Derived from Harmonic Theory

part of an ongoing research by

whatmusicreallyis.com

- The Profound Musical Meaning of All Numbers and Number Systems
- True Prime Numbers: The Pattern
- Geometry, or Frozen Music
- Trigonometry, the Study of Harmonic Relationships in Triangles
- The Colors of Music and Music of Color
- What “Vortex Based Math” Really İs
- Modular Arithmetic and the 12-hour Clock
- Numerology's Sum of Digits Is Just the Special Case of Base 10, Modulus 9
- TO DO: base n, mod n-1 or base m+1, mod m
- Torus Creation from This Math

of All Numbers and Number Systems

This section shows how all numbers and number systems come from the (Musical) Harmonic Series. The reasoning behind this fact might be of little value to the innocent music lover and mathematician when taken out of its original context. Therefore, it is assumed the reader has knowledge about the True Harmonic Series in its completeness: its meaning for music, what it really is, where can it be found in nature and how different harmonics can be isolated and emphasized on instruments that allow it. With other words, it is assumed the reader knows “Where All Music Comes From” and is well acquainted with “The Physics of Harmonics”.

It is important to realize that the numbers we use every day are part of a system of numbers, which is not the only system of numbers in existence. In fact, there are as many systems as there are numbers, and the one we use is not particularly special.

When we write 1 2 3 4 5 6 7 8 9 10 ... we use a numbering system called “Base 10” or “decimal”, which comes from the Harmonic Series.

**All integers (natural, or whole numbers) come from the Ascending Series of Harmonic Overtones, while the decimals and recurring decimals of rational numbers together with the infinite and non-recurring decimals of irrational numbers come from the Descending Series of Harmonic Undertones.**

In simple words, all figures to the left of the comma (in some countries point) are overtones, while all figures to the left of the comma are undertones. Another way of saying it is: everything before comma overtones, everything after comma undertones. This applies to left-to-right writing; for right-to-left, the opposite is true.

As an example, the number 1,5 (in some countries written 1.5) is a decimal expansion and sum of following bundles: 0 units of
(or 10 to the 1^{st} power), 1 unit of 1's (or 10 to the 0 power) plus 5 units of
(or 10 to the −1^{st} power).

Our everyday numbering system, the Base 10, comes from the equal progression (also known as “geometric” progression or series) generated by the number 10. It is strictly a series of powers of 10, in both ascending and descending directions.

The number 365,242189 means 5 bundles of 1, plus 6 bundles of 10, plus 3 bundles of 100. Plus 2 bundles of 1/10, plus 4 bundles of 1/100, plus 2 bundles of 1/1.000, and so on. With other words:

365,242189 = 3×10^{+2} + 6×10^{+1} + 5×10^{0} + 2×10^{-1} + 4×10^{-2} + 2×10^{-3} + 1×10^{-4} + 8×10⁻^{-5} + 9×10^{-6}

365,242189 = 3×100 + 6×10 + 5×1 + 2÷10 + 4÷100 + 2÷1.000 + 1÷10.000 + 8÷100.000 + 9÷1.000.000

But this is just Base 10. There are of course other Bases, and all of them are an expression of the Harmonic Series.

For example, the number 24,041666666... having 6 as a recurring decimal in Base 10, also written as 24,0416 or, like in the table below 24,0416, can be represented in any other base like this (from top to bottom, bases 16, 12, 10, 7, 4 and 2):

This number 24,0416 (in Base 10) is an arbitrary addition of and .

Although represented separately, all number lines are part of the same Series of Harmonics. This can be verified by checking whether the number 16 for example sits on the same vertical projection with 4^{2} and 2^{4}.

**To conclude...**

The True Prime Numbers Pattern is discussed below in Base 10, but these properties are valid and remain the same no matter the Base used.

Are All About Undertones

Continued fractions are another possible representation of real numbers in terms of a sequence of integers, aside from decimal expansion. An archaic word for a continued fraction is anthyphairetic ratio.

*Source: http://mathworld.wolfram.com/ContinuedFraction.html*

Continued fractions are all about undertones, since by definition, their structure implies the fundamental 1 divided by an integer, for a continued number of times (usually forever). That's because musically, any integer divided by 1 is, frequency-wise, an undertone harmonic of the descending series.

Since all figures to the right of the comma (in some countries point) are undertones as demonstrated above, and this way of writing numbers (called decimal expansion in Base 10) can also be represented by continued fractions, it follows that continued fractions are also undertones.

