DR. DOR ABRAHAMSON

 

In his video, Dr. Abrahamson spoke of a few simple ratios and provided a demonstration on his Cello.  The first note struck was “A440” and can be referred to as the “fundamental tone” (or frequency) in his demonstration. 

The note “A440” is often used as a benchmark in music; it has a frequency of 440 cycles per second (440 Hertz) and is 9 “semi-tones” above “Middle C” (C₄) on a piano keyboard. 

Each time Dr. Abrahamson “shortened” the string on his Cello by a factor of ¹/₂ , the frequency of the resulting note was double that of its predecessor. 

The notes struck, their description and the ratio of each note’s frequency to that of A440 are shown below.

Table 1 

NOTE......MUSICAL DESCRIPTION.......FREQUENCY.........RATIO TO “A440”        

A440…..…Fundamental Tone..…...........440(2)⁰……………...440:440 = 1:1

A880……..Second Harmonic ………........440(2)¹………….…..880:440 = 2:1

A1760…....Fourth Harmonic….…….........440(2)²……….…....1760:440 = 4:1

A3520…....Eighth Harmonic……..……..…440(2)³……………..3520:440 = 8:1

 

 The illustration below provides a geometric representation of A440, A880 and A1760 shown above.

The function defining the graph below appears as follows:

h₁(s) = 440 (2)^(s/12)

 

 Graph 1 

 

The frequencies representing each of the notes in the graph above are measured on the vertical axis.  The number of semi-ones from A440 is measured along the horizontal axis.  As shown in the graph, every set of 12 semi-tones advanced horizontally constitutes a doubling in the frequency of each successive note displayed.

The three notes shown in the geometric progression above are once again represented in the illustration below, this time as sinusoidal functions. 

The frequency of each of these notes is measured horizontally here and determines the period of each function.  The vertical scale measures the volume of each note; the volume is assumed to be constant throughout and, hence, the amplitude remains unchanged.

 

Graph 2

 

The defining functions for the three graphs shown above are documented below.

 

A₁ = 4sin(2π*440t)  à black

A₂ = 4sin(2π*880t)  à red

A₃ = 4sin(2π*1760t)  à blue dotted

 

 

Phase Shift

 Had we described the defining equation from the perspective of A880, the defining equation would be “tweaked” slightly as shown below.

h₁(s) = 440 (2)^(s/12)

becomes

h₁(s) = 880 (2)^((s-12)/12)

 

Additionally, if A880 had been the “fundamental tone”, it would be desirable to position that note directly on the vertical axis.  The point representing A880 in the graph above would therefore be “slid” 12 units leftward from its original position.  The defining equation for this new function, having A880 as the fundamental tone, would appear as follows.

h₂(s) = 880 (2)^((s-0)/12)

 

The “s-0” in the numerator of the exponent above reflects the fact that this note, A880, is now occurring 12 semi-tones “sooner” than in the previous case when A440 was the fundamental tone.  This phase shift is responsible for the function h₂(s) having a higher starting position as compared to that of h₁.

The illustration below shows the original case in which A440 was the fundamental tone (dotted) and the phase shift resulting in the new fundamental tone, A880, shown in red.

 

Graph 3

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Harmonic Series

 Since Dr. Abrahamson shortened the Cello string each time by a factor of ¹/₂ with each trial, the frequency of each note was double that of the previous one, explaining why the third, fifth, sixth and seventh harmonics are missing from Table 1. 

Consequently, each note struck was precisely one octave higher than its predecessor. 

 

A harmonic series consists of notes whose frequencies are “integral multiples” of that frequency held by the fundamental tone; these would be multiples of 440 hertz in our example.

In terms of wavelengths, a harmonic series would therefore be produced by creating notes whose wavelengths are the “multiplicative inverses” of the integer multiples referred to above.

