### A starting point in music theory

In Introducing the Western Music Language, we saw how we can derive the scale notes from the harmonics of one string. When considering subsequent divisions of a string by whole numbers, we actually observe a manifestation of the harmonic mediating proportion.

The series 1/2, 1/3, 1/4, 1/5, 1/6, ... is harmonic, ie **b = 2ac/(a+c)**

The notes that are created this way explain why the major chord is so pleasing to the ear. I did outline in a little textbook how we can derive all of the notes (chromatic scale) we use in western music just using this principle. However, when we look at a lot of ethnic music, we observe the 5 note pentatonic scale. This scale, by the way, is responsible for the blues and its numerous derivations.

The pentatonic scale can be wholly constructed using geometric proportion, and it can be derived by an even simpler method. To fully understand this process it is helpful to construct following contraption:

The string should be tightened as close as possible to the surface, but not so close that it buzzes when plucked. Aim for a good deep sound. The intervals (distances) between the notes (sounds) we will create are not dependent on the tightness of the string but on the relative subdivisions of the length of the string.

Note also that the 2 pegs on top are 81 cm apart. This is convenient because 81 is a multiple of 3 which will come in handy as we will see later on.

Now mark off the distances on your instrument as shown on the diagram below. Then press the string down with your left finger on each of these nodes, sounding the string with your right hand

If we press down halfway the string (at 40.5 cm), we will get the
same note, but at a higher level. This is the octave. If we now press
halfway the distance of this half (at 1/4), the same note will sound
again at a higher level. We can theoretically go on like that ad infinitum,
each time attaining the same note but at a higher level.

The image below is not to scale.

In other words, we are creating the following series:

1/2 of 1 | = | 1/2 * 1 | = | 1/2 | |

1/2 of 1/2 | = | 1/2 * 1/2 | = | 1/4 | |

1/2 of 1/4 | = | 1/2 * 1/4 | = | 1/8 | |

1/2 of 1/8 | = | 1/2 * 1/8 | = | 1/16 |

Because each subsequent ratio is created by multiplying it by the
same amount (in this case 1/2), the series is said to be **geometric**.

Ok, sounds good mathematically, but doesn't sound very exciting
to our ears. Not a great variety of notes! What would the next logical
step be? The same as above but using the next subdivision that is possible:
division by 3. It turns out that the first subsequent divisions of a
string by three creates our pentatonic scale. Though the principle is
the same as above, in order for us to create a proper pentatonic scale,
we also need to employ the important discovery we made above, namely
that **doubling or halving the length of a string produces the same
note, be it on a different level**

Why do we want to use this? Because we want to keep all our new-to-be-found notes within the same level. And in order to keep all our new-found notes within the octave, all we need to do is ensure that all the markations end up on the first half of the string. Note that this first half marks the octave (the same note on a higher level). Logically, all the notes that lie beyond this markation will repeat the ones below. Note: This is yet another example (metaphor) to depict the infinity of numbers on any length of string. Doing the following experiment will make this clear.

To avoid confusion, always press down with the LH and pluck the string
with your RH. Now verify that **all notes in B are present in A, but
on higher levels.**

Also note that we measure from the **right**, as we did in the
experiment for deriving the octaves. This is important since we need
to get the ratios from the string-length last obtained (by means of
pressing down the string at those nodes).

Getting back to our intention of subsequently dividing the string by 3 ... mathematically,

1/3 of 1 | = | 1/3 * 1 | = | 1/3 | |

1/3 of 1/3 | = | 1/3 * 1/3 | = | 1/9 | |

1/3 of 1/9 | = | 1/3 * 1/9 | = | 1/27 | |

1/3 of 1/27 | = | 1/3 * 1/27 | = | 1/81 |

Now translate the found ratios so they are greater than 1/2, this way we remain within the octave: Remember, we can only multiple by multiples of 2 in order to obtain the same notes, be it on different levels:

1/3 < 1/2 | ==> | 1/3 * 2 | = | 2/3 | (2/3 > 1/2) | sol | |

1/9 < 1/2 | ==> | 1/9 * 8 | = | 8/9 | (8/9 > 1/2) | re | |

1/27 < 1/2 | ==> | 1/27 * 16 | = | 16/27 | (16/27 > 1/2) | la | |

1/81 < 1/2 | ==> | 1/81 * 64 | = | 64/81 | (64/81 > 1/2) | mi |

The reasoning used for finding each of these notes is as follows (using the example of 1/9):

- 1/9 is smaller than 1/2, therefore multiply with 2
- 2/9 is smaller than 1/2, therefore multiply with 2 again
- 4/9 is smaller than 1/2, therefore multiply with 2 a third time
- 8/9 is greater than 1/2, therefore this is our note.

Do this with the other ratios as well, and sito, there's our five notes (including the full length, do): do, sol, re, la, mi

All we need to do now is to convert it to our instrument. Since the string is 81 cm long, we use the found ratios to calculate the distances:

2/3 of 81 | = | 54 cm | |

8/9 of 81 | = | 72 cm | |

16/27 of 81 | = | 48 cm | |

64/81 of 81 | = | 64 cm |