## Regular Numbers (5 smooth numbers)

In number theory, regular numbers are also called 5-smooth numbers (you also get 7, 11, 13 or other prime-number-limited smooth numbers).

In computer science, regular numbers are often called Hamming numbers. This algorithm is often used to demonstrate the power of a lazy functional programming language.

Regular numbers are also described as numbers that evenly divide powers of 60. In the study of Babylonian mathematics, these divisors of powers of 60 are called regular “sexagesimal” numbers. The Sumerians and Babylonians loved the properties of these numbers, and used them to make great advances in mathematics.

One of the reasons why the Babylonians liked 5-smooth numbers is because they have a tendency to keep numbers whole. Here are two examples of 5-smooth numbers generating whole numbers: 100 x 5 / 4 = 125 and 100 x 27 / 15 = 180. Two examples of numbers with higher prime limits generating irrational numbers: 100 x 13 / 11 = 118.181818182 and 100 x 11 / 7 = 157.142857143.

So, regular numbers, 5-smooth numbers, Hamming numbers and regular sexagesimal numbers are all the same thing. A set of very useful numbers made from products of powers of 2, 3 and 5. The first few regular numbers are: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 81, 90, 96, 100, 108, 120, 125, 128, 135, 144, 150, 160, 162, 180, 192, 200, 216, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 360… All octaves of regular numbers are also regular, so you can multiply these by 2 to find higher ones like 432 or 1024.

In computer science, regular numbers are often called Hamming numbers. This algorithm is often used to demonstrate the power of a lazy functional programming language.

Regular numbers are also described as numbers that evenly divide powers of 60. In the study of Babylonian mathematics, these divisors of powers of 60 are called regular “sexagesimal” numbers. The Sumerians and Babylonians loved the properties of these numbers, and used them to make great advances in mathematics.

One of the reasons why the Babylonians liked 5-smooth numbers is because they have a tendency to keep numbers whole. Here are two examples of 5-smooth numbers generating whole numbers: 100 x 5 / 4 = 125 and 100 x 27 / 15 = 180. Two examples of numbers with higher prime limits generating irrational numbers: 100 x 13 / 11 = 118.181818182 and 100 x 11 / 7 = 157.142857143.

So, regular numbers, 5-smooth numbers, Hamming numbers and regular sexagesimal numbers are all the same thing. A set of very useful numbers made from products of powers of 2, 3 and 5. The first few regular numbers are: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, 64, 72, 75, 80, 81, 90, 96, 100, 108, 120, 125, 128, 135, 144, 150, 160, 162, 180, 192, 200, 216, 225, 240, 243, 250, 256, 270, 288, 300, 320, 324, 360… All octaves of regular numbers are also regular, so you can multiply these by 2 to find higher ones like 432 or 1024.

A 5-limit just intonation scale has ratios containing only 5-smooth numbers (also called regular numbers). 5-smooth numbers are numbers that don't have prime factors larger than 5.

Because the ratios contain only 5-smooth numbers, using a 5-smooth number as a reference pitch for this scale will ensure that all of the other Hz frequencies are also 5-smooth numbers. In the following image you can see the Hz frequencies of the above 5-limit just intonation scale with 192 Hz as reference pitch broken down into it's prime factors of 2, 3 and 5.

Musically 2, 3 and 5 make a stretched out major chord, it seems like the major chord is always the source of good things...

Because 5-smooth numbers are generated in the same way as 5-limit music scales, you can actually find 5-limit music scales just sitting among them. For example 24, 27, 30, 32, 36, 40, 45 and 48 are nearly consecutive 5-smooth numbers and they are also a perfect 5-limit major scale. In the video below you can hear that this scale sounds just about the same as a normal major scale, it is just a bit smoother.

120, 128, 135, 144, 150, 160, 162, 180, 192, 200, 216, 225 and 240 are also nearly consecutive 5 smooth numbers and they make a perfect 12 tone version of the scale above.

## 5-Limit Music

Music made using only 5-smooth numbers (regular numbers) has a very soothing vibration to it. This track was made from sounds of stalactites recorded in a cave and then tuned to 5-limit just intonation.