58.2 Diagram. Beats- Prime Tone, Upward Interval, Downward Interval Helmholtz

"When two simple tones, making a small interval, generate beats, the number of beats in a second is the difference of their vibrational number.

Our calculation and the rule based on it shew that if the amount by which one of the tones is put out of tune remains constant, the number of beats increases according as the interval is expressed in larger numbers. Hence, for 6ths and 3rds the pitch numbers of the tones must be much more nearly in the normal ratio, if we wish to avoid slow beats, than for octaves and unisons. On the other hand, a slight imperfection in the tuning of 3rds brings us much sooner to the limit where the beats become too rapid to be distinctly separable. If we change Unison c'' c'', by flattening one of the tones, into the semitone b' c'', on sounding the notes together there results a clear dissonance of 33 beats",

58.3

"the number which, as before observed, seems to give the maximum of harshness. But to obtain 33 beats from 5th f' c", it is only necessary to alter c" by a quarter of a tone. if it is changed by a semitone, so that f' c" becomes f' b', there results 66 beats, and their clearness is already much injured. To obtain 33 beats the c" must not be changed in the 5th c" g" by more than one-sixth of a tone, in the 4th c" f" by more than one-eighth, in the major 3rd c" e" and major 6th c" a" by more than one-tenth, and in the minor 3rd c" e" flat by more than 1 / 12th. Conversely, if in each of these intervals the pitch number of c" be altered by 33, so that c" becomes b' or d"flat we obtain the following number of beats":

58.4 Diagram. Beats Helmholtz

58.5 Beat Partial of 33 Vibrations Helmholtz

"The prime tones of the notes in this example are all partials of C, which makes 33 vibrations in a second, hence their own pitch numbers and those of their upper partials are multiples of 33; consequently the difference of these pitch numbers, which gives the number of beats, must always be 33, 66, or some higher multiple of 33":

58.6 Diagram. Beat Partials of 33 Vibrations Helmholtz

"The following general view of the partials of the first 16 harmonics of C 66 (with the exception of the 11th and 13th...), will shew generally how they affect each other in any combination. The number of vibrations of each partial of each harmonic is given, whence the beats can be immediately found":

58.7 Diagram. Finding Beats Due to Combinational Tones Helmholtz

"Beats of the first two combinational tones c' + e' and e' - g' in tempered intonation; 5 2/3 in the second. These are plainly audible in all qualities of tones, if the tones themselves are not too weak.

Beats of the major Third c' + e' alone, 10 1/2 in the second, which, however, are not plainly audible unless the qualities of tone employed have high upper partials.

All these beats occur twice as fast when the chord lies an Octave higher, and half as fast when it occurs an Octave lower.

For the same reason the beats of the third and forth kinds, arising from the Thirds, which are clearly audible on harsher qualities of tone..., are decidedly disturbing in the middle positions, even in quick time"...

Sensations of Tone.

58.8 Diagram. Dissonant Intervals Helmholtz

58.8.1

58.9 3rd and 6th Beats Helmholtz

58.10 Diagram. Tones of the Major Scale Helmholtz

"The above schemes shew that in the justly intoned major and minor scales, three kinds of Thirds occur, and their inversions give three kinds of Sixths. These are:

- The justly-intoned major Third 5/4, [12, cents 386, roughness 8], and its inversion
the minor Sixth 8/5, [28, cents 814, roughness 20], both consonant.

- The justly-intoned minor Third 6/3, [11, cents, 316, roughness 20], and its inversion the major Sixth 5/3, [29, cents, 884, roughness 3], also both consonant.

58.11

- The Pythagorean minor Third, 22/27, [9 cents, 294, roughness 26], between the extreme tones of the key, d and f. If we use d sub1 in place of d, this interval would occur between b sub 1 and d sub 1:

58.12 Diagram. Dissonance Roughness Table Helmholtz

58.13 Diagram. Beats and Dissonance Helmholtz

58.14.1

* The interval is found as the ratio of the pitch number to the same increased by the number in the next column to it; thus for C it is 64 : 64 + 16 = 4 : 5, and for g' it is 384 : 384 + 60 = 96 :111"...