 

That is to say, a harmonic series is produced by multiplying the wavelength of a fundamental tone by the rational numbers 1, ¹/₂, ¹/₃, ¹/₄, ¹/₅, ¹/₆, ¹/₇, ¹/₈, etc. to obtain the wavelengths of each successive overtone in the harmonic series.

 

The table below shows the linear progression in the harmonic series whose fundamental tone is A440. 

Table 2

Note      DESCRIPTION                     FREQUENCY       PERIOD

A_{4.............}Fundamental_{.............................}1*440=440_{..............}1* (¹/₄₄₀ )

A_{5.............}Second Harmonic_{....................}2*440=880_{.............}¹/₂*(¹/₄₄₀ )

E_{6.............}Third Harmonic_{.........................}3*440=1320_{.............}¹/₃*(¹/₄₄₀ )

A_{6.............}Fourth Harmonic_{......................}4*440=1760_{.............}¹/₄*(¹/₄₄₀ )

C#_7...........Fifth Harmonic_{..........................}5*440=2200_{.............}¹/₅*(¹/₄₄₀ )

E_{7.............}Sixth Harmonic_{.........................}6*440=2640_{.............}¹/₆*(¹/₄₄₀ )

G_{7.............}Seventh Harmonic_{..................}7*440=3080_{.............}¹/₇*(¹/₄₄₀ )

A_{7..............}Eight Harmonic_{........................}8*440=3520_{.............}¹/₈*(¹/₄₄₀ )

 

 

The illustration below provides a visual representation for the linear progression of the notes shown in Table 2.

 

Graph 4

 

The period shown in Table 2 becomes significant when graphing the sinusoidal functions representing each note and can also be helpful in calculating wavelengths of the frequencies of those notes.

For our purposes here, the relative “periods” of each note will be adhered to rather than determining their respective wavelengths.  The conversion is a very simple one, involving ratios once again.  For example, assuming the speed of sound is 343 m/s, the wavelengths of the frequencies above would be determined as follows:

wavelength = ^{343 m/s}/_{Frequency (s)}

OR

wavelength = 343 m/s * Period (s)

 

Either way results in the following:

 

wavelength (A440) = ^{343 m/s}/₄₄₀   à 0.7795 meters

wavelength (A880) = 343 m/s * ¹/₂*(¹/₄₄₀ ) s  à 0.3800 meters

 

Each of the notes described in Table 2 can easily be represented as a sinusoidal function.

 

The reason I prefer to represent these functions in terms of their respective periods (as opposed to their wavelengths) is two-fold: 

First, the speed of sound varies with altitude, atmospheric conditions, etc. and second, the requirements put forth in our curriculum lean towards identifying periods of cyclic functions as opposed to wavelengths.  

As shown directly above, the shift from period to wavelength is an easy one.

 

 

 

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On a very special day each year, we are privileged to hear examples of the Harmonic Series in the form of a Bugle call.

"The Last Post" and "Reveille" (known as "Rouse" by some) are played each year to honor the fallen at services during Remembrance Day (also known as Poppy Day, Armistice Day, Veterans' Day).

 

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Octave Series

The twelve-tone equal temperment found in the Octave Series was favored over the Harmonic Series by many during the Baroque era, including J. S. Bach.  Not only did it fit nicely into the keyboard design, but the twelve-tone equal temperment also allowed for much more intricacy in the compositions of the time.  

 

 

Referring to the diagram above will be of assistance for the remainder of this post.

 

While the harmonic series is arithmetic in nature, each “octave series” can be described as a geometric progression, comprised of 12 semi-tones.

Since the frequency ratio of corresponding notes in successive octaves is 2:1, it is reasonable that the ratio between consecutive semi-tones must therefore be “2 ^(1/12)” or “(¹²√2)”.

