Divertimento

Other ways to deal with beating

Just intonation is not the only way to deal with annoying beating in chords. Here I list some that require a simpler keyboard than the just intonation keyboard described on this website. They will be illustrated by renderings of the modulation by Max Reger.

19 Tone Equal Temperament

An octave can be divided into 19 equal steps. A moderately complicated keyboard can be constructed for that, much like the pedal keyboard described on this website. An advantage over 12 tone equal temperament is that the beating rates of minor and major thirds are almost zero (minor thirds) and much lower (major thirds). A disadvantage is that pure fifths beat more.

(19 tone equal temperament)

31 Tone Equal Temperament

An octave can be divided into 31 equal steps. A keyboard can be constructed for that. For a description and pictures, check the internet (terms: 31 tone equal temperament keyboard Fokker). An advantage over 12 tone equal temperament is that the beating rates of minor and major thirds are almost zero (major thirds) and much lower (minor thirds). A disadvantage is that pure fifths beat more, but not very fast. An advantage over just intonation is that the syntonic comma is a non issue in this tuning, because it reduces to a prime with a pitch ratio of 1/1 just like in 12 and 19 tone equal temperament.

(31 tone equal temperament)

Tremolo

Here is a rendering of the fragment in 12 tone equal temperament with a single reed stop, just to illustrate the fast beating:

(12 tone equal temperament, no tremolo)

In the following fragment, for the upper voices an extra reed stop with a tremolo effect has been drawn:

(12 tone equal temperament, with tremolo)

The tremolo effect to a certain extent masks the fast beating. Accordions have similar means to accomplish this effect.

Modified Fifth Partial

As we have seen, an ideal major third has a pitch ratio 5/4, a minor third has 6/5. In both, the factor 5 indicates that the fifth partial sine wave is crucial for causing the beating if the intervals do not have the ideal pitch ratios. So what if we use an ordinary organ keyboard with 12 tone equal temperament, eliminate the 5th, 10th, 15th etc. partial sine waves from all stops, and tune the 1 3/5' pitch stops a bit higher so they do not beat with a 12 tone equal temperament major third?

(12 tone equal temperament, 5th partials removed/retuned)

This is a solution for electronic organs only. I do not see a way to remove selected partials from physical organ pipes.

Hammond Organ

In 1934 Laurens Hammond went even further and chose all partial sine waves to be in the 12 tone equal temperament scale. In his system, the fifth partial, for instance, does not have a pitch exactly 5 times the pitch of the first partial (the base pitch). Instead, it is a factor 5.0396842 = 2^28/12. Thereby the fifth partial coincides with a keyboard tone 28 semitones higher: no beating.

Hammond included the first 10 partial sine waves but skipped the 7th. The fragment below includes all 10:

(12 tone equal temperament, 10 partials retuned to keyboard tones)

As expected, this rendering contains no beating. Actually, no dissonant could produce any beating in this 'tuning'. To restore some of the liveliness of a real organ sound, a Leslie-box is sometimes used, which distorts the sound much like in the following fragment:

(12 tone equal temperament, 10 partials retuned to keyboard tones, Leslie-like)

Hammond Organ 2.0?

A possible reason why Laurens Hammond did not include partial sine waves number 11 and higher, is that many of those would have to be retuned quite a bit to have them coincide with a nearby keyboard tone. For instance, the exact pitch of the 11th partial of C (256 Hz) is 11 x 256 Hz = 2816 Hz, which is about halfway the keyboard tones F 41 semitones up (2733.752 Hz) and F♯ 42 semitones up (2896.309 Hz).

So what if we allow partials to be retuned to a finer grid as in 24 tone equal temperament? To prepare the illustration of that, first here is the Reger fragment in ordinary 12 tone equal temperament with 32 partial sine waves. The volume of partial sine wave number n is 0.7ⁿ. The fragment starts with the consecutive tones of a C major chord:

(12 tone equal temperament, 32 partials)

The same in just intonation:

(just intonation, 32 partials)

Finally, you can compare this with Hammond 2.0, so to speak:

(12 tone equal temperament, 32 partials, each retuned to nearest 24 tone equal temperament step)

My first impression is that this sounds somewhere halfway the ordinary 12 tone equal temperament and the just intonation versions.

Vox Humana

For years, the software has been playing wavetables with a frequency spectrum that was uniform across a five octave organ keyboard. Then I read in [Py 2011] a partial specification of a formant in a Vox Humana stop. A stop has a formant if a certain fixed region of the frequency spectrum - say 2000 to 3000 Hz - stands out, irrespective of which particular pipe is played. That is a classic formant. The human voice also has some of those. They enable one to differentiate between ooh, aah and eeh sounds etc. Then there are traveling formants. In those, the formant's frequency region is not entirely fixed, but moves up slowly with the tone played. In the Vox Humana stop used below, a low C at 64 Hz has a formant pitch of 1344 Hz, which is partial sinewave no 21 (64 Hz x 21 = 1344 Hz); a high C at 1024 Hz has a formant pitch of 3072 Hz, which is partial sinewave no 3 (1024 Hz x 3 = 3072 Hz).

Cesar Franck wrote Choral in E major for organ with a section specifically to be played with a Vox Humana stop and with the use of a tremolo. The piece is heavilly chromatic as you can see in the matrix below. Will Franck survive just intonation? Or rather: will just intonation survive Franck? Here is one battle:

Self Test

Here is a simple test which tells you in two minutes whether just intonation is good for you.