Sensations of Tone.

58.15 Comments on Prof. Tyndall

..."Prof. Tyndall undertakes to determine the consonances of the octave, fifth, fifth, and major third for two simple tones, without employing combination-tones. He writes as follows- "bearing in mind the beats and the dissonance vanish when the difference of the two rates of vibration is 0; that the dissonance is at its maximum when the beats number 33 per second; that it lessens gradually afterwards and entirely disappears when the beats amount to 132 per second- we will analyze the sound of our forks, beginning with the octave. Here our rates of vibration are:

512 - 256; difference = 256. It is plain that in this case we can have no beats, the difference being too high to admit of them.

"Let us now take the fifth. Here the rates of vibration are- 384 - 256; difference = 128.

This difference is barely under the number 132, at which the beats vanish; consequently the roughness must be very slight indeed.

"Taking the fourths, the numbers are- 384 - 288; difference = 96.

Here we are clearly within the limit where the beats vanish, the consequent roughness being quite sensible.

"Taking the major-third, the numbers are- 320 - 256; difference = 64.

Here we are still further within the limit, and accordingly the roughness is more perceptible. Thus we see the deportment of our four tuning-forks is entirely in accordance with the explanations which assigns the dissonance to beats.

It will not be difficult to test the value of the above reasoning. Starting from the rate of 256 vibrations selected by Prof. Tyndall, all that can be deduced from his definition of beats and dissonance at the head of the extract is that the maximum of dissonance will fall on the interval 256 : 256 + 33. i.e. 256 : 289; and all intervals larger than 256 : 256 +132, i.e. 256 : 388, will be free from dissonance. These numbers indicate almost exactly a whole-tone and a fifth respectively. Each of these results is contrary to experience: the dissonance of a whole-tone is less harsh than that of a half- tone; and intervals greater than a fifth are by no means equally free from dissonance. Moreover, it follows that the determination of the octave by this reasoning is delusive, for the process would bring out, as perfect consonances, a seventh or a flat ninth, which are extreme discords, just as readily as an octave. If we apply the same method to other parts of the scale than that to which Prof. Tyndall has restricted himself, the results are even more remarkable. Thus, starting from the higher octave of 256, viz. 512, the maximum roughness falls on 545, a half-tone, and dissonance ceases after 644, which lies between a major third and a forth. For the next octave, i.e. starting from 1,024, dissonance ceases before we reach the interval of a whole-tone. If we take the lower positions on the scale we obtain opposite results. With 128 as our fundamental note, the maximum dissonances falls on 161, slightly above a major third, while roughness extends to 260, just beyond and octave.

With 64 the worst dissonance is 97, just above the fifth, and roughness reaches 196, another octave higher. Starting from 32 the worst dissonance 65 is just above the octave, and roughness is not got rid of until 164, two octaves and a major third from the fundamental note....

...These conclusions are so utterly at variance with facts, that the method by which they have been obtained must be pronounced erroneous. In fact, Prof. Tyndall is himself a witness that this is so; for in speaking of the octave, he remarks that if this interval be slightly impure, beats of the fundamental tone are heard. Now this does not square with his own theory; for suppose two simple tones 513 and 256 vibrations per second, which would form an impure octave: the difference is 257, which is as much "too high" as 256 was in the case of the pure octave. This interval, should, therefore, give no beats, and an impure octave be as harmonious as a pure one. But according to Helmholtz's view, the first combination-tone of 513 and 256 viz. 257, will produce one beat per second with the fundamental tone, as stated, but not satisfactorily explained by Prof. Tyndall. This amounts to a practical admission by him that the beats of two simple primaries are not adequate for the determinations of their consonances, and that recourse must be had for this purpose to the combination-tones".

