In this mathematical exploration, the students attempts to study the mathematical relationship of notes within chords inn harmony and discordant chords. Furthermore the musical chords are representated with the help of multiple sinosodial curves, so that they can be broken down and their mathematical relation can be understood.

Table of content

*(► Examiner's comments)*

**Modeling musical chords using sine waves**

**Modeling musical chords using sine waves**

**Investigation**

I started by working out how to model a single note. I decided to use “Middle” A *(►A. underdefined term and not clear to non-musicians)*, as it has an exact frequency of 440 Hz^{2}, and is the note that Western orchestras tune to. I wanted to make the sound wave of this note equal to the sine wave, which meant making 220Hz equivalent toππ *(► typing error) *. This produces the following graph:

__1. The Major Chord__

Using an A major chord, I can look at the relationship between the notes in a major chord. The following graph demonstrates how this chord travels through the air:

*(►A. Lack of clarity for non-musicians)*

Equation 1: y=sinx

Equation 2: y=sin(554.37/440x)

Equation 3: y=sin(659.26/440x)

Equation 4: y=sin2x

*(►A. No mathematical explaination on how these are determined)*

*(►B. Erroneous notation, it should be 554.37x/440)*

I altered the wavelengths, or frequencies, of the graphs by changing the coefficient of x. In order to get the most accurate graph, I used the original frequencies of the other notes^{3}. The table below shows more information about the notes in this chord.

Note | Semitones away from A |
Frequency (Hz) | Approximate frequency ratio (Hz/440) |

A | 0 | 440.00 | 1 |

C# | 4 | 554.37 | 1.25 |

E | 7 | 659.26 | 1.50 |

A | 12 | 880.00 | 2 |

*(►B. No explaination of how approx frequency ratio values are determined. ►B. Graphs not related to tables)*

The frequency ratios in this chord appear to be neatly spaced, with the third gap in the chord equalling the sum of the first 2 gaps in the chord. The ratio of the gaps is 1:1:2. This may indicate the start of a pattern. Similarly, it could be that the third gap is double the first or second gap. I will now look at different chords to see if this conjecture holds true. The gaps in the ratios do not directly relate to the difference in semitones, as the semitones do not have equal frequencies-the frequencies of semitones diverge as the frequencies increase, and the notes get higher in pitch.

*(►C. Student created own way of looking for patterns in the chords)*

__2. The Minor Chord__

Minor chords are also considered to be in harmony. Again using an A chord, I have modeled a minor chord and looked at its features. These waves look very similar to the major chord waves, as only one note has changed; the C# has gone down by a semitone.

Therefore the graphs are very similar. The graph for the minor chord looks like this:

*(►E. Using trignometric graphs but nothing is done with them. Limited understanding of notes played together)*

Equation 1: y=sinx

Equation 2: y=sin(523.27/440x)

Equation 3: y=sin(659.26/440x)

Equation 4: y=sin2x

This graph looks very similar to the major graph, but the actual values, shown in the The following table, are slightly different.

Note | Semitones away from A |
Frequency (Hz) | Approximate frequency ratio (Hz/440) |

A | 0 | 440.00 | 1 |

C | 3 | 523.25 | 1.20 |

E | 7 | 659.26 | 1.50 |

A | 12 | 880.00 | 2 |

Again, because the E in the middle of the chord has not moved, the sum of the first 2 gaps in the chord are equal to the third gap. The ratio of the gaps is now 2:3:5. However, to test this theory properly, we need to look at chords that differ from the original major chord more dramatically.

__3. The First inversion major chord__

An inversion chord is the same as a major chord, but the notes are in a slightly different order. A first inversion chord starting on A will actually be in the key of F major. The graph of such a chord looks like this:

Equation 1: y=sinx

Equation 2: y=sin(523.27/440x)

Equation 3: y=sin(698.46/440x) *(►Incorrect value(698.46) compared with the table)*

Equation 4: y=sin2x

Note | Semitones away from A |
Frequency (Hz) | Approximate frequency ratio (Hz/440) |

A | 0 | 440.00 | 1 |

C | 3 | 523.25 | 1.20 |

F | 8 | 659.26 | 1.60 |

A | 12 | 880.00 | 2 |

The table of values for this chord shows that my original conjecture does not hold true for all chords. However, the gaps are still in a neat ratio of 1:2:2.

__4. The Second inversion major chord__

This chord, like the first inversion, is another different rearrangement of the notes in a major chord.

