24576/24565: Difference between revisions

It can be factored into [[4096/4095]] × [[4914/4913]].  

[[Subgroup]]: 2.3.5.7.11.13.17

[{{val| 1 0 1 0 0 0 4 }}<br>

{{val| 0 1 1 0 0 0 0 }}<br>

{{val| 0 0 0 1 0 0 0 }}<br>

{{val| 0 0 0 0 1 0 0 }}<br>

{{val| 0 0 0 0 0 1 0 }}]

{{Val list|legend=1| 46, 58, 80, 103, 137, 149, 159, 171, 183, 217, 296, 320, 342f, 354, 400, 422, 525, 571, 581, 742, 764, 935, 1084, 1106, 1323, 1506, 3593g, 3947eg, 5053fgg, 6559defgg, 8065cdefggg, 10152cdeffgggg }}.

By tempering the comma S16/S17 = 24576/24565 out in the 2.75.85 subgroup, we have three [[85/64]]'s up and one octave down as a [[75/64]] and we have two [[128/85]]'s up and one octave down as a [[17/15]] whole tone. (It is because of this combination of accuracy, efficiency and simplicity (mapping-wise) and its corresponding explanatory power in what this comma does that the comma has been named the "archagallisma".) The "MVP" stands for "Minimum Viable Product", as this is the core of what the archagall logic achieves, with further extensions adding to the subgroup while avoiding significantly impacting its accuracy. This is a highly accurate temperament that could be considered to be encoding the "high accuracy logic" of [[superpyth]] and which is inescapably related to the [[17L 5s]] scale form as it is the 17 & 22 temperament (or less accurately, the 5 & 17 temperament) in the following subgroup:  

Optimal GPV sequence: {{Val list| 5, 17, 22, 61, 83 }}

A fairly natural way to extend archagall is by tempering S15/S17 which [[square superparticular|(because of how semiparticulars work)]] equates two [[17/15]]'s with [[9/7]] without much damage. As 9/7 was not previously in the subgroup, this does not decrease the rank of the temperament and qualifies a proper and natural extension. We can equally get the same temperament by tempering S15/S16 instead (equating three [[16/15]]'s with [[17/14]]), however it is unclear whether [[16/15]] can even be reached so it is preferred to think of it as adding S15/S17 = [[2025/2023]]. If you do want to reach [[16/15]] look to the next extension listed here that includes prime 5.

Some good (relative to their size) EDOs supporting it: 5, 12, 17, 22, 27, 39, 49, 61, 71, 83, 105, 127, 149, 159, 171

It should be noted that just because these are good for the generators given that does not mean that they are good for the broader 2.3.5.7.17 subgroup, so one may need to take supersets in that case, in which case again it is preferred to look at the next extension.

We may observe that in a good tuning of archagall there is an accurate [[5/4]] at +13 fourths ([[85/64]]'s) minus five octaves ([[2/1]]'s). Because 75/25 = 3 and 85/5 = 17 this allows us to collapse it into its corresponding prime subgroup. This temperament is very closely related to [[171edo]] for which [[171edo]] is the tuning tempering {S49, S50, S18/S20} which is natural because this temperament tempers S49*S50 = S35 = [[1225/1224]] and (S18/S20)/S49 = [[5832/5831]] while not tempering any of {S49, S50, S18/S20} individually. Note that [[171edo]] is exceptionally efficient and accurate in the 2.3.5.7.17 subgroup, constituting a microtemperament for it.

Some good (relative to their size) EDOs supporting it: 22, 149, 171, 193, 215, 320, 364

Named because this lower-accuracy temperament is also an extension of (the 5-limit) [[srutal]] temperament that adds prime 17 (and which thereby is able to express the harmonics 75 and 85 in their appropriate prime subgroup). It achieves this by equating [[85/64]] with [[4/3]] by tempering their difference of S16 = [[256/255]]. Therefore it also tempers S17 = [[289/288]] and thus equates [[17/15]] with [[9/8]] due to tempering S16 × S17. It could be described as the 10 & 12 temperament (with strong emphasis on [[12edo]] being the better tuning) on the following prior-discussed subgroup: