Talk:Small comma
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No! :(
I wanted the commas and the Unnoticeable ones back together. I don't like the way it's done now.
- PiotrGrochowski May 24, 2017, 09:21:21 AM UTC-0700
"Commas" ?
What's with all these large "commas? I think they should be moved somewhere else.
- genewardsmith January 29, 2015, 12:43:39 AM UTC-0800
What do you think is a *large comma*? Greater than 50 cent?
- xenwolf January 29, 2015, 03:41:53 AM UTC-0800
As good a definition as any.
- genewardsmith January 29, 2015, 04:19:52 PM UTC-0800
George Secor and Dave Keenan used some comma definitions depending on size in a certain introduction article for Sagittal notation*. For example, intervals larger than half an apotome (~56.8 cents) were called large dieses. Not sure how large a large diesis can be, though...
- sagittal.pdf ("Xenharmonikon article") from the Sagittal homepage http://sagittal.org/
- Gedankenwelt January 29, 2015, 05:58:43 PM UTC-0800
So, maybe the biggest commas should not exceed the half tone (about 100ct), or, as to be not too western, 120ct (the half of a 5th octave)?
- xenwolf January 30, 2015, 12:46:27 AM UTC-0800
Well, there are several options. For example:
- Since the article is called "Comma", we could borrow their definition of a comma (between ~11.7 and ~33.4 cents), and exclude everything that is larger (small, "normal" and large diesis), while optionally including or not including everything that is smaller (kleisma, schisma, schismina).
- We could exclude everything that is larger than a "normal" diesis (> ~56.8 cents), since iirc this the definition with the largest cents value they gave.
- We could define an upper bound for large dieses, and exclude everything above. It probably wouldn't hurt to ask them about their opinion.
- We could simply use a more or less arbitrary cents value (like 120 or 133.33 cents), and exclude everything above.
- Gedankenwelt January 30, 2015, 05:49:12 AM UTC-0800
I was just thinking a little bit about extraterrestrial music. Isn't it reasonable to assume universal hearing range limitations for physical reasons? Putting all animals together (humans included) there is a range of slightly more than 13 octaves. After this digression, I think we should not be too restrictive with commas...
...on the other hand, commas as big as fourths seem absurd to me (to be honest)
- xenwolf January 30, 2015, 08:08:32 AM UTC-0800
Would people be OK with a cutoff of 100 cents?
- genewardsmith January 30, 2015, 08:52:10 AM UTC-0800
@xenwolf: Those large "commas" may seem absurd when looking at their size in just intonation. But on the other hand, they can become very small in certain temperaments, and tempering them out may lead to useful results, so I think it's not completely far-fetched to call them commas.
Let's take the 49-comma |78 -49> (~404 cents), for example: It may be a large interval in just intonation, but it gets fairly small in a typical superpyth tuning. And despite its large size, tempering it out means the fifths become only ~8 cents sharp (leading to 49-tet), so the "damage" to important JI intervals is much smaller than one might expect. I think tempering out the 49-comma is musically much more meaningful than tempering out 9/8, even though the latter is only half as small.
- Gedankenwelt January 30, 2015, 09:17:43 AM UTC-0800
I think any reasonable cutoff will work for now. If we'll realize later that another cutoff would be better, we can still change the list.
But where would we put the large "commas"? We could create a new article called "Large Commas", where only "commas" are allowed that are larger than 100 cents, but there should be further restrictions. For example, we may only allow commas which, if tempered out, define an equal temperament (patent val assumed) if a suitable prime limit is given, or something like that.
I also discovered an article where some methods to define the quality of a comma are specified:
http://xenharmonic.wikispaces.com/ABC%2C+High+Quality+Commas%2C+and+Epimericity
Would a generous restriction of the interval's epimericity lead to useful results?
- Gedankenwelt February 02, 2015, 06:26:10 AM UTC-0800
The boundary between our large dieses and our small semitones is at half a pythagorean apotome plus half a pythagorean comma, approximately 68.6 cents. All our boundaries are at the square-roots of 3-prime-limit ratios as follows. These boundaries allow commas to be named systematically using only their prime factors greater than 3.
0 cents
schismina
[-84 54>/2 ~= 1.8075 cents (half pythagorean schisma = half Mercator's comma)
schisma
[317 -200>/2 ~= 4.4999 cents
kleisma
[-19 12>/2 ~= 11.7300 cents (half a pythagorean comma)
comma
[27 -17>/2 ~= 33.3825 cents (half a pythagorean large-diesis = half a pythagrean limma minus half a pythagorean comma)
small-diesis
[8 -5>/2 ~= 45.1125 cents (half a pythagorean limma = half a pythagorean apotome minus half a pythagorean comma)
(medium-)diesis
[-11 7>/2 ~= 56.8425 cents (half a pythagorean apotome)
large-diesis
[-30 19>/2 ~= 68.5725 cents (half a pythagorean apotome plus half a pythagorean comma)
small-semitone
[-49 31>/2 ~= 80.3025 cents
limma
[-3 -2>/2 ~= 101.9550 cents
large-semitone
[62 -39>/2 ~= 111.8775 cents
apotome
[-106 67>/2 ~= 115.4925 cents
schisma-plus-apotome
[317 -200>/2 + [-11 7>
[295 -186>/2 ~
118.1849 cents
kleisma-plus-apotome
[-19 12>/2 + [-11 7>
[-41 26>/2 ~
125.4150 cents
etc up to double-apotome (with limma-plus-apotome also called whole-tone).