Over a steady A tone, you hear a rising 'scale' whose tones have pitches which are 8 to 16 times that of an A which is three octaves lower (not actually sounding). The 'scale' tones therefore correspond to partial sinewaves of that low A. Notes 8, 9, 10, 12, 15 and 16 are also in the 5-limit just intonation system. Note number 14 is very close to one. But particularly number 11 and 13 are not. They are foreign to the scales used in mainstream Western music, here emphasised by the non standard note head. This is how it sounds:

If you hear nothing special about notes 11 and 13, you may like just intonation right away, but it may not sound as an improvement over 12 tone equal temperament to you.

If you find notes 11 and 13 sound odd, wrong, out of tune, unusual, ugly, irritating, distracting or the like, the self test is as follows: listen to this fragment ten times. If at the end of that, 11 and 13 sound as bad as they did in the beginning, just intonation is probably not for you. On the other hand, if you start to get used to 11 and 13 and maybe you could even imagine that music could be made with them, there is a chance that you will (once) appreciate music in just intonation.

Just to be sure: just intonation on this website does not contain notes as unusual as notes 11 and 13. But notes 10 and 14 are part of just intonation and those are noticeably different from their 12 tone equal temperament representation.

Enharmonic Equivalence

On ordinary keyboards, a single key may represent different notes, such as D♯ and E♭. The notes in such a pair are said to be enharmonically equivalent. Composers have exploited this by writing chords in which a particular note should be for instance D♯ when the chord is related to the previous chord, but at the same time should be E♭ in connection to the subsequent chord. Obviously that is impossible in (tempered) just intonation, where these notes have slightly different pitches. In rendering such music, we shall have to make a choice, here between D♯ and E♭.

Here are the notes of an enharmonic modulation from E minor to B♭ major as given by [Sc 1972], p127.

So let's hear how this sounds with the notes above:

Now we change the D♯ in the chord marked x in the score into E♭:

For comparison, here is the same in 12 tone equal temperament:

(modulation E minor to B♭ major, 12 tone equal temperament)

Loosely related is the following cadence, that has the note G♯ being followed by A♭. This can be faithfully rendered in just intonation, because the notes occur in succession, rather than at the same time.

The melodic interval G♯ - A♭ is burried in the other voices, which is sensible, for if you emphasize it, it sounds as follows (agravated by the reverberation):

(interval G♯ - A♭ emphasized, tempered just intonation)

In just intonation, this G♯ - A♭ interval has a pitch ratio 128/125 (41.05886 cents), a little less than half a semitone.

For comparison, here is the original in 12 tone equal temperament:

(cadence with G♯ followed by A♭, 12 tone equal temperament)

Odd Organ Stop

Many organs have a mixture stop with breaks. A mixture stop has multiple ranks, each of which reinforces a different partial sinewave of the base tone. Usually a mixture pipe reinforces (an octave of) the base tone, or (an octave of) the third partial sinewave. The word 'break' here means that the lower tones of the keyboard compass have quite high pitched partial sinewaves reinforced and when one plays rising scales along the keyboard, there are moments where a mixture rank 'breaks' and for instance the eighth partial sinewave is not reinforced anymore, but instead the rank starts to reinforce the fourth partial sinewave.

Now that we have a software 'organ', we can easily experiment with other mixture stop compositions. Here is a rendering of the first 10 measures of Trio Sonata in C minor, BWV 526 - Largo by J.S. Bach where the first melody in fast notes has an 8' stopped flute plus a mixture stop sounding with six ranks of stopped flutes, which reinforce (octaves of) the 3th, 5th, 7th, 9th and 11th partial sinewaves of the base tone.

(BWV 526 - Largo, 10 measures)

And here is a C major scale across the five octave compass of an organ keyboard with the same stops:

(C major scale)

You can now clearly hear the breaks and also that this stop combination must not be used in the bass part of the keyboard.

The breaks are as follows:

Partial sinewave numbers translated to traditional organ stop pitches expressed in feet:

For a long time I have been avoiding such sound experiments, as I tried to keep the sound as organ-like as I could, so the strange sound would not distract interested people from the essence of this website: just intonation. But when I dropped my hesitation (April 2020), I thought it might sound just fine on a real organ. Or as an electronic addition in a hybrid organ.

Beating and Reverberation

As we saw and heard earlier (Background 1: What is beating?), two sine waves with slightly different pitches beat heavily, in the sense that there are regular moments when they completely cancel one another out. Now when one such sine wave sounds in a hall with heavy reverberation, after a while, when all the wall reflections of the original sound and all reflections of reflections have settled, the combined result can only be ... just another simple sine wave. That is a property of sine wave sounds.

If that is so, how would two beating sine waves sound in a reverberating hall? There is no escaping the conclusion that they would sound like two simple sine waves that beat, without any trace of reverberation. Let's hear it:

(beating sine waves reverberating)

This sounds most convincing with speaker boxes - not headphones. During the first seconds, you can hear the effect of reverberation: the wall reflections are still settling. But after that, the sine waves beat while the hall's reverberation seems to be silent. Until the tones stop and the reverb takes over again.

This refutes the simple notion that reverberation would just smear out all sounds.

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