Equation 1: y=sinx

Equation 2: y=sin(587.33/440x)

Equation 3: y=sin(739.99/440x)

Equation 4: y=sin2x

*(►A. Repetition of the same work)*

Note | Semitones away from A |
Frequency (Hz) | Approximate frequency ratio (Hz/440) |

A | 0 | 440.00 | 1 |

C | 5 | 587.33 | 1.33 |

F# | 9 | 739.99 | 1.67 |

A | 12 | 880.00 | 2 |

Interestingly, in this chord, the frequencies of the notes are evenly spaced, as the ratio of the gaps is 1:1:1.

I will now look at some discordant notes and see how they compare with the chords I have already looked at.

__5. The Augmented 4__^{th}

This chord is also known as “The Devil’s Chord”, as it is considered to be the most the unpleasant sounding chord in music. As it is only 2 notes, it cannot be modeled in exactly the same way as the other chords, but can still be graphed:

Equation 1: y=sinx

Equation 2: y=sin(622.25/440x)

Note | Semitones away from A |
Frequency (Hz) | Approximate frequency ratio (Hz/440) |

A | 0 | 440.00 | 1 |

D# | 6 | 622.25 | 1.41 |

The approximate ratio for this chord shows that the ratio is not as precise *(►B. poor use of terminology) *as with the other chords. However, there is not necessarily a difference in chords that are in harmony and discordant chords. This chord is not directly comparable to the other chords, as it does not have 3 notes like the others.

*(►D. Some reflection)*

__6. The Augmented 6__^{th}

I will now look at the augmented 6^{th}, another discordant chord. This chord, however, has 3 notes, so it can be more easily compared to the other chords I have looked at. An Augmented 6^{th} is also known as a dominant 7^{th}, which was one of the first discordant chords to be used in music in the West.

Equation 1: y=sinx

Equation 2: y=sin(523.25/440x)

Equation 3: y=sin(739.99/440x)

Equation 4: y=sin2x

Note | Semitones away from A |
Frequency (Hz) | Approximate frequency ratio (Hz/440) |

A | 0 | 440.00 | 1 |

C | 3 | 523.25 | 1.2 |

F# | 9 | 739.99 | 1.67 |

A | 12 | 880.00 | 2 |

In this chord, the gaps in the ratios are not easily relatable to each other, suggesting that discordant sets of notes are not as natural as those that sound harmonic.

*(►D. Reflection on results)*

__“Perfect Intervals”__

The intervals are known as a “perfect fourth” and “perfect fifth” are, as the name suggests, considered to be the most harmonic intervals. Interestingly, the graphs of these intervals show that the notes in the perfect fourth cross at exactly 3π, and the notes in the perfect fifth cross at 2π.

The graph of a perfect fourth:

Equation 1: y=sinx

Equation 2: y=sin(587.33/440x)

The graph of a perfect fifth:

Equation 1: y=sinx

Equation 2: y=sin(659.26/440x)

*(►A,D. Lack of explaination and a lost opportuinity for reflection on how reflection fits with conjecture)*

**Conclusion**

This evidence would suggest that notes which are traditionally considered to be “in harmony” are mathematically “neater” than those that are not in harmony. The mathematical nature of the notes in traditional harmony suggests that the sound waves are more pleasing to our brain because of the way the notes fit together. The ratios of frequencies of chords in harmony are more concise *(►B. More poor use of terminology)* than the discordant notes. The sound waves produced by oscillators are in the same form as notes produced by mechanical instruments, so the same note ultimately sounds the same, or very similar, to our ear. For this reason, this theory does not only apply to sound coming to electronic instruments but every kind of instrument.

**Evaluation**

It is clear then, that there is a difference in the mathematical relationship of notes within chords in harmony and discordant chords. This means, that perhaps in the future music can be created entirely mathematically. It may be that maths can help determine which chords, and sequences of chords, will work together and be most pleasing to the ear.

This investigation was very limited, as I was not able to research properly how electronic instruments work, and how they differ from mechanical instruments. I also would have liked to try to find a mathematical formula connecting the number of semitones to the frequency ratios. The evidence here suggests that notes in harmony fit into a mathematical context, whereas discordant notes do not. However, this is just a theory and would have to be tested more thoroughly. To extend this investigation, I would like to look at the area between the waves in relation to the semitone difference and frequency ratios.

*(►D. Superficial reflection)*