The EQUALLY TEMPERED SCALE

North Suburban HAMMOND ORGAN Society

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.

The pitch difference from absolute true pitches that this tuning system introduces is the reason why when harmonics are derived from the tempered scale, which is the basis of the Hammond system, the use of tempered scale tones does not always work. All pitches, with the exceptions of octavely-related harmonics such as the 2nd, 4th, 8th, 16th, etc. are not exact whole number multiples of the fundamental frequency for any pitches. The third, 6th and 9th harmonics when derived from the tempered scale are very close to the true harmonics that they represent and are perfectly usable in the harmonic synthesis system. The 5th harmonic is off by an error of 14%. It is still usable in most instances, although if a complex tone with true harmonics that has a strong natural 5th harmonic is sounded together with a synthesized tone with a fifth harmonic taken from the tempered scale, there will be a noticeable clash which becomes worse as you ascend the scale.

When instrument makers use the 7th harmonic out of the tempered scale, the error is around 33% and this is usable only with a strong vibrato, if the synthesized tone is sounded together with a tone which has true harmonics. It does work on the Hammond X66 most of the time because tempered scale harmonics do not clash with each other. However, on the X66, if a drawbar combination which includes the 7th harmonic is sounded together with any of the complex tones (Hammond bright tones) there will be a noticeable clash and an out-of-tune sound. Fortunately, the Hammond vibrato will obliterate this, but this begins to show the reason why tempered scale harmonics are limited. (And this does not really work if the vibrato is off.) The next odd harmonic that is significantly different is the 11th harmonic. This pitch is off by nearly a quarter of a tone; (About 49%). It is too far off to work under any condition which is why you will not find the 11th harmonic on any X66, even though the X66 does give you the 10th and 12th harmonics, which have a much smaller pitch error. In some very elaborate pipe organs, the 11th harmonic is provided. However, on a pipe organ, this is provided by a separate rank of pipes which is correctly tuned to sound the true eleventh harmonic. But it will not work on any compromise harmonic synthesis scheme where harmonic pitches are taken from the same frequencies which are tempered scale fundamental pitches for higher notes of the scale.

Most of the last two paragraphs is mainly of academic interest, but it explains why some harmonic pitches are not found on Hammond organs. Originally, Hammond felt that even the seventh harmonic was too far off, but it can work, as long as it is not sounded simultaneously with a tone that contains natural or true harmonics, and with the vibrato on!

 

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