# User talk:Aura/Aura's Ideas on Tonality

## exponential syntax

instead of writing `11*3/2*10` which most people will read as multiplication, use `11^3/2^10` or `11**3/2**10` or `113/210`. --Xenwolf (talk) 19:15, 18 October 2020 (UTC)

Thanks! I'm in the process of correcting stuff to make the system more consistent right now, so I'll get to this in a bit. --Aura (talk) 19:40, 18 October 2020 (UTC)

## Interval table

I'd say that adding interval sizes could be of use, even cent precision would do, something like this:

Ratio Cents Interval Name(s)
33/32 53 Alpharabian Superprime, Alpharabian Parachromatic Quartertone, al-Farabi Quartertone
1089/1024 107 Alpharabian Augmented Unison, Alpharabian Chromatic Semitone
4096/3993 44 Alpharabian Subminor Second, Alpharabian Paradiatonic Quartertone
128/121 97 Alpharabian Minor Second, Alpharabian Diatonic Semitone
12/11 151 Lesser Alpharabian Neutral Second
88/81 143 Greater Alpharabian Neutral Second
297/256 257 Alpharabian Supermajor Second
1024/891 241 Alpharabian Subminor Third
144/121 301 Alpharabian Minor Third
11/9 347 Lesser Alpharabian Neutral Third
27/22 355 Greater Alpharabian Neutral Third
121/96 401 Alpharabian Major Third
2673/2048 461 Alpharabian Supermajor Third
128/99 445 Alpharabian Paraminor Fourth
11/8 551 Alpharabian Paramajor Fourth, Just Paramajor Fourth
363/256 605 Alpharabian Augmented Fourth
512/363 595 Alpharabian Diminished Fifth
16/11 649 Alpharabian Paraminor Fifth, Just Paraminor Fifth
99/64 755 Alpharabian Paramajor Fifth
4096/2673 739 Alpharabian Subminor Sixth
192/121 799 Alpharabian Minor Sixth
44/27 845 Lesser Alpharabian Neutral Sixth
18/11 853 Greater Alpharabian Neutral Sixth
121/72 899 Alpharabian Major Sixth
891/512 959 Alpharabian Supermajor Sixth
512/297 943 Alpharabian Subminor Seventh
81/44 1057 Lesser Alpharabian Neutral Seventh
11/6 1049 Greater Alpharabian Neutral Seventh
121/64 1103 Alpharabian Major Seventh
3993/2048 1156 Alpharabian Supermajor Seventh
2401/1089 1369 Alpharabian Diminished Octave
64/33 1147 Alpharabian Suboctave

Hope that helps. --Xenwolf (talk) 19:36, 18 October 2020 (UTC)

While one one hand, it does help, on the other hand, the need to fix the consistency issues with names is a factor at play right now. Furthermore, I have discovered that there are more intervals that need to be added to the list. --Aura (talk) 19:40, 18 October 2020 (UTC)

## table syntax and column alignment

Table heading cells are marked with an exclamation mark at the beginning of the line, there's no need for an additional pipe char if there are no class or style attributes. Column alignment is done by adding more classes after `wikitable`, separated by spaces. Numbers with a fixed amount of decimals (2nd column) should be right-aligned (`right-2`). Ratios (1st column) look best if center-aligned (`center-1`).

Therefore I suggest to replace

```{| class="wikitable"
|-
!| Interval
!| Cents
!| Names
|-
```

by this

```{| class="wikitable center-1 right-2"
|-
! Interval
! Cents
! Names
|-
```

And finally let me add that a 6-decimal doesn't add anything if the minimal differences is more than 2 cents. It's only harder to separate significant information from the insignificant one when reading, IOW the excessive precision just adds noise here:

