Expansion of harmonic series.

[ryank@12]

I was just curious;
Is there any organ that reinforces its mutations with mutations derived from the harmonic series of the mutation? For instance:
5-1/3' unison (but labeled quint)
2-2/3' octave
1-7/9' quint
1-1/3' super-octave
1-1/15' tierce
and
3-1/5' unison (but labeled tierce)
1-3/5 octave
1-1/15' quint
4/5' super-octave
16/25' tierce

Has this ever been done? How exactly would it sound? I would assume that it sounds like a complete mess, but it makes sense in theory.

[Havoc]

I doubt it is done and probably not as you put it. You can find in some organs 5-1/3, 3-1/5, 2-2/3 and 1-3/5 on the same manual (french classical ones) but they have a more or less defined use, they are not there to build a plenum from them. Sometimes you can even find a 10-2/3 but that has an even more special use (to make a compound 32). And the 1-1/3 is also common.

The only place I have seen other mutations in in descriptions of German home organs. There you find things like 2/5, 1-3/5 + 1-1/7 or 1/3. But it looks as these are more used to get a reed sound using labials which are easier for a home organ.

[Clarion]

Has this ever been done? How exactly would it sound? I would assume that it sounds like a complete mess . . .

Bagpipes!!

[regeron]

It is an interesting coincidence that the fifths in the harmonic series are represented by fractions over 3, and the thirds in the harmonic series are represented by fractions over 5.

If you take the fifths, and re-write the fractions, 5 1/3= 16/3; an octave higher is 2 2/3= 8/3; 1 1/3=4/3; 2/3= itself; 1/3= itself
If you take the thirds, and do the same, 3 1/5= 16/5; 1 3/5=8/5; and there are options of 4/5, 2/5 and 1/5, though I would hate to be responsible for tuning them. You will see that the numbers move along just like unison-octaves: 16', 8', 4', 2', 1'

The purpose of the mutations is to reinforce harmonics that are already present in an unison or octave-sounding stop. The idea of have mutations for the mutations is interesting but would be of very limited use.

When Havoc mentions their use on German home organs, 2/5, 1 3/5 and 1/3 would not be that unusual in that they are terz [third] and quint [fifth] mutations. The 1 1/7 would not be that common. Though I have seen 7th, 9th, and 13th stops; they tend to be used for solo registrations.

If you think about it, they're all based on simple fractions. And if you start with an 8' stop....
1/1 x 8' = 8' stop
1/2 x 8' = 4' stop
1/3 x 8' = 8/3' [2 2/3'] stop
1/4 x 8' = 2' stop
1/5 x 8' = 8/5' [1 3/5'] stop
1/6 x 8' = 8/6' [1 2/6 or 1 1/3'] stop
1/7 x 8' = 7/8' stop
1/8 x 8' = 1' stop
1/9 x 8' = 8/9' stop

The Pythagoreans loved the beauty of the simple fractions, and the organ's stops help to demonstrate that. You could take this another step further and replace all the 1's with 2's, in the fractions, but you would simply get the same relationships, but based on the 16' series.

The theory behind a 10 2/3' stop is that a normal 32' stop would include harmonics at the 16' and 10 2/3' levels. The assumption is that, in a backwards kind of way, if you are hearing the 16' and the 10 2/3' stops, there must be a 32' underneath, creating them. A bit like... "If there's smoke, there must be fire."

[ungern2]

I think the Atlantic city convention hall organ has a bunch of higher order harmonic mutations.

Ungern

[davidecasteel]

The theory behind a 10 2/3' stop is that a normal 32' stop would include harmonics at the 16' and 10 2/3' levels. The assumption is that, in a backwards kind of way, if you are hearing the 16' and the 10 2/3' stops, there must be a 32' underneath, creating them. A bit like... "If there's smoke, there must be fire."

That is true, but it is also true that the difference frequency between a 16' pitch and the corresponding 10 2/3' pitch is at 32' pitch. (Another way of putting it is that the "beat tone" would occur at the 32' frequency.)

Your conjecture also finds expression in the sound made by tubular chimes: they have a very odd set of partials, but those include frequencies that are the 4th, 5th, and 6th harmonics of a frequency f, which is not the root tone of the chime; however, the ear "hears" the frequency f even though it is not sounding.

David

[VaPipeorgantuner]

the 32-foot pitched "resultant" works, because of the (additive) frequency differential between the prime (unison) pitch and the quint (fifth), that when played together "result" in a perceived pitch an octave below the unison pitch. (by the way, you can hear this phenomenon in mixtures when you are tuning them). The harmonic series of the 32' pitch does include the the 10 2/3 quint pitch...as ell as the 8' 6 3/5' 5 1/3' 4 1/7' and 4' and 3 5/6' etc. you can construct a "grand cornet" resultant using the pitches at 16 foot CCC, 8 foot CC, EE, BB-flat and D for low C on the pedals...it is particularly effective with a bright-toned 16 pedal reed stop.

[michaelhoddy]

the 32-foot pitched "resultant" works, because of the (additive) frequency differential between the prime (unison) pitch and the quint (fifth), that when played together "result" in a perceived pitch an octave below the unison pitch.

Wow, I just always thought that the resultant worked because the 16' and 10-2/3' pitches were the second and third harmonics, thus causing the ear to infer the presence of the prime. I'd never heard anything about frequency differential. Or is this just another way of saying the same thing?

[davidecasteel]

They definitely are not the same thing. The difference frequency is real and can be measured with the proper instruments. The assumption that a 32' exists because the 2nd and 3rd harmonics of it are heard is purely psychological and cannot be measured. Both, however, do have their impact on the listener.

David

[j reimer]

The difference frequency is real and can be measured with the proper instruments.
David

I agree totally with David's main point, for in this matter we are talking about resultant frequencies perceived because of the nonlinearity of our hearing. However, I would be very interested to know what instruments could possibly be used to quantify such perceptions.

John Reimer