The simplest continued fraction generates the ‘Golden Number’ proportion. All of its elements are 1. Supposedly this is why the ‘Golden Mean’ is so wide-spread in nature.^{26}

Writing this number as 1,618... is not only incorrect because of the inherent undertone fault in the way all positional notation numeral systems are used (as demonstrated above), it is alo wrong since 1,618... is the hypothetical value of convergence where the continued fraction would reach infinity. By definition, infinity has no end, so converging or reaching infinity's end is not only a pleonasm (gramatical error), but also a mathematical error.

A quite special structure is exhibited by the [non-simple] continued fraction of the Euler-number *e* = 2.71828 … This continued fraction contains the sequence of all natural numbers and the sequence of all musical [sonic distances]. The [...] Continued-fraction for *e* is composed of the [undertone series].^{26}

The simple continued fraction for pi does not show any obvious patterns, ...

*source: http://mathworld.wolfram.com/PiContinuedFraction.html*

... but clear patterns do emerge in the beautiful non-simple continued fractions

*source: http://mathworld.wolfram.com/PiContinuedFraction.html*

TO DO: express the left equalities in terms of Tau instead of Pi. π|4=τ|8 & 2|π=4|τ.

**To conclude...**

The (simple or non-simple) continued fraction representation of any number is given by

- a succesion of numbers written as an overtone (harmonic of the ascending series / number over the fraction bar / numerator in integer form) and
- an addition or/and subtraction of multiple undertones (harmonics of the descending series / numbers under the fraction bar / denominators in integer form) which, no matter if it goes on forever or not, in most cases displays an orderly progression pattern.

The Pattern

A Single Number and Its True Primes

**every number can be considered a scaled version of 1**

http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/

image copyright: http://betterexplained.com/articles/rethinking-arithmetic-a-visual-guide/

Operation | Old Concept | New Concept |

Addition | Repeated counting | Sliding |

Multiplication | Repeated addition | Scaling |

Exponents | Repeated multiplication | Growing for amount of time |

Table taken from: http://betterexplained.com/articles/understanding-exponents-why-does-00-1/

Odd and Even

**Simple numbers can be divided into 2 groups: • Odd and •• Even Numbers.** All other numbers are combinations of these.

**• Odd Numbers are divisible only by 1, themselves and combinations of themselves. According to other definitions, Odd Numbers are not divisible by 2.****•• Even Numbers are divisible by 2.**

The natural progression of simple numbers, in consecutive order, has Odd and Even Numbers one after another, in this particular order, every iteration. The pattern is very simple, and it repeats exactly every 2^{nd} number: Odd - Even. Or: 1 - 2.

• | •• | 1 | 2 |

• | •• | 1 | 2 | |

• | •• | 3 | 4 | |

• | •• | 5 | 6 | |

• | •• | 7 | 8 |

The table below allows us to verify this. Natural or whole numbers 1 to 64 are cataloged according to the specific type they belong to; the prime factorization (sub-component break down) of each is listed. It will be noticed that:

**• Odd Numbers are represented by the number 1.**They are only divisible by the numbers they include: 1, themselves, and other Odds, and also combinations of these. For example: 5, 7, 11, 13 (highlighted in light red) are Simple Odds or Simple True Primes, as each contains only one number that can be divided just by 1 and itself. 25, 49, 55, 63 (highlighted in light orange) are Composite Odds, because they contain more than one such number. Note that the Odd Numbers from the table below, marked with 1 red dot • and highlighted as described, are not divisible by 2.**•• Even Numbers are represented by the number 2.**They are always divisible by 2, because they only include (the scaling) factor 2, together with other Odds. All Even Numbers from the table are marked with 2 blue dots ••.