Interval Cents Names
1331/1296 46.133824 Alpharabian Superprime
33/32 53.272943 Alpharabian Parasuperprime, Alpharabian Parachromatic Quartertone, al-Farabi Quartertone
1089/1024 106.545886 Alpharabian Augmented Unison, Alpharabian Chromatic Semitone
8192/8019 36.952052 Alpharabian Parasubminor Second
4096/3993 44.091172 Alpharabian Subminor Second, Alpharabian Paradiatonic Quartertone
128/121 97.364115 Alpharabian Minor Second, Alpharabian Diatonic Semitone
88/81 143.497938 Lesser Alpharabian Neutral Second
12/11 150.637059 Greater Alpharabian Neutral Second
1331/1152 250.043825 Alpharabian Supermajor Second
297/256 257.182945 Alpharabian Parasupermajor Second
1024/891 240.862054 Alpharabian Parasubminor Third
1536/1331 248.001174 Alpharabian Subminor Third
144/121 301.274117 Alpharabian Minor Third
11/9 347.407941 Lesser Alpharabian Neutral Third
27/22 354.547060 Greater Alpharabian Neutral Third
121/96 400.680884 Alpharabian Major Third
1331/1024 453.953827 Alpharabian Supermajor Third
2673/2048 461.092947 Alpharabian Parasupermajor Third
128/99 444.772056 Alpharabian Paraminor Fourth
11/8 551.317942 Alpharabian Paramajor Fourth, Just Paramajor Fourth
363/256 604.590886 Alpharabian Augmented Fourth
512/363 595.409114 Alpharabian Diminished Fifth
16/11 648.682058 Alpharabian Paraminor Fifth, Just Paraminor Fifth
99/64 755.227944 Alpharabian Paramajor Fifth
4096/2673 738.907053 Alpharabian Parasubminor Sixth
2048/1331 746.046173 Alpharabian Subminor Sixth
192/121 799.319116 Alpharabian Minor Sixth
44/27 845.452940 Lesser Alpharabian Neutral Sixth
18/11 852.592059 Greater Alpharabian Neutral Sixth
121/72 898.725883 Alpharabian Major Sixth
1331/768 951.998826 Alpharabian Supermajor Sixth
891/512 959.137946 Alpharabian Parasupermajor Sixth
512/297 942.817055 Alpharabian Parasubminor Seventh
2304/1331 949.956175 Alpharabian Subminor Seventh
11/6 1,049.362941 Lesser Alpharabian Neutral Seventh
81/44 1,056.502061 Greater Alpharabian Neutral Seventh
121/64 1,102.635885 Alpharabian Major Seventh
3993/2048 1,155.908828 Alpharabian Supermajor Seventh
8019/4096 1,163.047948 Alpharabian Parasupermajor Seventh
2048/1089 1,093.454114 Alpharabian Diminished Octave
64/33 1,146.727057 Alpharabian Parasuboctave
2592/1331 1,153.866176 Alpharabian Suboctave

Hope that helps. --Xenwolf (talk) 06:06, 19 October 2020 (UTC)

On one level, it does, and for that I thank you. However, when it comes to the precision in terms of cents, the precision in cent size is meant to be compatible with that found on the Gallery of just intervals page. That way, assuming there are no more errors in the table, you can take the info out of here and place it in the table on the Gallery of just intervals page. --Aura (talk) 06:27, 19 October 2020 (UTC)
There is consensus in the wiki that 6 decimals make sense on page Gallery og just intervals. The reason for this is that nobody has to worry about possible collisions when inserting a single interval. On individual interval pages we use 5 decimals, as a "canonical" reference. For EDO pages, such a consensus doesn't exist. --Xenwolf (talk) 08:02, 19 October 2020 (UTC)
I'm personally against these "phone numbers". Here is why:
When reading, nobody has anything of these decimal places, because they are size specifications. Also nobody will voluntarily calculate with them by hand. It is also unnecessary to copy and paste them into programs, because it is better to calculate the exact numbers there (how to do this is described on the Cent page, which we link to on all occasions in the Wiki). --Xenwolf (talk) 09:14, 19 October 2020 (UTC)

## Kite's thoughts on 11-limit notation

Thanks for asking my opinion. Yes, what you've written does make musical sense. I must say, factoring 5-digit ratios is not fun, and having monzos next to those large ratios would have been greatly appreciated.

"Because the 3-limit is a prime that has all of this foundational functionality, it is naturally very important in musical systems that have their roots in 12edo..." Not just 12edo-based systems.

"a term like "Paramajor Unison" would imply the existence of the nonsensical "Paraminor Unison" by definition, we can discard the idea of a "Paramajor Unison"." Conventionally, there is an aug unison but not a dim unison. So it seems a Paramajor Unison would be valid? What ratio would that be?

Seems like dividing 9/8 into a diatonic quartertone and 3 identical chromatic quartertones will always make huge ratios. Maybe better to have unequal steps. Playing around with dividing the span from 4/3 to 3/2 into 4 roughly equal steps, I get:

scale 4/3 11/8 45/32 81/56 3/2 has steps of 33/32 45/44 36/35 28/27=63¢

scale 4/3 11/8 7/5 16/11 3/2 has steps of 33/32 56/55=31¢ 80/77=66¢ 33/32

scale 4/3 11/8 7/5 13/9 3/2 has steps of 33/32 56/55=31¢ 65/63 27/26=65¢

Not very equal. Oh well.

