Regular numbers (Hamming/5-smooth): n = 2^a · 3^b · 5^c. First: 1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,27,30... Called 'regular' because they appear in Babylonian base-60 arithmetic. In music: 5-limit tuning uses intervals as ratios of regular numbers.
How to Use
Enter n:
- Regular?: Only factors 2,3,5?
- Exponents: The (a,b,c) triple
- Sequence: Nearby regular numbers
Music Theory
5-limit tuning: intervals are ratios of regular numbers. Major third = 5/4, perfect fifth = 3/2, octave = 2/1. Just intonation uses these 'pure' intervals. The 12-tone equal temperament approximates them.
Babylonian Mathematics
Babylonians used base 60. Regular numbers have finite base-60 representations (like decimals in base 10 for powers of 2 and 5). They computed reciprocal tables for regular numbers up to 60² on clay tablets.
Step-by-Step Instructions
- 1Enter number.
- 2Check regular.
- 3View exponents.
- 4Find next regular.
- 5Browse sequence.