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Monday, April 29, 2019

Pipe Organ Mutations and Arithmetic

Introduction

This post shows how to use arithmetic to predict what pitch 2 2/3', 1 3/5', and 1 1/3' stops will sound relative to the fundamental 8' pitch. (It also works for 5 1/3' and other mutations.) If you hate arithmetic, already understand mutations, or are bored with organs, this is probably tl;dr (too long, don't read).


Two days ago I attended an American Guild of Organists (AGO) "Super Saturday" of lectures and music. Over the years since I started playing the organ, the pitches at which pipes sound has become second nature to me:
  • 8' (pronounced "eight foot") stops sound at the same pitch as one would hear on (say) a piano. (They are named that way because an open flue pipe played by the lowest key on an organ keyboard measures 8 ft. from the top of the opening near the bottom of the pipe to the opening at the top of the pipe.)
  • 4' stops sound an octave higher than 8' stops.
  • 2' stops sound an octave higher than 4' stops.
  • 1' stops sound an octave higher than 2' stops.
  • 16' stops sound an octave lower than 8' stops.
The 2 2/3', 1 3/5', and 1 1/3' stops—called mutations—are the subject of this blog post.
A few of the stop tabs on my home practice organ. The 16' 8', 4', 2', 1' on the tabs indicate the pitch at which they sound. Note that this particular organ also has a 2 2/3' and a 1 3/5' stop on the Swell. (The Great has a 1 1/3' stop—not shown.)


(Organists can skip this part—scroll down to "Mapping mutations to harmonics")

Playing a single note on a musical instrument produces—in addition to the main (8') frequency—additional frequencies called harmonics. The relative strength of the various harmonics is part of what makes (for example) a flute sound like a flute, an oboe sound like an oboe, or a cello sound like a cello.
The notes of the first 8 harmonics of the C below middle C
Note that the 1st harmonic corresponds to 8' pitch, the 2nd harmonic to 4' pitch, the 4th harmonic to 2' pitch, and the 8th harmonic to 1' pitch.

By selecting combinations of stops at different pitches, organists can create a variety of sounds on their instruments.

Mapping mutations to harmonics (The main point of this post)

Mutations are stops that don't sound exact octaves above or below the first harmonic.

Here is how to map 2 2/3', 1 3/5', and 1 1/3' stops to the harmonic series:


2 2/3'

Convert the whole number to a fraction

2 2/3 = 6/3 + 2/3 = 8/3

The denominator identifies the harmonic.

2 2/3' stops sound at the 3rd harmonic above the 8' (indicated by the numerator) pitch. 

If we select an 8' stop and a 2 2/3' stop, then depress the key one octave below middle C, we'll hear the C below middle C and the G above middle C (the 3rd harmonic), sounding at the same time.



1 3/5'


1 3/5 = 5/5 + 3/5 = 8/5

The 5 in the denominator indicates the 5th harmonic above the 8' (indicated by the numerator) pitch.

If we select an 8' stop and a 1 3/5' stop, then press the C below middle C, we'11 hear the C below middle C and the E that is 2 octaves and a major 3rd higher. (the 5th harmonic)



1 1/3'


This one is slightly trickier.

1 1/3 = 3/3 + 1/3 = 4/3

The 3 in the denominator indicates the 3rd harmonic above the 4' (indicated by the numerator) pitch.

Convert the numerator to an 8 be multiplying 4/3 by 2/2:

4/3 = 4/3 x 2/2 = 8/6

1 1/3' stop sounds at the 6th harmonic above the 8' pitch.

If we select an 8' stop and a 1 1/3' stop, and press the C below middle C, we'll hear the C below middle C and the G that is 2 octaves and a perfect 5th higher (the 8th harmonic).


Postscript

Non-organists might wonder about the stops in the picture above that have Roman numerals (the IV and the II above). These are outside the scope of this post, but I'll briefly mention that those stop tabs control multiple ranks (sets) of pipes.

The Viola Celeste II 8' controls two ranks (sets) of pipes (or sound generators) that are slightly out of tune with each other. (Other organs often name the celestes differently, but they involve two sets of pipe slightly out of tune, producing a lush sound.)

The Mixture IV controls four sets of pipes that add brilliance to a principal chorus. The harmonics are generally octaves and 5ths. To avoid excessive shrillness, the highest notes of the mixture generally "break back"—go down in pitch in a cascading manner—as the keys ascend to the highest notes of the keyboard.

Mutations and mixtures (excluding the two-rank celestes mentioned earlier) are used in conjunction with stop(s) sounding at the 1st (and higher) harmonics. Multi-rank mixtures supplement a principal chorus (e.g. at 8', 4', 2') and reeds to add bite to the organ sound.