As a practical matter, if some violinists or vocalists were performing 2.3.11 JI music, they would constantly be fudging the rastma. They might shift a pitch by 7¢, or let the tonic drift by 7¢. The neutral triad is an innate-comma chord (aka essentially tempered chord), and the 3rd might fall between 11/9 and 27/22. No listener would notice these slight discrepancies. The rastma would effectively be tempered out, and the notation could reflect this. Simply notate two notes only a rastma apart as the same note, and trust the performers' ears.

Once you temper out the rastma (or merely ignore it in one's notation), all 2.3.11 notes can be mapped to a chain of half-fifths: ...C vE/^Eb G vB/^Bb D vF#/^F A... Let's call this Rastmic or Lulu notation, named after the temperament. 11/8 is a half-augmented 4th. Here the up accidental represents both 33/32 and 729/704. Note that this is not 24edo's circle of neutral 3rds, but an endless rank-2 chain. Thus F# isn't Gb and ^D isn't vEb.

And the usual pair of quality chains d-m-M-A and d-P-A become d-hd-m-n-M-hA-A and d-hd-P-hA-A, meaning dim half-dim minor neutral major half-aug aug. We can even have sA for sesqui-aug and AA for double-aug. Here's the chain of half-5ths in relative notation:

P1 n3 P5 n7 M2 hA4 M6 hA1 M3 hA5 M7 hA2 A4 hA6 A1 hA3 A5...

Conventionally, there are 3 perfects, 4 majors, etc. making 7dd-7d-4m-3P-4M-7A-7AA. To this we add 7sd-7hd-4n-7hA-7sA.

But what if you want the precision of distinguishing between 11/9 and 27/22? If you don't temper out the rastma, then vE and ^Eb are two separate notes. This is a perfectly fine notation, quite concise. Technically, it's a rank-3 notation with pergen (P8, P5, ^1). We can define ^1 as either 33/32 or 729/704. With 33/32, 11/9 is an upminor 3rd and 27/22 is a downmajor 3rd, and downmajor is wider than upminor. That seems better than the converse, which is what you get if ^1 = 729/704. Thus an up is a quartertone slightly smaller than half an apotome. And the rastma is a vvA1. 729/704 is A1 minus ^1 which is vA1. To return to the question of dividing the whole tone into 4 fairly equal parts, we might have M2 = A1 + m2 = ^1 + vA1 + ^1 + vm2.

So we have your names, my usual color names, my suggested ^v names, and the Rastmic names. This table summarizes them all:

