In order to know the precise frequencies of both pure tones and pure colors, *Wholly Science* goes back to the groundbreaking work of Pythagoras, who was born around 570 BCE on the Greek island of of Samos, and died about 495 BCE. In his work, Pythagoras found the numerical logic underlying many phenomena of the perceived reality. Key to understanding the numerical logic underlying pure musical tones and pure colors is the **tetractys**.

The tetractys is a triangular figure consisting of four hierarchical rows. The top row at the apex of this triangle has one point, the second row two, the third one three, and the fourth row at the bottom of this triangle has four points. Adding up these points of the tetractys gives (1+2+3+4=) 10. Just think of the bowling cones, in order to remember the ordering of the tetractys (or vice versa).

The word *octave* means ‘whole of eight’. There are indeed eight whole tones starting with the lower tone Do and ending with the higher tone Do. The frequency of the higher tone Do is exactly twice as fast or high as the lower tone Do. Actually, this harmonic ratio of 1 : 2 is between every first and eighth tone, as *octave* also means ‘eighth’.

Only an octave of eight pure tones (Do, Re, Mi, Fa, Sol, La, Si, and Do again) contains four *quints* and four *quarts*. A *quint* (meaning ‘fifth’) is the harmonic ratio of 2 : 3 between a first and a fifth tone, while a *quart* (meaning ‘fourth’) is the harmonic ratio of 3 : 4 between a first and a fourth tone. Together, there are eighth harmonic ratios (4 quints + 4 quarts = 8), which gives a double meaning to the ‘whole of eight’ of each pure octave. Only with the correct Pythagorean tuning, there are eighth harmonic ratios in each octave.

Tone |
Hz |
Note |

Do |
512 | C |

Si |
486 | B |

La |
432 | A |

Sol |
384 | G |

Fa |
341^{1}/_{3} |
F |

Mi |
324 | E |

Re |
288 | D |

Do |
256 | C |

The above table shows the frequencies in cycles per second or Hertz (Hz) of an octave with the correct Pythagorean tuning, starting with the tone Do (of note C) at 256 Hz. Many have heard of the natural frequency of **432 Hz** for the note A (or tone La), but hardly anyone knows the correct frequencies of the other pure tones as well.

Tone |
nm |
Color |
THz |

Do |
384 | Ultraviolet | 780 |

Si |
405 | Violet | 740 |

La |
455 | Blue | 658 |

Sol |
512 | Green | 585 |

Fa |
576 | Yellow | 520 |

Mi |
607 | Orange | 494 |

Re |
683 | Red | 439 |

Do |
768 | Infrared | 390 |

When we examine the frequencies of pure colors, we again find the tetractys. The above table shows these frequencies in terahertz (THz), which is a trillion cycles per second), and as well in nanometer (nm), which is a millionth of a millimeter. The trinity of primary colors consists of red, yellow, and blue. In between there are the colors of orange, green, and violet. These are the six pure colors between the lower Do of infrared and the faster Do of ultraviolet, corresponding to the notes Re, Mi, Fa, Sol, La, and Si.Finally, the Golden Section cuts the visible spectrum in the blue part, dividing the colors cyan and indigo.

There is much more to say about this, but for that, you need to read the third chapter of *Wholly Science* – On understanding it all.