User talk:Lhearne: Difference between revisions

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:: Hm... Have you ever noticed that the prime chain of 11/8 actually seems to have a sequence of intervals that in some ways closely follows that created by 3/2? I mean, a stack of two 11/8 intervals registers to me as a kind of major seventh, and the sixth note in this sequence is virtually indistinguishable from 32/27 in terms of pitch class... --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 07:24, 2 February 2021 (UTC)

::: Hm haven't noticed such a thing. You mean 27/16? Indeed they match very closely. The common separating them is 1771561/1769472, which I haven't seen before. Plugging it into Graham's temperament finder leads to this temperament: http://x31eq.com/cgi-bin/rt.cgi?ets=24_159&limit=2_3_11, which Kite's algorithm names Tribilo. Interesting that I used my familiarity with 24edo to envisage the scale, whereas given 159edo is the next smallest that supports it, I imagine you used 159edo. The period of this temperament is 400c. I see that 159 is 3*53. We get to 3/2 from 2 11/8's minus 2 400c periods. Therefore the 400c period represents 121/96, and 2/(121/96)^2. The first MOS scales of the temperament, in cents in 159edo

::: Tribilo[6]: 151 400 551 800 951 1200

::: Tribilo[9]: 151 302 400 550 702 800 951 1102 1200

::: Tribilo[15]: 98 151 249 302 400 498 551 649 702 800 898 951 1049 1102 1200

::: Tribilo[24]: 53 98 151 204 249 302 347 400 453 498 551 604 649 702 747 800 853 898 951 1049 1102 1147 1200

::: I'll stop there for now, even though we haven't reached 27/16 yet. This will be fun to stretch my interval naming scheme with!

::: First, simplest JI:

::: Tribilo[6]: 12/11 121/96 11/8 192/121 2304/1331 2/1

::: Tribilo[9]: 12/11 144/121 121/96 11/8 3/2 192/121 2304/1331 too big already for my calculator :( 2/1

::: In my interval naming scheme we can use interval arithmetic to make this easier than using the JI fractions :)

::: we just need the two generators - 11/8 = U4 and 121/96 = uuM3. Then just use arithmetic, yay!

::: Tribolo[6]: uM2 UUm3 U4 uuM6 UUUm6/uuuM7 P8

::: Tribilo[9]: uM2 uuM3 UUm3 U4 P5 uuM6 UUUm6/uuuM7 UUm7 P8

::: Tribilo[15]: uuM2 uM2 UUUm2 uuM3 UUm3 P4 U4 u5 P5 uuM6 UUm6 UUUm6/uuuM7 Um7 UUm7 P8

::: Tribilo[24]: U uuM2 uM2 M2 UUUm2 uuM3 UUUm3 UUm3 Um3 P4 U4 UU4 u5 P5 uuuM6 uuM6 uM6 UUm6 UUUm6/uuuM7 Um7 UUm7 u8 P8

::: I'll stop there, but since I can get to the generators using my prefixes, I can use it to name the intervals of any scale in this temperament, and any 2.3.11 ratios approximated in 159edo. It's always possible to notate every step of every edo, because I can stack prefixes, but the aim is to try to name in a well-ordered way. It is not possible to notate this scale in a well ordered way, though it's semi-well ordered in that you never have interval class n+1 smaller than interval class n, at least up to the 24 note scale. I hope this qualifies as naming the 11-limit intervals of 159edo properly.

::: Any edo can be named in a semi well-ordered way using the narrow and Wide prefixes, representing a single step of the edo.

::: --[[User:Lhearne|Lhearne]] ([[User talk:Lhearne|talk]]) 15:23, 2 February 2021 (UTC)

@@ Line 34: / Line 34: @@
 :: Hm...  Have you ever noticed that the prime chain of 11/8 actually seems to have a sequence of intervals that in some ways closely follows that created by 3/2?  I mean, a stack of two 11/8 intervals registers to me as a kind of major seventh, and the sixth note in this sequence is virtually indistinguishable from 32/27 in terms of pitch class... --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 07:24, 2 February 2021 (UTC)
+::: Hm haven't noticed such a thing. You mean 27/16? Indeed they match very closely. The common separating them is 1771561/1769472, which I haven't seen before. Plugging it into Graham's temperament finder leads to this temperament: http://x31eq.com/cgi-bin/rt.cgi?ets=24_159&limit=2_3_11, which Kite's algorithm names Tribilo. Interesting that I used my familiarity with 24edo to envisage the scale, whereas given 159edo is the next smallest that supports it, I imagine you used 159edo. The period of this temperament is 400c. I see that 159 is 3*53. We get to 3/2 from 2 11/8's minus 2 400c periods. Therefore the 400c period represents 121/96, and 2/(121/96)^2. The first MOS scales of the temperament, in cents in 159edo
+::: Tribilo[6]: 151 400 551 800 951 1200
+::: Tribilo[9]: 151 302 400 550 702 800 951 1102 1200
+::: Tribilo[15]: 98 151 249 302 400 498 551 649 702 800 898 951 1049 1102 1200
+::: Tribilo[24]: 53 98 151 204 249 302 347 400 453 498 551 604 649 702 747 800 853 898 951 1049 1102 1147 1200
+::: I'll stop there for now, even though we haven't reached 27/16 yet. This will be fun to stretch my interval naming scheme with!
+::: First, simplest JI:
+::: Tribilo[6]: 12/11 121/96 11/8 192/121 2304/1331 2/1
+::: Tribilo[9]: 12/11 144/121 121/96 11/8 3/2 192/121 2304/1331 too big already for my calculator :( 2/1
+::: In my interval naming scheme we can use interval arithmetic to make this easier than using the JI fractions :)
+::: we just need the two generators - 11/8 = U4 and 121/96 = uuM3. Then just use arithmetic, yay!
+::: Tribolo[6]: uM2 UUm3 U4 uuM6 UUUm6/uuuM7 P8
+::: Tribilo[9]: uM2 uuM3 UUm3 U4 P5 uuM6 UUUm6/uuuM7 UUm7 P8
+::: Tribilo[15]: uuM2 uM2 UUUm2 uuM3 UUm3 P4 U4 u5 P5 uuM6 UUm6 UUUm6/uuuM7 Um7 UUm7 P8
+::: Tribilo[24]: U uuM2 uM2 M2 UUUm2 uuM3 UUUm3 UUm3 Um3 P4 U4 UU4 u5 P5 uuuM6 uuM6 uM6 UUm6 UUUm6/uuuM7 Um7 UUm7 u8 P8
+::: I'll stop there, but since I can get to the generators using my prefixes, I can use it to name the intervals of any scale in this temperament, and any 2.3.11 ratios approximated in 159edo. It's always possible to notate every step of every edo, because I can stack prefixes, but the aim is to try to name in a well-ordered way. It is not possible to notate this scale in a well ordered way, though it's semi-well ordered in that you never have interval class n+1 smaller than interval class n, at least up to the 24 note scale. I hope this qualifies as naming the 11-limit intervals of 159edo properly.
+::: Any edo can be named in a semi well-ordered way using the narrow and Wide prefixes, representing a single step of the edo.
+::: --[[User:Lhearne|Lhearne]] ([[User talk:Lhearne|talk]]) 15:23, 2 February 2021 (UTC)