Newsletter 28: Friday 10 July 2026

Arithmetic Musings #15

Even Numbers and time - in particular musical harmonic frequencies. 

Here are some pebble patterns using Even numbers on the left and Odd numbers on the right, where a new one is introduced into the even pattern.

In the next picture, even-numbered lines of pebbles are formed into areas, building into larger areas.

The first diagram there is a rectangle 1 high by 2 wide;  ratio 1:2;
next diagram the diagram has grown to 2 high by 3 wide;  ratio 2:3;
next diagram the diagram has grown to 3 high by 4 wide;  ratio 3:4;
next diagram the diagram has grown to 4 high by 5 wide;  ratio 4:5;
next diagram the diagram has grown to 5 high by 6 wide;  ratio 5:6.

 

 Pythagoras was interested in musical harmony and the ratios between lengths of vibrating strings.

When a solid, but flexible object, such as a taut string, is struck, shaken, plucked, bowed, etc., it is deformed slightly; this deflection pulls the neighbouring part with it, so that the deflection passes along like a wave. Then there is a correcting influence that pulls the stretched wire back to its original place, but it goes too far in the opposite direction.

This picture shows the alternating action as a travelling wave: 

If many things are vibrating at the same time, each making a sound, then the result may be an unpleasant cacophony or a pleasing harmony which we have come to associate with music. 

The energy of the blow or the bow is transformed into many different, simultaneous vibrations.  The vibrating surfaces cause the adjacent air to move in sympathy, at the same frequency.  This vibration is transmitted to our ears where we perceive it as a sound.  It is only a sound when we hear it.

Musicians are very selective about what and how they set things vibrating.

Stringed Musical instruments, such as a violin, cello, guitar, lute, harpsichord and piano have taut strings or wires that are flexible and of even heaviness. They are “tuned” to vibrate in a particular, predominant frequency, stimulated by the bowing, plucking or striking.  The horsehair violin and cello bow drags the string aside slightly, then releases it, initiating a lateral vibration.

One end of the vibrating string is attached to the hollow body of the instrument, which is caused to vibrate, thus stimulating the surrounding air which radiates to the air in our ears next to our eardrums, which mechanically transmit tiny vibrations to the liquid in the spiral cochlea (organised by our sentient body) where we perceive a tone, a musical sensation experienced by our sentient soul !!!

In a similar fashion the recorder, pipe, trombone, trumpet, oboe, bagpipes, etc. set enclosed lengths of air inside them into longitudinal vibration using the outbreath of the player. (scrutinise the back and forth movement of the red dot.)

The pitch of the tone created by a string is determined by 3 factors.

Factor	Low-pitched tones	High-pitched tones	Method
Linear weight of string/wire	Heavier weight	Lighter weight	Design
Tension in string/wire	Slacker	Tauter	Tuning
Length from bridge to fret	Longer	Shorter	Playing

 The length of a vibrating string can be altered by the player pressing the string on to a fret or the fingerboard.

A taut string/wire vibrates only in an exact, whole number of parts. When the part is large/long, then the tone is low. 

When it is just a single part, it is called the fundamental tone,  tonic or first harmonic.

When the string vibrates in two parts, the tone is called the second harmonic – it vibrates twice as fast as the tonic, that is at double the frequency.
similarly in 3 parts, at triple the frequency.

 

 

         

                   C        C        G       C        E         G      Bb      C        D      E          F          G       A       Bb      B       C

 

• Resonance:  When a guitar string or trumpet pipe vibrates, it resonates strongly at its primary pitch (the fundamental frequency) while simultaneously vibrating in fractional lengths (1/2, 1/3, 1/4) with less intensity. 

• Instrument Uniqueness, Timbre: These smaller, hidden frequencies are called overtones, or harmonics.  The specific blend and volume of these overtones determine the unique tone quality—or timbre—of every instrument.
• Watch this YouTube demonstration using a C-Horn it enables you to hear the sequence clearly
  

    https://www.youtube.com/watch?v=Qzvp4_igvyw&t=13s

• Basis of Harmony: The first few overtones of a fundamental note outline the major triad (the fundamental [1^st], 3^rd, and 5^th intervals) naturally.  These natural intervals form the basis for most Western musical harmony. 
  When several tones are played at the same time on different strings they form a “chord”;
  When tones are played in a sequence in time, they form a ‘melody’ – the jump from one tone to another is called an “Interval”.
• The Fundamental, 2^nd, 4^th, 8^th and 16^th harmonics form a very harmonious series: of Diapason (or Octave in modern terminology) interval series with the ratio 1:2. The Octave is the eighth note in a musical major scale.
• The interval between the 2^nd and 3^rd harmonics is the Diapente (or major Fifth) interval with the ratio 2:3.    
• The interval between the 3^rd and 4^th harmonics is the major Fourth  interval with the ratio 3:4.
• The interval between the 4^th and 5^th harmonics is the major Third interval with the ratio 4:5.
• The interval between the 5^th and 6^th harmonics is the minor Third interval with the ratio 5:6.

 

The Nodes are the parts of the string that twist a small angle, but do not move sideways or laterally.
The Antinodes are the parts of the string that move sideways but do not twist.
The form that they make is called a sine wave.
This scale was only replaced in 19^th Century by the 12 semi-tone equal-tempered Octave (based on the 12^th root of 2).  More about this when we consider the power series.

 More next week.  RB