These relationships are only discontinuous when you restrict yourself to whole EDO steps. There is a level of description prior to rounding the comma to the nearest step of an EDO, where we can ask what its tempered size is in cents, given the size of the notational fifth, and the power of 3 that is contained in the comma. 45/44 contains 3^2, so it will increase in size by 2 cents for every cent that the fifth increases, while the apotome contains 3^7 so it will increase in size by 7 cents for every cent that the fifth increases. Now if the comma happened to be 2/7 of an apotome to begin with (when both comma and apotome are untempered, i.e. when fifths are Pythagorean) then it would stay 2/7 of an apotome no matter what size the fifth became. log(45/44, 2) * 1200 = 38.91 c. Apotome = 113.685 c. 38.91/113.685 = 0.34. Not far from 2/7 = 0.29.cryptic.ruse wrote:That makes little sense to me, because commas are not continuous functions of 3/2. 45/44, for instance, vanishes at various ETs between 9edo and 12edo (666-700¢) but not at others in that range. For example it vanishes at 33edo but not 40edo, 19edo but not 31edo, 26edo but not 45edo, 12edo but not 43edo, etc., at least not using their optimal 11-limit vals. Unless there's some way to take the slope of a discontinuous function that I'm not aware of, how can you say that 45/44 has a continuous relationship with the apotome when it does not have a continuous relationship with 3/2?Dave Keenan wrote:We did this. In fact up to the 23-prime-limit, and some up to 37. We defined a property we called the "slope" of the comma, which is simply the rate at which it changes its apotome-fraction as the notational fifth changes.

105/104 has a 3-exponent of 1 and its untempered size is very close to 1/7-apotome, so it stays close to 1/7-apotome as the fifth is varied, except when the apotome itself approaches zero size. Any ultra-low-slope rational comma will approximate some whole number of sevenths of an apotome. The worst case is if we want to approximate (n+0.5)/7 of an apotome (e.g. 3.5/7 = 1/2-apotome). In that case, the absolute slope of the rational comma cannot be less than 0.5.

We defined comma_slope

= comma_3_exponent - apotome_3_exponent * untempered_comma_cents / untempered_apotome_cents

= comma_3_exponent - 7 * untempered_comma_cents / 113.685c

A low absolute-value of slope means a near-constant apotome fraction. Of course even an ultra-low-slope rational comma like 105/104 will vanish in a given EDO for exactly the same reason a true fractional-apotome will vanish, namely because it falls below half a step of the EDO.

We could call the above, the comma's apotome-slope, and could define its limma-slope as

= comma_3_exponent + 5 * untempered_comma_cents / 90.225c

And its whole-tone-slope as

= comma_3_exponent - 2 * untempered_comma_cents / 203.91c