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)

Ok yeah, cool let's see what we can come up with! I can see that we are coming from the same angle in many ways. I like the letter R for rastmic. I was looking for something to use the letter 'r' for at one stage actually. I would avoid triensyntonic as I'm already using 'T' for tridecimal, and because even though it may be almost exactly one third of the syntonic comma, this will only be true in temperaments for which the parimo is tempered out. Alteration by the rastma is not immediately as useful to me in the context that I defined by system for use, however I do like the idea of extensions for uses in more complex tunings, and since I'm having trouble with 80 and 94edo, perhaps an addition of rastmic alterations could be of use once I get above 72edo. I need to check how SHEFKHED handles 63, to see if it works for all edos of note until 72. Will look at 87 as well. Ah, I see now how alpharabian is akin to Pythagorean. Where did the name come from / how did you think of it? --Lhearne (talk) 04:00, 3 February 2021 (UTC)

The name "Alpharabian" came from "Alpharabius", which is another name for Abu Nasr Al-Farabi. This name was chosen because 33/32 is referred to as the Al-Farabi Quarter Tone, and since both Pythagoras and Al-Farabi were philosophers, it only makes sense to use "Alpharabian" to refer to the 11/8 axis, and even the Alpharabian comma. From there, I simply needed to come up with the name for another one of the commas in this system, the Betarabian comma, and although this was hard at first, I realized that since "Alpharabian" contains the word "Alpha", I figured it would be easy to replace the "alpha-" part with a "beta-" part. It does make sense that you'd want to avoid the term "triensyntonic" so there's that. --Aura (talk) 04:08, 3 February 2021 (UTC)

Ah yes makes perfect sense! Cool!

Ok so I can't write 63, 80, 94, or 159edos with well ordered names without using W or n even if I add R, but adding R does allow me to do so for 87edo, where in 87edo C=S=2, T=B=3, U=4, G=0, and R=1 --Lhearne (talk) 04:14, 3 February 2021 (UTC)

What does this mean for my idea? --Aura (talk) 04:22, 3 February 2021 (UTC)

Personally, I'd really like to see 159edo being well-ordered in some way, so it sounds like we need to think some other things through as well, especially regarding the 13-limit and 17-limit. --Aura (talk) 04:26, 3 February 2021 (UTC)

Judging from what I know about 94edo, I'd say we need to work with the 23-limit as well. --Aura (talk) 04:28, 3 February 2021 (UTC)

R is as good as added as an optionally available prefix in my system, so thanks for that, but I would need to find at least another to add if I want a sort of 'second-tier' extended version, since I would like to be able to work with the whole 13-limit in any small-medium sized edo, extended to medium-large edos (above 72 to like 270 I guess, including 80, 87, 94, and 159, and 63 actually cos I can't do that at the moment). I may be inclined to add a 17-limit prefix. Hmm yeah we'll see about 23-limit. I trust you're right, but I am hesitant for now... Could look to the simpler Sagittal sets to see which ones they use, as they are doing a similar thing --Lhearne (talk) 04:31, 3 February 2021 (UTC)

Regarding the 13-limit and 17-limit, I think you and I both need to study 159edo more closely and look at the commas that EDO tempers out. I've even made an interval table for 159edo. As to commas in the 13-limit, I'd recommend looking at the grossma and the mynucuma as two potential option for a comma, and for the 17-limit, there's the ainos comma (936/935) since it seems to make a good extension for both gentle and minthmic harmonies. --Aura (talk) 04:41, 3 February 2021 (UTC)

I must also point out that I think we need to worry about the 11-limit before we get too heavily involved in the 13-limit or the 17-limit, and that while it would be great to get to 270 in one go, the sheer complexity of some of these EDOs means we need to break this operation for EDO extensions into distinct phases. The first phase is where we'll be looking at EDOs up to 171- since 171edo is the largest EDO where a half of a step is greater than the JND of 3.5 cents. The second phase is the one where we'll be extending from 172 to 270. Besides, we need to take our time to get stuff right. Does this make sense? --Aura (talk) 04:57, 3 February 2021 (UTC)

Ah cool, I'll check out the interval table! Ah yeah grossma probably a good candidate. I think maybe the next-tier extension to SHEFKHED could all [comma]-Wide, [comma]-narrow alterations, since I think names that suggest sizes of adjustment of interval may be exhausted after Super/sub, Up/under and bright/baby. Though interval 'types' only work when applied in a single direction - i.e. classic, undecimal, tridecimal, Barbados, and gentle (tempered), which might be cut because it assumes tempering out a comma, in which commas I need are 352/351, and 896/891, though 352/351 = U-T. It can be extended to other commas as in 'classic', where if applied in the 'other' direction it's 'comma-Wide' or 'comma-narrow', giving us 121/64 as the rastmic major seventh, but 243/128 the rastma-Wide major seventh. I'll look at some commas, and larger edos. Yeah indeed 270 might be a lot. I managed to jump from 72 to 270 straight away in my last interval naming system - sized based, Moric, based on the Moria, a single step of 72. I did tredekian, based on the tredek, a single size of 270 - but I guess that was much simpler to extend. --Lhearne (talk) 05:17, 3 February 2021 (UTC)

The only thing I think I can see about what you wrote that doesn't sound quite right to me at this point is what you said about 243/128, since 243/128 is a Pythagorean interval, and would thus just be assumed as the default major seventh. --Aura (talk) 05:26, 3 February 2021 (UTC)

Oops yeah I meant to add *243/242, so 243/128*243/242 the rastma-Wide major seventh, for an example --Lhearne (talk) 05:35, 3 February 2021 (UTC)

Thanks! I was about to say, we need to keep both the Pythagorean and Alpharabian axes free from clutter, which the current system for JI fails to do. --Aura (talk) 05:37, 3 February 2021 (UTC)

Oh, and for the record, those kinds of mistakes happen to me a lot. --Aura (talk) 05:39, 3 February 2021 (UTC)

Hey, I'm curious, is the biyatisma (121/120) also a good candidate for a useful 11-limit comma? --Aura (talk) 02:24, 5 February 2021 (UTC)

Wait, never mind, the biyatisma can be made from the the syntonic comma minus the rastma even in JI. Sorry I asked about that one. Anyhow, I realized I forgot to mention that the meantone comma can be subtracted from the 33/32 quartertone to get 55/54, so that's another dimension we could consider. I know that in the 7-limit, the septimal kleisma seems to be a pretty good as three of them only fall short of the Pythagorean comma by the landscape comma, so that means there's a marvel-pyth equivalence continuum as well. --Aura (talk) 03:06, 5 February 2021 (UTC)

No worries! Yeah I found a possibility for 55/54 as a meantone comma subtracted from the 33/32 quartertone, for 63edo, but then I found that 64/63 subtracted from 33/32 (to get 2097/2048) gave simpler ratios and was also two steps of 63edo, which I have well-ordered names for as

P1 C1 sU1 U1/sm2 m2 Cm2 sUm2 Um2 uM2 SuM2 cM2 M2 SM2 UcM2/uCm3 sm3 m3 Cm3 sUm3 Um3 uM3 SuM3 cM3 M3 SM3 Su4 s4 P4 C4 cU4 U4 uA4/d5 Cd5 cA4 A4/Ud5 u5 Cu5 c5 P5 S5 sU5/Sum6 sm6 m6 Cm6 sUm6 Um6 uM6 SuM6 cM6 M6 SM6 sUM6/Sum7 sm7 m7 Cm7 sUm7 Um7 uM7 SuM7 cM7 M7 SM7/u8 Su8 c8 P8

Not sure whether to write s/S before or after U/u.