• | 1 | 1 | • | 25 | 5² | • | 49 | 7² | ||||

•• | 2 | 2 | •• | 26 | 2 × 13 | •• | 50 | 2 × 5² | ||||

• | 3 | 3 | • | 27 | 3³ | • | 51 | 3 × 17 | ||||

•• | 4 | 2² | •• | 28 | 2² × 7 | •• | 52 | 2² × 13 | ||||

• | 5 | 5 | • | 29 | 29 | • | 53 | 53 | ||||

•• | 6 | 2 × 3 | •• | 30 | 2 × 3 × 5 | •• | 54 | 2 × 3³ | ||||

• | 7 | 7 | • | 31 | 31 | • | 55 | 5 × 11 | ||||

•• | 8 | 2³ | •• | 32 | 2⁵ | •• | 56 | 2³ × 7 | ||||

• | 9 | 3² | • | 33 | 3 × 11 | • | 57 | 3 × 19 | ||||

•• | 10 | 2 × 5 | •• | 34 | 2 × 17 | •• | 58 | 2 × 29 | ||||

• | 11 | 11 | • | 35 | 5 × 7 | • | 59 | 59 | ||||

•• | 12 | 2² × 3 | •• | 36 | 2² × 3² | •• | 60 | 2² × 3 × 5 | ||||

• | 13 | 13 | • | 37 | 37 | • | 61 | 61 | ||||

•• | 14 | 2 × 7 | •• | 38 | 2 × 19 | •• | 62 | 2 × 31 | ||||

• | 15 | 3 × 5 | • | 39 | 3 × 13 | • | 63 | 3² × 7 | ||||

•• | 16 | 2⁴ | •• | 40 | 2³ × 5 | •• | 64 | 2⁶ | ||||

• | 17 | 17 | • | 41 | 41 | |||||||

•• | 18 | 2 × 3² | •• | 42 | 2 × 3 × 7 | |||||||

• | 19 | 19 | • | 43 | 43 | |||||||

•• | 20 | 2² × 5 | •• | 44 | 2² × 11 | |||||||

• | 21 | 3 × 7 | • | 45 | 3² × 5 | |||||||

•• | 22 | 2 × 11 | •• | 46 | 2 × 23 | |||||||

• | 23 | 23 | • | 47 | 47 | |||||||

•• | 24 | 2³ × 3 | •• | 48 | 2⁴ × 3 |

True Primes, Dual and Ternary

**Simple numbers can only be divided into 3 universal categories: • Primary, •• Secondary, and ••• Tertiary Numbers.** All other numbers are combinations of these.

**• Primary Numbers or True Primes are numbers divisible only by 1, themselves and combinations of themselves.****•• Secondary or Dual Numbers are divisible by Primary Numbers and 2.****••• Tertiary or Ternary Numbers are divisible by Primary Numbers, 2 and 3.**

The natural progression of simple numbers, in consecutive order, has Primary and Secondary Numbers changing places every other iteration. The pattern is simple, and it repeats exactly every 6^{th} number: Primary - Secondary - Tertiary, Secondary - Primary - Tertiary. Or: 1 - 2 - 3, 2 - 1 - 3.

• | •• | ••• | 1 | 2 | 3 | |

•• | • | ••• | 4 | 5 | 6 |

• | •• | ••• | 1 | 2 | 3 | |

•• | • | ••• | 4 | 5 | 6 | |

• | •• | ••• | 7 | 8 | 9 | |

•• | • | ••• | 10 | 11 | 12 | |

• | •• | ••• | 13 | 14 | 15 | |

•• | • | ••• | 16 | 17 | 18 |

The table below allows us to verify this. Natural or whole numbers 1 to 128 are cataloged according to the specific type they belong to; the prime factorization (sub-component break down) of each is listed. It will be noticed that:

**• Primary Numbers (True Primes) are represented by the number 1.**They are only divisible by the numbers they include: 1, themselves, and other True Primes, and also combinations of these. For example: 5, 7, 11, 13 (highlighted in light red) are Simple True Primes, as each contains only one number that can be divided just by 1 and itself. 25, 49, 55, 65 (highlighted in light orange) are Composite True Primes, because they contain more than one such number. Note that the True Prime Numbers from the table below, marked with 1 red dot • and highlighted as described, are not divisible by 2 nor 3.**•• Secondary (Dual) Numbers are represented by the number 2.**They are always divisible by 2, because they only include (the scaling) factor 2, together with other True Primes. Note that the Dual Numbers from the table, marked with 2 blue dots ••, are not divisible by 3 — because none of them contains this number.**••• Tertiary (Ternary) numbers are represented by the number 3.**These numbers are always divisible by 3, because the main (scaling) factor they include is 3, together with 2 and other True Primes. They are marked with 3 green dots**•••**.

• | 1 | 1 | • | 43 | 43 | • | 85 | 5 × 17 | ||||

•• | 2 | 2 | •• | 44 | 2² × 11 | •• | 86 | 2 × 43 | ||||

••• | 3 | 3 | ••• | 45 | 3² × 5 | ••• | 87 | 3 × 29 | ||||

•• | 4 | 2² | •• | 46 | 2 × 23 | •• | 88 | 2³ × 11 | ||||

• | 5 | 5 | • | 47 | 47 | • | 89 | 89 | ||||

••• | 6 | 2 × 3 | ••• | 48 | 2⁴ × 3 | ••• | 90 | 2 × 3² × 5 | ||||

• | 7 | 7 | • | 49 | 7² | • | 91 | 7 × 13 | ||||

•• | 8 | 2³ | •• | 50 | 2 × 5² | •• | 92 | 2² × 23 | ||||

••• | 9 | 3² | ••• | 51 | 3 × 17 | ••• | 93 | 3 × 31 | ||||

•• | 10 | 2 × 5 | •• | 52 | 2² × 13 | •• | 94 | 2 × 47 | ||||

• | 11 | 11 | • | 53 | 53 | • | 95 | 5 × 19 | ||||

••• | 12 | 2² × 3 | ••• | 54 | 2 × 3³ | ••• | 96 | 2⁵ × 3 | ||||

• | 13 | 13 | • | 55 | 5 × 11 | • | 97 | 97 | ||||

•• | 14 | 2 × 7 | •• | 56 | 2³ × 7 | •• | 98 | 2 × 7² | ||||

••• | 15 | 3 × 5 | ••• | 57 | 3 × 19 | ••• | 99 | 3² × 11 | ||||

•• | 16 | 2⁴ | •• | 58 | 2 × 29 | •• | 100 | 2² × 5² | ||||

• | 17 | 17 | • | 59 | 59 | • | 101 | 101 | ||||

••• | 18 | 2 × 3² | ••• | 60 | 2² × 3 × 5 | ••• | 102 | 2 × 3 × 17 | ||||

• | 19 | 19 | • | 61 | 61 | • | 103 | 103 | ||||

•• | 20 | 2² × 5 | •• | 62 | 2 × 31 | •• | 104 | 2³ × 13 | ||||

••• | 21 | 3 × 7 | ••• | 63 | 3² × 7 | ••• | 105 | 3 × 5 × 7 | ||||

•• | 22 | 2 × 11 | •• | 64 | 2⁶ | •• | 106 | 2 × 53 | ||||

• | 23 | 23 | • | 65 | 5 × 13 | • | 107 | 107 | ||||

••• | 24 | 2³ × 3 | ••• | 66 | 2 × 3 × 11 | ••• | 108 | 2² × 3³ | ||||

• | 25 | 5² | • | 67 | 67 | • | 109 | 109 | ||||

•• | 26 | 2 × 13 | •• | 68 | 2² × 17 | •• | 110 | 2 × 5 × 11 | ||||

••• | 27 | 3³ | ••• | 69 | 3 × 23 | ••• | 111 | 3 × 37 | ||||

•• | 28 | 2² × 7 | •• | 70 | 2 × 5 × 7 | •• | 112 | 2⁴ × 7 | ||||

• | 29 | 29 | • | 71 | 71 | • | 113 | 113 | ||||

••• | 30 | 2 × 3 × 5 | ••• | 72 | 2³ × 3² | ••• | 114 | 2 × 3 × 19 | ||||

• | 31 | 31 | • | 73 | 73 | • | 115 | 5 × 23 | ||||

•• | 32 | 2⁵ | •• | 74 | 2 × 37 | •• | 116 | 2² × 29 | ||||

••• | 33 | 3 × 11 | ••• | 75 | 3 × 5² | ••• | 117 | 3² × 13 | ||||

•• | 34 | 2 × 17 | •• | 76 | 2² × 19 | •• | 118 | 2 × 59 | ||||

• | 35 | 5 × 7 | • | 77 | 7 × 11 | • | 119 | 7 × 17 | ||||

••• | 36 | 2² × 3² | ••• | 78 | 2 × 3 × 13 | ••• | 120 | 2³ × 3 × 5 | ||||

• | 37 | 37 | • | 79 | 79 | • | 121 | 11² | ||||

•• | 38 | 2 × 19 | •• | 80 | 2⁴ × 5 | •• | 122 | 2 × 61 | ||||

••• | 39 | 3 × 13 | ••• | 81 | 3⁴ | ••• | 123 | 3 × 41 | ||||

•• | 40 | 2³ × 5 | •• | 82 | 2 × 41 | •• | 124 | 2² × 31 | ||||

• | 41 | 41 | • | 83 | 83 | • | 125 | 5³ | ||||

••• | 42 | 2 × 3 × 7 | ••• | 84 | 2² × 3 × 7 | ••• | 126 | 2 × 3² × 7 | ||||

• | 127 | 127 | ||||||||||

•• | 128 | 2⁷ |

The pattern in True Prime Numbers was also discovered by Anthony Morris after arranging the numbers 1 to 72 and their Mod 9 equivalents (see the Modular Arithmetic section below) into 6 columns, based on a 72 number sequence (“6 octaves of 12”) he discovered in music by using an invalid music theory of random numerological associations. The results were published under the title “The Re-Classification of Prime & Composite Numbers” on his website newunderstandings.com and also on academia.edu: https://www.academia.edu/4120168/The_Numerical_Universe_-_Part_1_-_Re-Classification_of_Prime_and_Composite_Numbers.

Working on it...

There's a little harmonic math trick I use to calculate angle values in degrees for regular polygons.

Say I'm being asked "How many degrees has one angle of a pentagon?" Firstly, penta means five. I subtract two from this number: 5−2=3 (I'll explain shortly why 2). This result is a harmonic, which I set on top of the initial value to get the ratio
,
which I multiply with the fundamental of the harmonic series, 180. The result:
x180=108 degrees in a pentagon angle.