Interval Cents Aura's Names Color Names ^v Names Rastmic Names
1331/1296 46.133824 Alpharabian Superprime, Alpharabian Subchromatic Quartertone Trilo Unison ^^^d1 hA1
33/32 53.272943 Alpharabian Parasuperprime, Alpharabian Parachromatic Quartertone, al-Farabi Quartertone Ilo Unison ^1 "
1089/1024 106.545886 Alpharabian Augmented Unison, Alpharabian Chromatic Semitone Lolo Unison ^^1 A1
8192/8019 36.952052 Alpharabian Parasubminor Second, Alpharabian Diesis Salo 2nd vm2 hd2
4096/3993 44.091172 Alpharabian Subminor Second, Alpharabian Paradiatonic Quartertone Satrilu 2nd vvvM2 "
128/121 97.364115 Alpharabian Minor Second, Alpharabian Diatonic Semitone Lulu 2nd vvM2 m2
88/81 143.497938 Lesser Alpharabian Neutral Second Ilo 2nd ^m2 n2
12/11 150.637059 Greater Alpharabian Neutral Second Lu 2nd vM2 "
1331/1152 250.043825 Alpharabian Supermajor Second Trilo 2nd ^^^m2 hA2
297/256 257.182945 Alpharabian Parasupermajor Second Lalo 2nd ^M2 "
1024/891 240.862054 Alpharabian Parasubminor Third Salu 3rd vm3 hd3
1536/1331 248.001174 Alpharabian Subminor Third Trilu 3rd vvvM3 "
144/121 301.274117 Alpharabian Minor Third Lulu 3rd vvM3 m3
11/9 347.407941 Lesser Alpharabian Neutral Third Ilo 3rd ^m3 n3
27/22 354.547060 Greater Alpharabian Neutral Third Lu 3rd vM3 "
121/96 400.680884 Alpharabian Major Third Lolo 3rd ^^m3 M3
1331/1024 453.953827 Alpharabian Supermajor Third Trilo 3rd ^^^m3 hA3
2673/2048 461.092947 Alpharabian Parasupermajor Third Lalo 3rd ^M3 "
128/99 444.772056 Alpharabian Paraminor Fourth Lu 4th v4 hd4
11/8 551.317942 Alpharabian Paramajor Fourth, Just Paramajor Fourth Ilo 4th ^4 hA4
363/256 604.590886 Alpharabian Augmented Fourth Lolo 4th ^^4 A4
512/363 595.409114 Alpharabian Diminished Fifth Lulu 5th vv5 d5
16/11 648.682058 Alpharabian Paraminor Fifth, Just Paraminor Fifth Lu 5th v5 hd5
99/64 755.227944 Alpharabian Paramajor Fifth Ilo 5th ^5 hA5
4096/2673 738.907053 Alpharabian Parasubminor Sixth Salu 6th vm6 hd6
2048/1331 746.046173 Alpharabian Subminor Sixth Trilu 6th vvvM6 "
192/121 799.319116 Alpharabian Minor Sixth Lulu 6th vvM6 m6
44/27 845.452940 Lesser Alpharabian Neutral Sixth Ilo 6th ^m6 n6
18/11 852.592059 Greater Alpharabian Neutral Sixth Lu 6th vM6 "
121/72 898.725883 Alpharabian Major Sixth Lolo 6th ^^m6 M6
1331/768 951.998826 Alpharabian Supermajor Sixth Trilo 6th ^^^m6 hA6
891/512 959.137946 Alpharabian Parasupermajor Sixth Lalo 6th ^M6 "
512/297 942.817055 Alpharabian Parasubminor Seventh Salu 7th vm7 hd7
2304/1331 949.956175 Alpharabian Subminor Seventh Trilu 7th vvvM7 "
11/6 1,049.362941 Lesser Alpharabian Neutral Seventh Ilo 7th ^m7 n7
81/44 1,056.502061 Greater Alpharabian Neutral Seventh Lu 7th vM7 "
121/64 1,102.635885 Alpharabian Major Seventh Lolo 7th ^^m7 M7
3993/2048 1,155.908828 Alpharabian Supermajor Seventh Latrilo 7th ^^^m7 hA7
8019/4096 1,163.047948 Alpharabian Parasupermajor Seventh Lalo 7th ^M7 "
2048/1089 1,093.454114 Alpharabian Diminished Octave Lulu 8ve vv8 d8
64/33 1,146.727057 Alpharabian Parasuboctave Lu 8ve v8 hd8
2592/1331 1,153.866176 Alpharabian Suboctave Trilu 8ve vvvA8 "

How to evaluate these? Well obviously the Rastmic notation carries less information and often isn't appropriate. How about the other three? The color names explicitly refer to the ratio. They are appropriate when there are other primes besides 2, 3 and 11. But if we are only using those primes, the ^v notation is my favorite.

There's nothing inherently "flawed" with your system. Although it's incomplete, because it doesn't address staff notation or chord names. Here's my thoughts on it:

I think a notation should be easy to learn. I dislike asking people to learn obscure names like Pythagorean, Dydimus, Archytas, Alpharabi, etc. It's just extra mental work, and there's enough of that already with microtonality! I also think a nomenclature should be reasonably concise. You have a lot of long words in your names.

Greater/Lesser is a problem because one can always add or subtract a rastma and get a new ratio slightly larger or smaller. I guess you solve this with Betarabic vs Alpharabic? This is the problem with cents-based nomenclatures. There's always more ratios in that cents range to deal with, and the nomenclature eventually breaks. I prefer monzo-based nomenclatures that don't break.

I think it's important to be able to work backwards from the name to the ratio or monzo. You can do that with both the color names and the ^v names. The latter are especially easy. The ^v part is 33/32 and the rest is a 3-limit ratio. So ^M3 means 81/64 times 33/32. But can one work backwards from your names? Is there a formula or an algorithm for that?

There should also be a simple process to get from the ratio to the name. Why are some ratios Alpharabian and some Betarabian? What simple test can be applied to say 729/704 to determine which category it falls into? What happens to the intervals that don't fit into either category?

IMO a good nomenclature/notation should make interval arithmetic easy. For example, what's an Alpharabian Minor Third plus an Alpharabian Supermajor Second? I have no idea, but I might guess an Alpharabian Super Fourth? But that's not in the table, so I guess that's wrong. Now let's try it this way: what's a vvM3 plus a ^^^m2? M3 + m2 = P4, ups and downs add up and cancel out logically, so we get ^4, an upfourth. What ratio is ^4? It's 4/3 (P4) times 33/32 (^1), so 11/8.