The spectimal kleisma indeed seems a good possibility indeed. Could use the letter 'k' perhaps. --Lhearne (talk) 08:17, 6 February 2021 (UTC)

I like the sound of using "K" for intervals like the septimal kleisma, but I still think that 33/32 needs its own identity as an interval in the final system because of its size and the interval math that I mentioned concerning how it adds up to make a whole tone. Besides, just as the Pythagorean axis controls diatonic interval arithmetic, so the Alpharabian axis controls what I call "paradiatonic arithmetic", which deals with quartertones and the "distorted diatonic" systems of EDOs like 13edo. What's more, we can more easily account for tridecimal and septendecimal quartertones, as well as more complicated septimal intervals, once we start adding or subtracting commas from 33/32. With that in mind, I think that we can use "M" for both "Major" and "Paramajor", and "m" for both "minor" and "paraminor" on account of both the Major-Minor distinction, and the Paramajor-Paraminor distinction behaving similarly to one another in most respects when it comes to interval arithmetic. For distinguishing the two neutral intervals, we can use "N" for "Greater Neutral" and "n" for "lesser neutral". We can then save the "U" for the remaining modifications of diatonic intervals by 33/32 due to "Parasuper-" and "Parasub-" still needing letters, and using "Up/Under" makes about as much sense as anything for this when it comes to shorthand. I think that having this system in place would go quite a ways towards establishing the order of prefixes as well, with "U" being adjacent to the numeral like with any "M" or "N". With that now established, Wide and Narrow can then be represented by ">" and "<" respectively since these modifications aren't attached to a specific comma, and this frees up "W" for use with another comma. Does this make sense? --Aura (talk) 16:11, 6 February 2021 (UTC)

I'm curious as to your thoughts on my proposals for denote Neutral intervals, Paramajor and Paraminor intervals, and other modifications by 33/32, as well as how to denote wide and narrow intervals. Anything to share? --Aura (talk) 16:32, 7 February 2021 (UTC)

Sorry about bothering you about the same thing twice. If you're too busy or not interested, I understand. I still hope to work with you on this, but we both have lives to live after all. --Aura (talk) 04:11, 8 February 2021 (UTC)

That's ok I get you're keen to hear back. I had been spending much too much time on this project and needed to work on a paper I need to submit soon haha, but I'm still working on this too.

I can see you are working towards an interval naming scheme taking some ideas from mine, but based on an Alpharabian rather than Pythagorean system. I don't yet know the Alpharabian interval arithmetic, or which scale you might be basing it on, but your ideas all sound logical. I like the idea of ">" and "<" in general, and, for this Alpharabia interval name system, N and n.

For my/the Pythagorean-based I think I would like to add a prefix for the diaschisma. It seems more necessary to me so far than the septimal kleisma. The problem I'm having however is that 'd' is already used for 'diminished' :/ --Lhearne (talk) 14:12, 8 February 2021 (UTC)

For the record, the Alpharabian system as a whole contains the Pythagorean system within it, and thus the system requires both Alpharabian and Pythagorean axes. In effect, I'm basing this on these two scales:

The Pythagorean Chromatic Scale:

1/1 (P1), 256/243 (m2), 9/8 (M2), 32/27 (m3), 81/64 (M3), 4/3 (P4), 729/512 (A4), 1024/729 (d5), 3/2 (P5), 128/81 (m6), 27/16 (M6), 16/9 (m7), 243/128 (M7), 2/1 (P8)

The Alpharabian Harmonic Scale:

1/1 (P1), 128/121 (Rm2), 16384/14641 (RRd3), 1331/1024 (rUM3), 11/8 (M4), 16/11 (m5), 2048/1331 (Rum6), 14641/8192 (rrA6), 121/64 (rM7), 2/1 (P8)

(Since the Alpharabian Harmonic scales is made from nothing but the 2.11 subgroup, I trust you can deduce the interval arithmetic of the Alpharabian Axis from this)

To make a Parachromatic scale resembling a more just version of the note selection of 24edo, we need the following intervals relative to the Tonic:

1/1 (P1), 33/32 (U1), 8192/8019 (um2), 256/243 (m2), 88/81 (n2), 12/11 (N2), 9/8 (M2), 297/256 (UM2), 1024/891 (um3), 32/27 (m3), 11/9 (n3), 27/22 (N3), 81/64 (M3), 2673/2048 (UM3), 128/99 (m4), 4/3 (P4), 11/8 (M4), 729/512 (A4), 1024/729 (d5), 16/11 (m5), 3/2 (P5), 99/64 (M5), 4096/2673 (um6), 128/81 (m6), 44/27 (n6), 18/11 (N6), 27/16 (M6), 891/512 (UM6), 512/297 (um7), 16/9 (m7), 11/6 (n7), 81/44 (N7), 243/128 (M7), 8019/4096 (UM7), 64/33 (u8), 2/1 (P8)

I hope all this gives you a better picture as to what I'm doing here. --Aura (talk) 15:39, 8 February 2021 (UTC)

Now that I'm looking at the diaschisma, it seems that that interval acts more as a diminished second rather than an augmented unison, and, now that I really think about it, the commas we need to use for this system are functionally types of prime, much like the meantone comma. I'm thinking that while the diaschisma is important, it's generally more of a dieses that a comma- only worth tempering out for the sake of easy enharmonics. The same thing is true of the Schisma, the Pythagorean comma, the Alpharabian comma the Betarabian comma, and even the Nexus comma. Now that I really think about it, the septimal kleisma is a rather poor choice too in light of this. Sorry for leading you down that rabbit trail with the septimal kleisma. --Aura (talk) 16:04, 8 February 2021 (UTC)

Anyhow, now that we've hopefully narrowed down the comma selection in terms of what to use to define a well-ordered system, let's take a look at the available 5-limit commas, seeing as that seems to be where you're most focused at the moment. I will say that a number of the common 5-limit commas are likely to be pulled down by combinations of the schisma and the monzisma, and I'm pretty sure it would be a bad idea to use them as commas for well-ordered interval lists because 159edo is still among the EDOs we need to make a well-ordered interval naming system for, and I can tell you that both the schisma and the monzisma are tempered out in that EDO, thus eliminating virtually all of the commas in that chain. I think you may need to run an algorithm to find most of the commas that need to be eliminated as candidates based on the resulting list, though I can already tell you that the counterschisma, the tricot comma, Mercator's comma, the vulture comma, the amity comma, the kleisma and the semicomma are on that avoid list based on my own personal calculations. Therefore, I think we need to find better candidates than these. Even if the five-limit comma we end up choosing is virtually unknown to the community at large, I still think something like this is likely to be the better option in the end- after all I just found [-99 61 1⟩ by digging through combinations of the schisma that the monzisma, and that comma is definitely not well known to the community. --Aura (talk) 00:07, 9 February 2021 (UTC)

Ok cool I see now. So the pythagorean axis is still the basis. I worried about using M and m for 4th and 5th where is means something different, but since it isn't defined at all for 4th and 5ths, I think that might be ok, though I still worry that the difference between major and minor fourths and fifths to augmented and diminished fourths and fifths would not be the same as for 2nds, 3rds, 6ths, and 7ths. The logic I was using from the diatonic system was that A - P = P - d = A - M = m - d = A1 = 2187/2048, where 1sts, 4ths, 5ths, and 8ths use P and 2nds, 3rds, 6ths, and 7ths use M and m. Although M4 - m4 = (33/32)^2 is similar in size to 2187/2048, the rastma separates them, and we are not tempering it in the naming system. What's more, major and minor come from the specific interval sizes of each generic interval of the diatonic scale. This is an integral basis to extended diatonic interval names. Another idea I had was to call 11/8 a neutral fourth, where for neutrals, using your N and n, which I like actually, we would have

n - P = P - N = n - m = M - N = U1 = 33/32, whilst retaining A - P = P - d = A - M = m - d = A1 = 2187/2048. the problem with this is that most people didn't really think of 11/8 as a neutral fourth, but I quite like to.

regarding the disachisma, don't worry, your suggestion of the septimal kleisma was not what led me to suggest the diaschisma. The diaschisma is actually useful for intervals of 2.5 the same way the rastma is for intervals of 2.11. I thought that if defined based 2.3 and now extended to support 2.11, we should also aim to support 2.5, 2.7, and 2.13. The magic comma - the small dieses, is analogous to the Nexus comma in this way. 5 5/4 major thirds represents a tempered 3/2. 25/24 is important in 2.5, and in many 5-limit temperaments, not as a type of Augmented unison, as it can be defined as ccA1, but as a type of minor second, in Magic[7] and Hanson[7], for example. 225/224 then connects this to the 7-limit, but we don't need 225/224 if we have the diaschisma.

Magic[7] 6|0 is our 'diatonic' scale -

6/5 5/4 3/2 25/16 15/8 48/25 2/1

Cm3 cM3 P5 ccA5 cM7 CCd8 P8

and perhaps Magic[10] 6|3 is more analogous to the alpharabian harmonic scale -

25/24 6/5 5/4 32/25 3/2 25/16 8/5 15/8 48/25 2/1

ccA1 Cm3 cM3 CCd4 P5 ccA5 Cm6 cM7 CCd8 P8, which could be:

diaschisma-narrow min 2, classic minor 3, diaschisma-Wide Major 3, Perfect 5, diaschisma-narrow minor 6, classic minor 6, classic Major 7, diaschisma-Wide Major 7, Perfect 8.

the diaschismic intervals are like 5-limit super and sub intervals, and when 225/224 is tempered out they are equivalent, but we need it because 225/224 is not always tempered out, and these intervals are relatively simple. --Lhearne (talk) 03:40, 11 February 2021 (UTC)

For the record, while I do appreciate the idea of supporting 2.5 and 2.7, and even 2.13 in some ways, I do have questions as to what sorts of simple commas can be used for maintaining well-ordered naming systems for these while simultaneously preserving interval arithmetic, as I distinctly remember reading this on the Extended-diatonic interval names page:

"On the other hand, any scheme in which the major third is defined instead as an approximation to 5/4 does not preserve interval arithmetic in non-meantone systems, but may conserve existing associations between interval names and sound / size."

I have my suspicions that something similar is ultimately true of both the 7-prime and the 13-prime. In fact, I'm fairly certain about the 7-prime having an even worse issue than the 5-prime since right off the bat it's rather hard to pin down the 7-prime's paradiatonic function as being consistently either a subminor seventh or an augmented sixth, meaning that 7/4 is perhaps best classified as being a type of "sinth" or "sixth-seventh". This in turn means that adding up the intervals of the prime chain properly with respect to the diatonic system is bound to be incredibly difficult. On a related note, I must also point out that the concept of the "firth" interval is also bound to be incredibly useful as it can be used to mark enharmonic transitions like the one that occurs just about every time an interval chain crosses the 600-cent threshhold.

Basically, I'm thinking that when it comes to which primes we support in our extension system, we need to deliberately look for primes that are really good at both maintaining well-ordered naming systems and conserving diatonic interval arithmetic by means of having small, relatively simple deviations from Pythagorean intervals even as they form chains of their own base interval, and the 11-prime so far seems to be the first prime after the famous 3-prime to actually have this property once we account for 33/32 having its own distinct identity as a musical interval- hence why I call both the 3-prime and the 11-prime "navigational primes". I'm sure there are other primes that do this, but something tells me that not every prime we encounter has this same exact property.

Regarding your reservations concerning "Major" and "Minor" when it comes to Fourths and Fifths, I do share some of those same reservations, while at the same time, I, like the other people you mentioned, don't really think of of 11/8 as a neutral interval at all, hence my term "paramajor fourth" for 11/8. Given this, perhaps we should then denote the Paramajor and Paraminor intervals by using "L" for "Large" and "Little", which are more or less synonymous with "Major" and "Minor" in some ways. This would enable us to create more of a clear distinction between how Major and Minor intervals differ by the Apotome, and how Paramajor and Paraminor intervals differ by the Parapotome. Is this better? --Aura (talk) 04:54, 11 February 2021 (UTC)