That is, if A440 is chosen as the starting point, then the 12 semi-tones leading to A880 would appear as follows:

 

Table 3

NOTE                     SEMI-TONES ABOVE A440            FREQUENCY (HERTZ)

A440.....................................0.........................................440(2)^⁰/₁₂ = 440.0000

A^#, B^b...................................1.........................................440(2)^¹/₁₂ = 466.1638

B...........................................2........................................440(2)^²/₁₂ = 493.8833

C₅.........................................3.........................................440(2)^³/₁₂ = 523.2511

C^#.........................................4.........................................440(2)^⁴/₁₂ = 554.3652

D...........................................5........................................440(2)^⁵/₁₂ = 587.3295

D^#, E^b....................................6........................................440(2)^⁶/₁₂ = 622.2540

E...........................................7........................................440(2)^⁷/₁₂ = 659.2551

F...........................................8........................................440(2)^⁸/₁₂ = 698.4565

F^#, G^b...................................9.........................................440(2)^⁹/₁₂ = 739.9888

G..........................................10.......................................440(2)^¹⁰/₁₂ = 783.9909

G^#.........................................11......................................440(2)^¹¹/₁₂ = 830.6094

A...........................................12......................................440(2)^¹²/₁₂ = 880.0000

 

 

These notes are represented geometrically in the following illustration.

 

Graph 5

 

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The information that follows provides further evidence of the presence of ratios contained within music.

 

A “harmonic interval” is one in which two notes (a dyad), having different pitches, are played simultaneously; each of these notes has a frequency (measured in hertz). 

Whether or not the resulting sound produced is a pleasing one depends on the ratios of these frequencies.  For example, the ratio of frequencies of the harmonic interval known as a “Perfect Fifth” is 3:2 while a “Perfect Fourth” is the result when the ratio of the two frequencies is 4:3. A ratio of 5:4 produces the harmonic interval known as a “Major Third”, a very pleasing sound and one found in many chords.

 

Incidentally, a perfect fifth above A440 (440 * ³/₂) results in a frequency of “660” which corresponds to the note “E” from the chart above.  A perfect fourth above this note, E660, will send us to A880.

440 * ³/₂ * ⁴/₃

440 * 2

880

It can therefore be said that a note in each successive octave is comprised of a perfect fifth followed by a perfect fourth above a fundamental frequency.

 

Although dyads (two note combinations) are referred to as “chords”, the most common types of chords are comprised of three notes (triads).  These are often characterized by their “root note”, the lowest frequency note of the triad.  For example, a “C Major Triad” is achieved by playing, simultaneously, the notes “C – E – G”. 

The ratios of frequencies of each consecutive pair of notes in this triad very closely approximate that connected with the “Major Third” harmonic interval.  While these ratios are not exactly equal to the Major Third ratio of 5:4 (G: E = 1.26, E: C = 1.9), they are within a range that is indistinguishable to the human ear.

 

 

The defining equations representing the C Major Triad are shown below.

 

C(s) = 4sin(2π*523.25s)  à  red

E(s) = 4sin(2π*659.26s)  à  blue

G(s) = 4sin(2π*783.99s)  à  black dotted

 

 

Graph 6

 

Again, there will be many individuals who scoff at what I’ve done here.  They will say I’m “lecturing” and not allowing students to discover these notions for themselves. 

Let me be perfectly honest here; we should be giving them something to go on. 

If students don’t even know where to start, they will become frustrated and walk away without giving it an honest try. 

There are SO many additional explorations available to students in the scenario above.  I've given them a starting point and some connections to what has previously been learned.  

I will challenge my students to come up with other major triads, for example or try different combinations of two or three note chords from each set of 12 semi-tones contained within each octave.  They can identify which chords produce a pleasing sound, determine the ratio of the frequencies of those notes and then produce wave functions and others that describe their findings.

Additional explorations can be made into how each octave is divided further into 1200 equals parts and how logarithms are used to determine the size of an interval (in cents) from one frequency to another. 

What appears in this post simply provides a starting point from which students can make further connections through exploration and discovery.

 

The End