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 11^{3}/2^{10}
. --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 | -v^{4}M2 | -d2 |
1771561/1769472 | [-16 -3 0 0 6> | 2¢ | Nexus comma | Tribilo comma | -v^{6}A2 | -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, v^{4}M2, v^{6}A2, v^{8}AA2... then v^{4}d3, v^{6}m3, v^{8}M3 (oops that's two v^{4}M2'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)