(To instantly get the total number of degrees inside this shape, I simply multiply the resulting harmonic with its fundamental: 3x180=540.)

Having 180 assigned to the 1/1 of the "scale" comes from the observation that the total number (arithmetic sum) of angle degrees in geometric shapes progresses harmonically like this:

harmonic | "frequency in angles" | number of angles (corners) | value of each angle |

1 | 180 | 3 | 60 |

2 | 360 | 4 | 90 |

3 | 540 | 5 | 108 |

4 | 720 | 6 | 120 |

5 | 900 | 7 | 128,571428 |

6 | 1080 | 8 | 135 |

7 | 1260 | 9 | 140 |

8 | 1440 | 10 | 144 |

9 | 1620 | 11 | 147,27 |

10 | 1800 | 12 | 150 |

Numbers after comma are repeating pattern of decimals. This table makes it clear why I subtract 2: because 2 is the constant difference between the harmonic and the number of angles or corners (and also sides) that particular shape has.

A generalized formula for this math trick is 180*[(n-2)/n] or 180 × , where n is the number of angles or corners a regular polygon has. For the pentagon example, this is 180 × = 108.

The numerical relations from the table above are best visualized on a logarithmic number line.

TO DO: remove extra data and optimize

The sonic number line, where what you see is how you hear, makes it visually obvious that all values are multiples of 180. Therefore, dividing them all by this value reduces the (already in-place) harmonic series to its simplest form.

TO DO: remove extra data and optimize

Having (musical unsimplified) ratios as values for interior angles is a powerful concept that instantly reveals information about that geometric shape:

- the undertone (or number under the fraction bar) is the number of angles/sides that gives the shape’s name;
- the overtone (or number over the fraction bar) is the sum of all interior angles, and also a harmonic of the fundamental value used for sonic angles measurement.

(If you're browsing around, wondering why I say overtone instead of “numerator” and undertone instead of “denominator”, check out Harmonic Genesis from “The Many Meanings of Every Ratio: How to İnterpret the 9 Different and Non-exclusive Meanings of Every Musical Fraction” chapter.)

**The most profound signification is that of finding the value of angles in an instant, by using the formula
, where n is the number of angles or corners a regular polygon has.**

Say we want to find out how many (sonic) degrees a hexagon has. Hexa means six, so we write this at the bottom of the fraction: . Next, we apply the formula and subtract 2 out of it: 6−2=4, which is the sum of interior angles, and which gets written on top: . That’s all folks!

We can of course complicate things, first by simplifying the fraction to , a ratio which does not tell much about the angular properties of the geometric shape. Secondly, we can multiply this (or the unsimplified) sonic value with 180 to get the value in standard degress: 120.

This works for regular polygons, that is, those shapes whose sides and angles are all equal. However, the sum of angles is the same no matter if the polygons are regular or not.

It works because every time a new side/corner is added, the interior angles add up to 1 sonic degree (or 180 regular degrees) more. This can be checked by dividing every shape into undivizible sub-shapes, by connecting one corner (vertex) to all others. The decomposition of polygons into triangles shows how, with every new side/corner, a new triangle appears. And since the sum of angles in every triangle is the same, all there's left to do is add them up.

To me, having 360° in a circle is as arbitrary as having 86.400 seconds in a day. Sure, they are composite numbers that make it easy to do some calculations, but look how 360 fails for 7- and 11-angled shapes.