I (the author of this 1870 text) claim to have shown that the method by which Prof. Tyndall appears to determine the consonances of simple tones is erroneous, and the determinations themselves fallacious. I proceed to point out the defect which vitilates his reasoning. He enunciates but one condition for the production of audible beats, that their number should not exceed 132 per second. Helmholtz lays down a second, quite as important- that the tones producing them should not differ too much in pitch. "These beats", he writes, "are powerful when their interval amounts to a half-tone or a whole-tone, but weak and audible only in the lower portions of the scale, when it is equal to a third, and they diminish in distinctness as the interval increases. Here we see at once why it is futile to determine consonances of a fifth or an octave by the beats of two simple primaries- viz. that for these wide intervals the beats are imperceptible.

Let us now proceed to Prof. Tyndall's theories of consonance for composite sounds. Taking the octave C' C'', its two fundamental tones and their over-tones answer respectively to the following rates of vibration-

. |
. |
1 |
: |
2 |
. |

Fundamental |
. |
264 |
: |
528 |
Fundamental |

Over-Tones |
1 |
528 |
: |
1056 |
. |

. |
2 |
793 |
: |
1584 |
. |

. |
3 |
1056 |
: |
2112 |
. |

. |
4 |
1320 |
: |
2640 |
. |

. |
5 |
1584 |
: |
3168 |
. |

. |
6 |
1845 |
: |
3696 |
. |

. |
7 |
2112 |
: |
4224 |
. |

. |
8 |
2376 |
: |
4752 |
. |

. |
9 |
2660 |
: |
5280 |
. |

Here again, the neglect of the second condition for the production of audible beats
is at the root of the error"....

Nature

March 3, 1870

58.16 Beats and Damping

"Sympathetic bodies which do not damp readily, such as tuning forks, consequently require two exciting tones which differ extraordinarily in pitch, in order to show visible beats, and the beats therefore must be very slow. For bodies readily damped, as membranes, strings, &c., the difference of the exciting tones may be greater, and consequently the beats may succeed each other more rapidly"...

58.17 Beat Changes with Increased Speed

"The question is: What becomes of the beats when they grow faster and faster?.....Most acousticians were probably inclined to agree with the hypothesis of Thomas Young, that when the beats became very quick, they gradually passed over into a combinational tone (the first differential)"...

Sensations of Tone.

58.18 Beats and Tone Interval

..."we may also adduce the testimony of our senses, which teaches us that a much greater number of beats than 30 can be distinctly heard. To obtain this result we must pass from the slower to the more rapid beats, taking care that the tones chosen for beating are not too far from each other in the scale, because audible beats are not produced unless the tones are so near to each other in the scale that they can both make the same elastic appendages of the nerves vibrate sympathetically. The number of beats, however, can be increased without increasing the interval of the tones, if both tones are taken in the higher octaves"...

Sensations of Tone.

58.19 Beats / Ordinal Numbers / Loudness

"Now, for the tones most used in music, partials with a low ordinal number are loudest, because the intensity of partial tones usually diminishes as their ordinal number increases"...

Sensations of Tone.

58.20 Helmholtz Questions about Beats

Objective Beats, Subjective Beats and Differential Tones; Helmholtz, pg. 531,

Subjectivity of Summational Tones; Helmholtz pg. 532,

Can Beats generate Tones? First Beat of Intermittence; Helmholtz pg. 533,

Can Beats generate Tones? Secondly, Beats of Interference; Helmholtz pg. 533,

Would a Tone generated by Beats be louder than the Primaries?; Helmholtz pg. 534,

Beats and Beat-Notes heard Together; Helmholtz pg. 535,

Beat-Notes and Beat Tones; Helmholtz pg. 535,

Beats and Combinational Tones; Helmholtz pg. 532,

Beats according to Koenig and Bosanquet; Helmholtz pg 529,

Upper and Lower Beats and Beat-Notes; Helmholtz pg 529,

Koenig's Explanation of Summational Tones; Helmholtz pg. 536,

Koenig's Theory for the Origin of Beat-Notes; Helmholtz pg. 536.

Sensations of Tone

© Copyright. Robert Grace. 1999