What's an Alpharabian Parasupermajor Second plus a Greater Alpharabian Neutral Second? What do parasupermajor and greater neutral add up to? Paramajor? Supermajor? I don't know, but in ^v, it's ^M2 + vM2 = M3.

Now it may well be that you know your system well enough that you can answer these questions easily. But can you explain to someone not familiar with your system how you arrived at the answer? Can you work out a general process for that, other than translating each name to a monzo, adding the monzos, and then translating back?

More examples:

24057/16384, the Betarabian Parasuperaugmented Fourth becomes ^A4, upaugmented 4th.

352/243, the Lesser Betarabian Paraminor Fifth --> updim 5th

1944/1331 (the Greater Betarabian Paraminor Fifth?) --> triple-down aug 5th

Ratio Monzo Cents Aura's Name Color Name ^v Name Rastmic Name
243/242 [1 -5 0 0 2> 7.139 Rastma Lulu comma vvA1 P1
131769/131072 [-17 2 0 0 4⟩ 9.18177 Alpharabian comma Laquadlo comma -v4M2 -d2
1771561/1769472 [-16 -3 0 0 6> Nexus comma Tribilo comma -v6A2 -d2
264627/262144 [-18 7 0 0 2⟩ 16.32089 Betarabian comma Lalolo comma -vvm2 -d2

Some of these commas are negative 2nds. As is the pythagorean comma, and also 50/49 and 225/224. The ^v notation helps one find new commas. Just take a smallish 3-limit interval and add two downs for every 12-edo semitone it contains. So vvm2, v4M2, v6A2, v8AA2... then v4d3, v6m3, v8M3 (oops that's two v4M2's).

I'm not a fan of using large edos to measure everything else, so I'll skip the 159edo vs. 205edo discussion.

"Conventionally, there is an aug unison but not a dim unison. So it seems a Paramajor Unison would be valid? What ratio would that be?" Well, Major and Minor never occur unpaired in conventional music, and "Paramajor" and "Paraminor" are coined in analogy to Major and Minor respectively. This is why I reject the idea of a "Paramajor Unison". Rather, I refer to a "Parasuperprime", which, ideally, has 33/32 as the ratio.
"Seems like dividing 9/8 into a diatonic quartertone and 3 identical chromatic quartertones will always make huge ratios. Maybe better to have unequal steps." Nope. Why? Because the Apotome has a ratio of 2187/2048- that's four digits for each number, just like with the ratios in my current system.
"Greater/Lesser is a problem because one can always add or subtract a rastma and get a new ratio slightly larger or smaller. I guess you solve this with Betarabic vs Alpharabic? This is the problem with cents-based nomenclatures. There's always more ratios in that cents range to deal with, and the nomenclature eventually breaks. I prefer monzo-based nomenclatures that don't break." I can understand that.
"There should also be a simple process to get from the ratio to the name. Why are some ratios Alpharabian and some Betarabian? What simple test can be applied to say 729/704 to determine which category it falls into? What happens to the intervals that don't fit into either category?" The terms are "Alpharabian" and "Betarabian", and yes, there is a rudamentary test in place. Alpharabian intervals are generally either directly in the 2.11 subgroup, or, they differ from a nearby Pythagorean interval by either 33/32, 1331/1296, 8192/8019, 4096/3993, or 243/242. The system for denoting Betarabian intervals is less complete, but so far, Betarabian intervals usually differ from Alpharabian intervals by a rastma. When dealing with intervals that are distant by additionl rastmas, I would just use "Rastimic", "Birastmic", "Trirastmic", and so on.
"As a practical matter, if some violinists or vocalists were performing 2.3.11 JI music, they would constantly be fudging the rastma. They might shift a pitch by 7¢, or let the tonic drift by 7¢. The neutral triad is an innate-comma chord (aka essentially tempered chord), and the 3rd might fall between 11/9 and 27/22. No listener would notice these slight discrepancies. The rastma would effectively be tempered out, and the notation could reflect this. Simply notate two notes only a rastma apart as the same note, and trust the performers' ears." On the contrary, I've met a music director who can hear a difference between 440 Hz and 438 Hz- a difference of less than 8¢, and I know that 7¢ is above the average JND, so performers fudging the rastma would likely drive this director batty. The main reason for me not tempering out the rastma is that it is useful in modulating to quartertone-based keys Jacob-Collier-style. --Aura (talk) 11:03, 1 November 2020 (UTC)