In this small article I will show you some ways of thinking about the scales in music.
There is a lot of rubbish floating around on the Web that contains incorrect calculations etc.
My aim is to explain the concepts in both an understandable *and* precise way.

**definition 1**: A *scale* is a finite sequence of points that divides an octave in parts.

**definition 2**: The *chromatic note system* is the standard note system in the western world. An
octave is divided into 12 notes, each a half step apart.

For illustration, let's see some scales visualized on a guitar or a bass fretboard. Think of the
E string. The green boxes indicate notes that belong to the scale while the white boxes indicate
the notes that do not belong to the scale. If you think about the real instruments,
please note that the fretboards shown here are *not* drawn using correct proportions.

major (or ionian) scale

mixolydian scale

natural minor (or aeolian) scale

harmonic minor scale

blues scale

Now, using the chromatic note system, how many ways are there to divide an octave?
If you examine the scale shapes above, you can see that each of them contains
the same starting note (i.e. the *tonic*) and the ending note (i.e. the nearest *octave*
following the tonic). This is a direct consequence of our **definition 1**. Because we
are dividing an octave, we must *always include the tonic and the next octave*.

octave interval

In these examples,
assuming standard E tuning, the tonic is on the 1st fret and so it is **F**, but that fact is *not* important to our investigation.
We could start using any note and it would make no difference. This is because all the tonalities are *isomorphic*, i.e. the structures
of octaves are always the same.

Because the starting notes and the ending notes are always included, to find the different combinations, it
suffices to examine *only the inner blocks* of the fretboard. We proceed to take the following two steps:

- instead of using colours green and white, we use
*binary digits*1 and 0 to signify whether a note belongs to a scale or not - drop the starting note and the ending note

For example, using the binary notation, the above scales and the octave interval look like this.
Note that the starting and the ending note have *not* been dropped yet:

major | 1010110101011 |

mixolydian | 1010110101101 |

natural minor | 1011010110101 |

harmonic minor | 1011010110011 |

blues | 1001011100101 |

octave interval | 1000000000001 |

Below we can see the scales after dropping the first and last binary digit. Note
how the *length* of all the binary digit strings goes down from 13 to 11:

major | 01011010101 |

mixolydian | 01011010110 |

natural minor | 01101011010 |

harmonic minor | 01101011001 |

blues | 00101110010 |

Now, the important realization is this: the binary digit strings of length 11 correspond *exactly to the white area
of the octave interval above*.

We can easily count the number of *all binary strings having length 11*, using
the familiar formula: 2 to the power of 11. The result is 2048.

However, there is one problem. Our count includes the string **00000000000**, but when
we add the starting note and the ending to it, we get **1000000000001**. That is the
*octave interval*, and it is probably best to exclude it, because an interval is, well,
just an interval, and perhaps not a proper scale. So to get the correct result, we count
2048 - 1 to arrive at the final result: There are *2047 different ways* to divide
an octave using the chromatic note system, and so *2047 different scales*. Of course
only a small part of these scales has been named.

What is sound? According to a physical definition, it is vibration in the air at certain
wavelength, the frequency of which is measured using Hertz or Hz. For example, the
tuning, or *pitch*, of a certain **A** note could be 440Hz, 441Hz or 442Hz, meaning that many
*vibrations per second*. We will now ignore the fact that natural sounds consist of
not only the base frequency, but of additional harmonic multiples too.

Assuming a 440Hz tonic **A**, the octave from that **A** to another higher pitched **A** is the frequency range between
440Hz and 880Hz. It is important to understand that the *chromatic note system* is only one possible way of
splitting the octave. What if we included *quarter steps*, like they do
in some non-western cultures? How many scales would we have then?

**definition 3**: The *quarter note system* is an alternative way of dividing an octave
into parts. In this system the notes are a quarter step apart.

Quarter note fretboard with 26 frets shown

Now the octave from **F** to **F** would look like this:

Quarter note fretboard octave from F to F

Using the same argument as with the chromatic note system, we will now investigate only the white blocks *
inside* the green blocks. We have 23 white blocks, hence all our binary digit strings will be 23 digits long.
So we count 2 to the power of 23, getting 8388608. And just like we ignored the all-zeroes bit string of length 11, we can
also ignore the **00000000000000000000000** of length 23. So our final result concerning the quarter note system
is that there are 8388608 - 1, *8388607 different ways and thus 8388607 scales* in there. That is a huge amount, almost
8 and a half million combinations, especially when compared to our earlier chromatic system result.