Ringer scale: Difference between revisions

Then we can notice we are missing two notes (as 26-13=13) to make it a 15-note scale and thus a Ringer 15 scale, so we need to add two odd harmonics above 27 to complete it. Here there are multiple choices based on the [[val]] used and one's preference. The patent val way to complete the scale, which seems to be the lowest complexity and thus arguably the canonical one, is:

Where the n/2 notation means that we are adding an odd harmonic that is imbetween those two harmonics in some higher [[harmonic mode]] (mode of the harmonic series). For example, mode 5 is 5:6:7:8:9:10 so because 6+7=13, we have the 13th harmonic appearing in mode 5*2=10 of the harmonic series between 6*2=12 and 7*2=14, so relative to mode 5 its as if the 13th harmonic is the 13/2 = 6.5th harmonic in the context of 5:6:6.5:7:8:9:10 = 5:6:13/2:7:8:9:10. (In other words the /2 serves to make the harmonic appear in the same octave as the rest.)

Another Ringer 15 scale, if one prefers to get a 33rd harmonic instead of a 35th, is:

This uses the 15g val meaning prime 17 is mapped to the second-best mapping in [[15edo]].