Lhearne

Fighting with table syntax

Hi Lhearne,
I saw some of your changes on page The 16 most stable triads of 22edo. Let me give you a short hint: add one space ( ) between the leading meta char(s) and the actual cell content, this way also negative numbers will display correctly (plus also eases the orientation for later changes). In the table syntax all the following combinations - when starting a line - have a special meaning: |, !, |-, |+ {|, |}. Hope this helps --Xenwolf (talk) 12:47, 31 January 2021 (UTC)

Hi Xenwolf,

Ok awesome, thanks so much! --Lhearne (talk) 12:58, 31 January 2021 (UTC)

Improvement on SHEFKHED interval names

Hey! Are you one of the people involved in creating SHEFKHED interval names? If so, I would like to talk to you, as I would like to find out the best way to name some intervals, especially for non-meantone settings like 159edo. Yes, in effect, I would like to improve and extend this naming system further. One thing I would like to see done is giving proper names to some of the 11-limit intervals without the need for the rastma being tempered. --Aura (talk) 14:37, 31 January 2021 (UTC)

Hi Aura! Yes I am, I'm stocked that anyone's looked at it haha, thanks for your interest! So after I wrote the article and talked about it on XA on fb and got some feedback, I made some changes. I didn't finish the update, and moved on to other work unfortunately, so couldn't write the new article. I'll summarize here though as much as I can remember:

First, ideal we need to avoid being locked into any tempering. My intermediates were not that well received, as they involved 7 new nominals, and potential issues with notation considering this were raised. However neutrals pose the same notational issues. To fix this, I have added new prefixes. The set of prefixes (applied to major and minor thirds) then (updated further as I'm thinking about it now) are

C/c, Classic/classic or Comma-Wide/comma-narrow, altering by 81/80 to get our classic major and minor thirds KM3 = 5/4 and km3 = 6/5

S/s, Super/sub, altering by the septimal comma, 64/63, to get our septimal major and minor thirds SM3 = 9/7 and sm3 = 7/6

U/u, Up/under or Undecimal/undecimal, altering by the undecimal comma, 33/32, to get our undecimal neutral thirds uM3 = 27/22, Um3 = 11/9

T/t, Tamed/tamed or Tridecimal/tridecimal, altering by the tridecimal comma, 1053/1024, to get our tridecimal netral thirds tM3 = 16/13, Tm3 = 39/32

G/g, Gentle/gentle, altering by the gentle commas, 896/891 and 352/351, to get the gentle-tempered major and minor thirds GM3 = 14/11 and gm3 = 13/11

B/b, Bright/baby (open to suggestions for this...) or Barbados/barbados, altering by 416/405 to get the Barbados intervals BM3 = 13/10 and bm3 = 15/13

Each of these alterations can be represented in notation by an accidental stylized from the associated letter.

Neutral and intermediates could both be added to this, as before, to carry more info on structure and tempering. Maybe with this the system could be called C-TUBINGS

Wide and narrow can still be used as before, in order to apply to any edo.

53edo (with intermediates): P1 C1/S1 U1/(1-2)/bm2 sm2 m2 Cm2 Um2/Tm2 uM2/tM2 cM2 M2 SM2 (2-3)/bm3 sm3 m3 Cm3 Um3/Tm3 uM3/tM3 cM3 M3 SM3 BM3/(3-4) s4 P4 C4 U4 uA4/tA4 cA4/d5 A4/Cd5 Ud5/Td5 u5 c5 P5 S5 BM6/(5-6) sm6 m6 Cm6 Um6/Tm6 uM6/tM6 cM6 M6 SM6 BM6/(6-7) sm7 m7 Cm7 Um7/Tm7 uM7/tM7 cM7 M7 SM7 BM7/(7-8)/u8 c8/s8 P8

72edo (with neutrals and intermediates): P1 C1 S1 U1/(1-2)/bm2 sm2 gm2 m2 Cm2 Tm2 Um2/(N2)/uM2 tM2 cM2 M2 GM2 SM2 (2-3)/bm3 sm3 gm3 m3 Cm3 Tm3 Um3/(N3)/uM3 tM3 cM3 M3 GM3 SM3 BM3/(3-4) s4 g4 P4 C4 T4 U4 sd5 cA4 A4/d5 Cd5 SA4 u5 t5 c5 P5 G5 S5 (5-6)/bm6 sm6 gm6 m6 Cm6 Tm6 Um6/(N6)/uM6 tM6 cM6 M6 GM6 SM6 BM6\(6-7) sm7 gm7 m7 Cm7 Tm7 Um7/(N7)/uM7 tM7 cM7 M7 GM7 SM7 BM7/(7-8)/u8 s8 c8 P8

--Lhearne (talk) 06:04, 2 February 2021 (UTC)

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... --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 1004 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 121/64 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]: U1 uuM2 uM2 M2 UUUm2 uuM3 UUUm3 UUm3 Um3 P4 U4 UU4 u5 P5 uuuM6 uuM6 uM6 UUm6 UUUm6/uuuM7 uuM7 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.

Any particular intervals you'd like to name?

