of an instrument, the tuning is within ± 0.5 or a half cycle per second, it will be very good indeed.
With the exception of octaves, there are no other exact pitch intervals in the equally tempered scale. Octavely related pitches have a 1:2 frequency ratio. True fifth-related pitches have a 1:1.5 ratio. If we were to play Middle A and the next higher E, for example, that musical interval is known as a fifth. It should have an exact 1:1.5 frequency ratio, so that if A is 440 Hz, then E should be (1.5)(440) = 660 Hz.
If, however, we are working with an equally tempered scale, A will of course be 440Hz, but E will be 659.26 Hz. The difference in pitch between the true E, and the equally tempered scale E in this case is only 0.74 Hz which is actually a very small error and is not really noticeable. But that slight discrepancy still lets that E work well with A, and it also works with any other key that we might wish to sound it with; thus the equally tempered scale lets us play in any key and still sound good. We trade off absolute perfection in any particular key for acceptable results in all keys and we also keep our keyboard instruments manageable, easy to play and affordable.
If we consider the normal musical scale and count the black as well as the white keys, we have 12 different tones or pitches in an octave, such as C, C#, D, D#, E, F, F#, G, G#, A, A#, and B. The next note of course would be the next C which has exactly twice the frequency of the first C. Therefore, the way to divide things evenly so that each successive pitch of the twelve is higher by the same ratio is to take the 12th root of two and use that number as the constant factor by which to multiply the frequency of any particular note or tone to get the next higher one.
Today, we have modern crystal-controlled electronic tuning devices which sound a very accurate series of pitches which we can use as standards or references to tune an instrument. Prior to the development of these devices, however, tuners would develop an equally tempered scale by ear, by sounding notes in pairs and listening for the slight amount of detuning necessary to get the desired result. Although this sounds quite formidable, instrument technicians could learn, in some cases fairly easily, what the various intervals should sound like and once having established an accurate C or an accurate A, could then easily get the remaining eleven notes. Even today many tuners still prefer to work by ear. This is, by the way, especially useful in piano tuning, because the upper harmonics of piano strings are not always exact, whole number integral multiples of the fundamental pitch, and an accurate "by ear" tuning can actually sound better then a tuning based on an electronic standard with perfect 1:2 octaves.
The actual creation of an equally tempered scale is done by sounding notes in progressive fifths and fourths. In the equally tempered scale, all fifths are slightly narrowed so that they are all off from a true 1:1.5 ratio by 0.112%. Likewise the fourths need to be widened so that they are off from a true 3:4 ratio by .114%. A very typical procedure used by many instrument tuners calls for working in the middle octave of a piano, for example and first accurately tuning Middle C as closely as possible to a standard. If A = 440 Hz, then Middle C, to be correctly tunied must have a frequency of 261.63 Hz.
Once the tuner has accurately tuned C, he next sounds the G which is a fifth above. The easiest way to tune this fifth accurately is first to tune it as close to exact as possible, so that it does indeed have a 1:1.5 relationship to C. Then, he proceeds to lower G very slightly. If G is not exactly equal to 1.5 times the pitch of C, or in this case (1.5)(261.63) = 392.45 Hz, then a slow beat will develop between the third harmonic of middle C and the second harmonic of Middle G. With practice, you can learn to hear this beat. As soon as the tuner slightly lowers the pitch of G, he will hear the beat between the aforementioned harmonics. When Middle G is tuned correctly in equal temperament with C, this beat rate will be about 0.9 beats per second. If we take C at 261.63 and multiply it by 1.5, we will get 392.45 Hz. But the tempered scale pitch should be 392 Hz which leaves a difference of 0.45, which we must multiply by two since we're dealing with the second harmonic of G. That's where we get 0.9 beats.
Once he has done that, the tuner then goes down a fourth to Middle D. Here, he first tunes Middle D exactly to the G that he just tuned, and then proceeds to lower D until he once again hears the beat, this time between the fourth harmonic of D and the third harmonic of G. To do this, he first tunes D dead on with G so that it comes in at 294 Hz. This is a precise 3:4 ratio between D and G. Then he lowers the D slightly until it is at 293.67 Hz. This will result in a beat between D's fourth harmonic and G's third harmonic of 1.32 beats per second.
All of the stuff in the above three paragraphs is strictly academic. In the middle octave of a keyboard instrument, it works out that the beat rates are very close to 10 beats in ten seconds for fifths, and 14 beats in ten seconds for fourths. (Actually each fifth and fourth is slightly different but who cares?) Believe it or not, you can secure a very nice-sounding, accurate equally tempered scale by using this ten-fourteen rule in the middle octave. Fifths beat ten beats in ten seconds; fourths beat 14 beats in ten seconds.
On a practical matter, it is not worth the time to tune each fourth and fifth to the exact, calculated beat rates, because a piano or organ tuner has to make a living and cannot afford to spend a whole day setting a theoretically perfect temperament correct to 3 decimal places, when the normal tolerance of the instrument he is tuning is not that close to begin with! With practice, it becomes after a while, very easy to recognize ten beats in ten seconds and fourteen beats in ten seconds, and then a tuner can secure a good temperament in about ten minutes or so. The discrepancy between a theoretically perfect equal temperament and that which you obtain with the ten-fourteen rule amonts to errors around a tenth of a percent, and as I just mentioned, there is no piano or pipe organ which does not over the course of a few minutes drift by more than that amount anyhow, just from normal temperature variations, and in the case of pipe organs, normal slight variations in the air pressure in the instrument and also atmospheric pressure as well as ambient temperature.
If you have access to a piano, pipe organ, or an electronic organ with individually tuneable notes and want to experiment, here's a handy chart of notes to play together and the beat rates between them. Stay entirely in the octave from Middle C to next higher B!
notes to hold |
beat rate in ten seconds | remarks |
Md C and G above |
10 |
Make sure Middle C is accurate first! Always tune interval true first, then LOWER, flatten the note you are tuning, in this case, G. |
G and D below |
14 |
flatten D |
D and A above |
10 |
flatten A |
A and E below |
14 |
flatten E |
E and B above |
10 |
flatten B |
B and F# below |
14 |
flatten F# |
F# and C# below |
14 |
2 successive fourths to keep us in the middle octave; flatten C# |
C# and G# |
10 |
flatten G# |
G# and D# below |
14 |
flatten D# |
D# and A# above |
10 |
flatten A# |
A# and F below |
14 |
flatten F |
F to Mid C |
14 |
Don't tune this one, just listen.
This is a check. If the rest of the procedure is done correctly,
this final fourth will be in tune and should be very close to
14 beats on the sharp side with Middle C in ten seconds. If not,
start at step one and repeat.
|
The subject of musical instrument tuning
is a complex one, but we hope that this cursory overview will be helpful
in understanding some of the technical discussions in the articles about
the various electronic organs and other instruments that we either have
or will have on this website.
