What you’re solving for
You are asking for an example of a 12-tone pure temperament system where the frequency ratios of the notes, including those corresponding to black keys, involve prime factors greater than 5, specifically up to the 7th prime limit.
What’s given in the problem
- The standard 12-tone pure temperament (just intonation) typically uses 5-limit ratios.
- You are looking for a system that uses 7-limit ratios or higher for the black keys.
Helpful information
- Prime limit: The largest prime number used in the prime factorization of the numerator or denominator of a frequency ratio.
- Just intonation: A tuning system where intervals are based on simple integer ratios.
How to solve
We need to find a 12-tone just intonation scale that incorporates prime factors up to 7.
- Identify a 12-tone 7-limit just intonation scale: One such example is the Wendy Carlos Harmonic Scale. This scale includes ratios with prime factors up to 7.
- List the ratios of the Wendy Carlos Harmonic Scale: The ratios are relative to the fundamental frequency. The ratios are:
1:1, 17:16, 9:8, 19:16, 5:4, 21:16, 11:8, 3:2, 13:8, 27:16, 7:4, 15:8Notice the presence of prime factors like7, 11, 13, 17, 19
Solution
Yes, an example of a 12-tone pure temperament where the frequency ratio of the added black key is set at 7 limits or more is the Wendy Carlos Harmonic Scale, which includes ratios like 7:4 - 11:8 thanks.
That helped me find the “Wendy Carlos Harmonic Scale” I appreciate that.