__Intervals
and Notation__

This section is dedicated to the mathematics involved in intervals and notation.

- Why are there twelve tones in an octave?
- Why do certain intervals or chords sound better than others?
- What does all this have to do with frequency/What frequency is middle C at?

*If you ever get lost in following the notes
up and down the scale, refer to this diagram:*

**1. Why are there twelve tones
in an octave?**

This is a fairly simple question. Going up one whole octave means doubling the frequency. In order for each individual sound to be included, there had to be twelve notes. When playing a chromatic scale, the average human ear cannot distinguish the fact that there are technically a number of notes in between each chromatic note. So when a chromatic scale (comprising all twelve notes in an octave) is played, we perceive that all the sounds between the first note and the last note (the first note doubled) are included.

**2. Why do certain intervals
or chords sound better than others?**

An interval or chord of a fifth, octave, or third (for example) sounds 'good' simply for the reason that when note values were 'assigned' it was decided that these intervals would be clear and 'nice-sounding'. Come to think of it, this isn't a particularly math-related question.

**3. What does this all have to
do with frequency?/What frequency is middle C at?**

The middle A above middle C is known universally to be at 440 Hz. When increasing any note by an octave, the Hz value is doubled, and when decreasing by an octave, it is halved. Therefore, A one octave higher is 880 Hz, and one octave lower is 220 Hz.

For many scientific reasons, an interval of frequencies at a ratio of 3:2 always sounds 'good'. This same 'good' sound is achieved by an interval of a fifth (four whole steps up). Thus, going up a fifth means x(3/2)=y, where x is the original note, and y is the transposed note. Using this, we can determine the frequency of any natural note. For example, one fifth above A-above-middle-C is E-more-than-an-octave-above-middle-C, at a frequency of 660 Hz [ 440(3/2)=660 ]. Thus, lower one octave is 330 Hz. A fifth above E-immediately-above-middle-C is B-almost-an-octave-above-middle-C, 330(3/2)=495 Hz, and so on.

The original question was the frequency of middle C. To do this, we continue, using the 'exhaustion method':

A-almost-an-octave-above-middle-C is at |
440 Hz |

Raise a fifth to
E-more-than-an-octave-above-middle-C |
440(3/2)=660 Hz |

Lower an octave to E-just-above-middle-C |
660/2=330Hz |

Raise a fifth to
B-almost-an-octave-above-middle-C |
330(3/2)=495 |

Lower one octave to B-immediately-below-middle-C | 495/2=247.5Hz |

Raise a fifth to F-immediately-above-middle-C | 247.5(3/2)=371.25 Hz |

Lower an octave to F-below-middle-C | 371.25/2=185.625 Hz |

Raise a fifth to Middle C |
185.625(3/2)=278.4375 Hz |

Therefore, Middle C is at a frequency of 278.4375
Hz |