A very interesting unit of angle measurement is the "turn", where one circle has 1 turn. It's also referred to as revolution, complete rotation or full circle. This is the same as having τ radians in one cycle (τ=2π or Tau = 2 Pi). This type of measurement is ideal when working with radians, which make sense of equations in terms of the mover.

However, if the arbitrary degrees are to be used, I prefer to assign the number 2 to one complete rotation / number of "degrees" in a circle. That's because in this way, the old 360 now equals 2, meaning that the angle of a straight line, previously 180, is now 1 (the fundamental), while the circle is the 2^{nd} harmonic instead of 360.

This looks very natural, and goes perfectly hand-in-hand with the above angle calculations in regular polygons. It also means that, instead of using archaic ratios for geometric calculations, regular fractions that imply harmonic relationships are used.

Harmonic Relationships in Triangles

Trigonometry is a branch of mathematics, which is a branch of Harmonic Theory, that deals with the relationships — that is, with the ratios — between sides and angles in triangles. Heaving a solid musical fundament about Harmonic Theory, which is the study of natural relationships, makes trigonometry self-evident.

One aspect I have been trying to avoid so far is the reciprocity between frequency (perceptual) and length (physical) ratios in music. Trigonometry is all about physical measurements and lengths, so all ratios are flipped. (3/2 as a frequency, written like this in the HarmoniComb, is written here upside down: 2/3. This doesn't change anything, as this ratio is still part of the Ascending Harmonic Series.)

Harmonics of the Ascending Series

The ratio of two sides of any right triangle with the same interior angles is always the same number independent of the size of the triangle, the triangle's orientation, or the units used to measure the sides of the triangles.

There are three ways to form a ratio of the three sides of a triangle (six if you count the inverse ratios also). The symbols "sin", "cos", and "tan" are the tags/labels we use to identify which ratio is which.

TO DO: fix arrows and set alt tag

Of these, sinus and cosinus are the main harmonic identifiers, since tangent is just their proportion (the ratio between them): tan(Φ) = .

Trigonometry is good at finding a missing side or angle in a triangle. The value of the right angle is always known, so with a few more values all others can be calculated according to the formulas. But what do the formulas *really* mean?

“Sinus” is just a fancy name given to a property of an angle, in this example Φ. That property is a ratio, a proportion between (the lengths of) two sides of the triangle: the one opposite that angle Φ and the one opposite the right angle (also called “hypotenuse”).

The same is true for the other trigonometric functions, although in their respective form: whatever the value of the unknown angle might be, it can be found knowing that two of the triangle's sides are in a certain ratio. Also, the sum of the two angles that are not the right angle equals the value of the right angle.

When the number above the fraction bar is smaller than the one below it AND the fractions represent physical lengths, the ratio belongs to the ascending series of harmonics a.k.a. overtones. **Sin and cos are always overtones, while tan can be either overtone or undertone, depending on context.**

Undertones of Sinus and Cosinus

By definition, undertones are inverses of overtones — and of course the other way around. Thus if the trig functions sin and cos are simply ratios of physical lengths which through their numerical form convey overtones, their inverses will be undertones.

csc(Φ) = =

sec(Φ) = =

cot(Φ) = =

Of these, cosecant and secant are the main harmonic identifiers, since cotangent is just their proportion (the ratio between them): cot(Φ) = .

When the number above the fraction bar is larger than the one below it AND the fractions represent physical lengths, the ratio belongs to the descending series of harmonics a.k.a. undertones. **Csc and sec are always undertones, while cot can be either undertone or overtone, depending on context.**

Many attempts have been made to correlate music and color. Some claim that the arbitrary and errorneous associations between the tones of a specific musical culture and the colors of a wrong color theory are somehow natural. This is why they are not:

The rest of this subchapter... COMING SOON! Until then, a quote from Hazrat Inayat Khan:

«Very often man gives great importance to color and tone so much so that he forgets that which is behind them, and that leads him to many superstitions, fancies and imaginations. Many people have fooled the simple ones by telling them what color belonged to their souls, or what note belonged to their lives. Man is so ready to respond to anything that can puzzle him and confuse his mind! He is so willing to be fooled! He enjoys it so much if somebody tells him that his color is yellow, or green, or that his note is C, D, or F on the piano. He does not care to find out why. It is like telling somebody: "Wednesday is your day, and Tuesday some other person's."»^{7}

... and a link to what I call “**What Color Really İs**”:

**Correct Color Theory** by Larry Robinson

Plus! Part of a conversation which forms the basis of this subchapter:

There are many problems with "transposing color into sound".