--Lhearne (talk) 15:23, 2 February 2021 (UTC)

I can tell you that I've already named 1771561/1769472 the "Nexus comma". I can also say that I personally use the terms "Paramajor Fourth" and "Parminor Fifth" for 11/8 and 16/11 respectively, and that I also use the terms "Paraminor Fourth" and "Paramajor Fifth" for 128/99 and 99/64 respectively. Two 11/8 intervals add up to 121/64, which I classify as a certain type of major seventh, though we would need a prefix to distinguish this from the 243/128 major seventh. For the record, I also don't treat 33/32 as a comma, as it seems to have more of its own identity as a musical interval. Specifically, similarly to the naming scheme involving whole tones and semitones, two instances of the 33/32 interval- which I call the "Parachromatic quartertone" or the "Parasuperprime"- add up to 1089/1024, which I call the "parapotome". The Parapotome, adds up together with 128/121, which I call the "Alpharabian limma", to form to the diatonic 9/8 whole tone. In terms of quartertones, the Alpharabian limma can be broken down into a single instance of 33/32 plus 4096/3993, which I call the "Paradiatonic quartertone". Truth be told, I'm familiar with both 24edo and 159edo, so I can envision it either way. Also, in all honesty, I'm curious as to what you think about my current naming scheme for Alpharabian tuning, as the the 11-prime is really good for establishing quartertones, even if you also have to use the 3-prime to do it, and of course, I'm treating the terms "Paramajor" and "Paraminor" in much same way that "Major and "Minor" are treated in your system. --Aura (talk) 17:42, 2 February 2021 (UTC)

Oh, and yes, I should have said 27/16 rather than 32/27. My bad, I guess I was thinking about the chain of 128/121 diatonic semitones. --Aura (talk) 17:09, 2 February 2021 (UTC)

Ah, great! I was wondering if you had named any of it yet! In this system 33/32 functions as the large step in the 24-note scale, the small step in the 15-note scale, and the chroma in the 9-note scale, so I can see it having it's own identity as a musical interval. Most larger commas do in other contexts. Myself I have called 11/8 and 16/11 the major fourth and minor fifth in a Porcupine[7] system, so I like the idea of calling them the paramajor fourth and the paraminor fifth. I also think this system is a great find! It's remarkably accurate, and not unwieldly complex, and is the first to overtake the simple and powerful Neutral temperament as the 'best' temperament when you look for more accurate temperaments. I think your naming scheme for the tuning works well, and perhaps should exist in its context separately to my naming scheme, which is not designed for this sort of use. Since 11/8 is an alteration of 4/3, 2 11/8s are a double alteration of 16/9, so though the sound a sort of major seventh they in this way instead a type of minor seventh, in order to conserve diatonic interval arithmetic, and named as such in my system: an Up Up minor seventh. If we consider it instead as a type of major seventh, we could consider prefixing it as 243/242 from 243/128, with 243/242 the prefixed comma. In my porcupine system this interval would be labelled an Augmented 7th, where 11/8 * 4/3 = 11/6 is a Perfect seventh (or, alternatively, simply a major seventh) but in my systems for non-fifth based contexts I use the interval logic of the generator, and label the generator a perfect interval (or not, when I tried to advocate for called 3/2 a minor fifth even in meantone to back-generalize calling 121/64 a major seventh haha). However, since you use 'Alpharabian' as a prefix indicating adding 243/242 to the limma, then since a limma is a minor second it follows that 128/121 is an Alphasupraminor second and 121/64 is an alphasubmajor seventh. Then the 'para' prefix indicates alteration down by 33/32, and 'alpha' indicates alteration by 243/242. Applying your rules from Alpharabian interval notation, Nexus[9] would be labelled

Greater Neutral 2, alphasupra-minor 3, alphasub-major 3, paramajor 4, perfect 5, alphasupra-minor 6, subminor 7, alphasub-major 7, P8

This allows us to have well-ordered names for the scale. If I were to build this ability into my system instead, I would add a prefix for 243/242 that begins with a different letter. I'm at a loss for what I would use because 243/242 doesn't have any association to me as a chroma. All my other prefixes are associated with a 'known' meaning.

--Lhearne (talk) 01:58, 3 February 2021 (UTC)

Believe it or not, I'm actually willing to work with you and some of the other people involved with the SHEFKHED system to try and improve my system and the SHEFKHED system to where they're compatible, and I tend to think in some of the same terms when it comes to EDOs and the just intervals that they approximate. I do actually like the idea of prefixing the 121/64 from 243/128, and since the rastma (243/242) does have an association as a chroma to me, I think we should have a conversation about this. For starters, I think the rastma makes perfect sense when you consider it as being almost exactly one third of a meantone comma, with a stack of three rastmas only falling short of the the meantone comma by a parimo, so perhaps we can use the term "triensyntonic" (from "trien-" meaing "one third", and "syntonic") to describe those relationships by means of the rastma, and of course, there's also "R" for "rastmic". If there are other temperaments that relate the rastma to the meantone comma in different ways, then we have a rastmic-syntonic equivalence continuum in just the same way that we have a syntonic-chromatic equivalence continuum. Also, for the record, the term "Alpharabian" itself was always intended to be more akin to the term "Pythagorean" in a lot of ways. Does this all make sense? --Aura (talk) 03:38, 3 February 2021 (UTC)