1. The common approach is to divide the frequencies of the color spectrum by 2 again and again (about 40 times in total) until the numbers fall into "complementary" frequencies of the audible range.

This is arbitrary and based on the assumption that any tone whose frequency gets halved or doubled is the same tone. This is "the very most" fundamental mistake of music: to base acoustics on faulty musical dogma. A tone whose frequency gets halved or doubled is NOT the same tone, period.

2. Even if a "correct" way of transposing were to be found, like for example demonstrating that powers of 2 are not part of the acoustical lie western music is based on, or using powers of Golden Ratio Φ instead of 2, or the Natural Logarithm e or whatever, we would end up with different slices of the same electro-magnetic spectrum which have no true physical correspondence since one of them is a simple harmonic while the other the sum of a (theoretically) infinite series of harmonics.

Here's what I mean by that: the color green for example is only one "wavelength" ~510,5 nm (nano meters) whose frequency is ~587,5 Thz (Terra Hertz). This is a single frequency, just like that of a simple "sine wave". If this frequency gets divided by 2^40 (which is wrong of course, but just for fun), the end result in the hearing range is 534,3281373... Hz. But this "transposed" frequency is actually the fundamental of a complex tone (as opposed to a single harmonic of sound or light), having its own series of harmonics which are perfect (integer) number multipliers of it:

1068,656275... (×2)

1602,984412... (×3)

2137,312549... (×4)

2671,640686... (×5)

... and so on. Or, it could be the fundamental of an inharmonic tone, meaning a musical sound whose harmonics deviate from perfect numbers due to physical properties of the oscillator like rigidness.

The "correct" way of interpreting this is taking the fundamental harmonic 534,3281373... Hz and realizing it is part of a complex tone who has many other harmonics we can hear, and a whole lot more we can't hear, but we experience differently — for example, we can see its 1.099.511.627.776th harmonic as the color green. But at this height of "pitch", harmonics are so close to one another that we can safely say all the color spectrum is in fact harmonics of a single fundamental frequency, no matter what that audible frequency might be.

3. a) There can be no one-to-one mapping between wavelength and RGB values. b) Color is a human perception through one of our senses, and it is not fixed. c) Gamma and Beta sensors in our eyes determine the intensity of the colors we perceive for each of wavelengths in the visual spectrum. All these (a, b and c) processes can be different from individual to individual.

With other words, my green example above at ~587,5 Thz is just how I perceive the "greenest" green, the one green I believe to be purest. Actually, this is also misleading since I was using the colors of my LCD screen to determine it, instead of passing natural sunlight through a prism and setting for a value like Helmholtz did (yea, he's also famous for light).

The main thing is, and I'm sure you noticed this too, we perceive different colors at different light-intensities. For example, the greenest green from light-splitting is way brighter than the green we are used to seeing in grass & trees, and this complicates things a whole lot more with the introduction of the black component which is (just as magenta?) extra-spectral and thus cannot be "transposed" to any sound.

Further research: "luminosity function" / "spectral sensitivity".

Having in mind there are only 3 number types, all others being combinations of these, it should be of no surprise that the 9 numbers used in Vortex Based Math are actually 3 reiterations of the 3 types of numbers mentioned above (since 9=3×3).

This section is under preparation. In the meantime, check out the Modular Arithmetic article from The Math Images Project on The Math Forum. For an extended explanation, see Fun With Modular Arithmetic article on Better Explained.

Is Just the Special Case of Base 10, Modulus 9

«Integration of Digits (Numerology) - 'Indigs' means integrated digits, the sum of 1+2+3+4 = 10. 1+0 = 1. (The indig of the series).» Quote by Buckminster Fuller.

Marko Rodin uses numbers in Base 10 - Mod 9. There is no surprise once you limit all conceivable numbers to Modulo 9, that patterns begin to repeat every 9 numbers. This in itself is no special, aside from the fact that 9 is a multiple of 3. You could use any Base n - Mod n-1 and still get nice patterns for doublings, Fibonacci, and so on. Try Base 7 - Mod 6 and map the numbers on a circle; a more difficult arbitrary one is Base 12 - Mod 11 (both presented below). Patterns appear wherever you create them.

To help calculate, use the following formula in an (Excel) spreadsheet: « =MOD(BASE(Y5;*n*)-1;*n-1*)+1 » where Y5 is the cell you want to calculate, *n* is the Base and *n-1* the Modulo. Example for the number 1024 in Base 3 Mod 2: « =MOD(BASE(1024;3)-1;2)+1 ». (Depending on your regional settings, you might need to replace ; with , in the formula.) Note that in this formula, as well as in all the examples given here, the number 0 (zero) is replaced by the highest number in the Mod series, which is actually the n from Mod n. For example, 9 mod 9 ≡ 9 and not 0.

Small numbers represent the value in Base 10.

TO DO: make circles concentric and tighten, arrange numbers and optimize

**The so-called “Pythagorean mathematics” or “Pythagoras’ lost mathematics”, also known as “sum of digits”, “integration of digits” or “digital root” — the primary mathematical operation in numerology — works for every Base n, Mod n-1 (or Base m+1, Mod m).**

For example, in the image above:

- 10 is 1 because 1+0=1 (in Base 4);
- 11 is 2, because 1+1=2 (in Base 4);
- 12 is 3, because 1+2=3 (in Base 4).

The peculiarity comes along with number 13, because we must calculate the sum of digits **in Base 4**. 1+3=4 in Base 10, whereas in Base 4, 1+3=10=1. Since our minds are stuck in Base 10, it's hard to do such calculations at once.

The trick is to do the math in Base 10 and use the small numbers under each value as a conversion table. So, 1+3=4 in Base 10. There is only one number 4 written with small numbers, and it sits under the figure 10. This means that 1+3=10 in Base 4. Now the summing of digits is repeated: 1+0=1 in Base 4.

In the image below:

- 15 is 6 because 1+5=6 (in Base 7);
- 16 is 7, because 1+6=7 (in Base 7);
- 20 is 2, because 2+0=2 (in Base 7).

Similarly to the process described above, 26 is 2 in Base 7 because 2+6=8 in Base 10, and the small 8 is under 11 (which means 8 in Base 10 is 11 in Base 7), and 1+1=2. This can be both confusing and frustrating at first.

TO DO: make circles concentric and tighten, arrange numbers and optimize

**That “there are only 9 numbers in existence” like some claim, is the direct result of summing digits (digital root) in Base 10.** If we were to use Base 4 instead, there would be only 3 numbers in existence and only 6 under Base 7 (as pictured above). If using Base 12, there would be only 11 numbers in existence (see graphic further down below).

TO DO: make circles concentric, arrange numbers and optimize

If we were to follow the pattern of True Prime Numbers on the diagram above, we'll be starting from 1 and tracing a line to 5, then to 7, then to the next True Prime Number which is 11 and which gets “reduced” to 2.

In this Base 10 - Mod 9 setup all Primary and Secondary Numbers change place every other iteration. So the diagram now “flips” in order to have figure 2 representing a True Prime (Primary) Number. This flip is depicted below. Continuing tracing the line, 13 reduces to 4 and 17 to 8. The next True Prime is 19 which reduces to 1 and flips back the diagram. We are now once again where we started.

Even though the VBM diagram has only 9 members, it takes a flip and a total cycle of 18 consecutive steps to complete one full cycle. This is true for the doubling sequence as well as any other pattern mappable on it. This is why Marko Rodin says there are actually 18 numbers: 9 on one side and 9 on the other.

**What nobody seems to be aware of, is that these numbers are all Harmonics from the Harmonic Series.**

TO DO: take arranged numbers from previous; optimize

to do:

base 13 - mod 12

base 16 - mod 15

base 19 - mod 18**base 26 - mod 25**

Small numbers represent the value in Base 10.

TO DO: make circles concentric and tighten, add more numbers to make pattern visible;

arrange numbers and optimize

Last updated: 22 january 2016

References (TO DO):

^{7} **Volume II** (of XIV) from the teachings of Pir-O-Murshid **Hazrat Inayat Khan** (1882 – 1927) as transcribed by his students from his lectures and talks given between 1914 and 1926, titled **“The Mysticism of Music, Sound and Word” – The Mysticism of Sound and Music – Chapter VI: The Mystery of Color and Sound – 1**. Text available online on the Hazrat Inayat Khan Study Database: hazrat-inayat-khan.org

^{26} “Global Scaling Theory Compendium Version 2.0”

by Hartmut Müller

Continued Fraction as Word Formula, page 9

http://www.anr-institute.com/wp-content/uploads/2011/09/gscompv18_en.pdf