- d.keenan November 02, 2016, 06:12:37 PM UTC-0700
Go comma
I added this comma and I asked myself for its name, but suddenly the Japanese board game Go came to my mind, and its 19x19 fields that are so surprisingly close to 360... there was no reason to hesitate... ;)
- xenwolf May 19, 2013, 03:44:36 PM UTC-0700
You might as well throw in the "Boostma" of 106/105: Two months of unlimited data plus premium voicemail minus a $60 and $45 card on Boost Mobile. (Now, SoftBank, if you could please lower your premimum voicemail to $2.49 so we can pay for it easier...)
- bootmii December 03, 2016, 08:08:51 PM UTC-0800
You might as well throw in the "Boostma" of 106/105: Two months of unlimited data plus premium voicemail minus a $60 and $45 card on Boost Mobile. (Now, SoftBank, if you could please lower your premimum voicemail to $2.49 so we can pay for it easier...)
- bootmii December 03, 2016, 08:08:52 PM UTC-0800
Precision of cent values
Should we better use a unified amount of decimals, such as 4 or 5. In case of commas 4 seems a pragmatic decision for me. (Please don't ask for the precision overkill of the freqency ratios ;-) )
- xenwolf June 01, 2010, 05:41:30 AM UTC-0700
Pythagorean Integer Cents comma is funny. But it's actually 8.9998269225164511026534525240378976445421538045454906193629101364253293539449946526860196588218981646410059857998... cents
- PiotrGrochowski August 13, 2016, 09:07:54 AM UTC-0700
Pythagorean Integer Cents comma is funny. But it's actually 8.9998269225164511026534525240378976445421538045454906193629101364253293539449946526860196588218981646410059857998... cents
- PiotrGrochowski August 13, 2016, 09:07:54 AM UTC-0700
Pythagorean Integer Cents comma is funny. But it's actually 8.9998269225164511026534525240378976445421538045454906193629101364253293539449946526860196588218981646410059857998... cents
- PiotrGrochowski August 13, 2016, 09:07:55 AM UTC-0700
Calculated 5–limit commas
I was in a calculation of 5–limit commas 4294967296 integer limit below 100 cents. The calculation was blazingly fast but was in a rudimentary format.
List 1:
1600000/1594323 6.153558074133514
81/80 21.5062895967165
131072000/129140163 25.70612688291476
20000/19683 27.65984767084646
128/125 41.05885840550059
6561/6400 43.012579193433 (square of 81/80)
43046721/41943040 44.96629998136541
1638400/1594323 47.21241647963268
250/243 49.166137267562604
648/625 62.565148002217086
531441/512000 64.51886879015092 (cube of 81/80)
3486784401/3355443200 66.4725895780748
20480/19683 68.71870607634492
25/24 70.67242686427875
43046721/40960000 86.025158386866 (fourth power of 81/80)
409600000/387420489 96.37855374719493
6561 3125 84.07143759893216 3 3125 29.613568458779582 129140163 15625 15.352731522582985 59049 15625 98.33227453512521 243 15625 8.107278862061662 1 15625 82.11771681100117 4782969 78125 76.825984938408 19683 78125 13.399010734655548 387420489 390625 55.31969534169292 1594323 390625 34.905300331368494 729 390625 78.77970572634041 3 390625 11.445289946721005 129140163 1953125 56.41158992808357 59049 1953125 57.27341612962533 243 1953125 32.951579543438925 4782969 9765625 35.767126532910254 19683 9765625 54.457869140154 9 9765625 59.227136917559164 387420489 48828125 14.260836936190913 1594323 48828125 75.9641587368705 729 48828125 37.7208473208384 3 48828125 52.504148352227276 129140163 244140625 97.47044833358558 59049 244140625 16.214557724123324 243 244140625 74.01043794894235 1162261467 1220703125 84.93326380046966 4782969 1220703125 5.291731872591754 19683 1220703125 95.51672754565459 9 1220703125 18.168278512052893
List 2: 2197265625 39.674568108773656 263671875 30.997858755506513 4271484375 9.491569158785751 17578125 80.73342651427424 512578125 80.1639960230716 2109375 10.060999649994073 34171875 31.56728924670915 15625 82.11771681100117 553584375 53.073578843429914 253125 60.611427214286095 4100625 39.10513761757102 16875 51.11985805549466 66430125 17.59884802085594 273375 72.62614765220974 125 41.05885840550059 1076168025 3.907441575864823 4428675 94.1324372489305 2025 19.55256880878551 7971615 88.27127488513042 32805 1.9537207879324114 135 92.17871646099525 129140163 66.76498528842103 531441 23.46001038464749 243 90.22499567306284 1 0
List 1 repeats 3 numbers: a power of 3, a power of 5 and a cents value. The fractions have to be manually multiplied with powers of two for octave reduction.
List 2 repeats 2 numbers: a fraction part and a cents value. The fraction part must be counterparted with a power of two.
There may also be redundant squares, cubes, etc. of the fractions. Hope we can get a complete comma list from it!
And someone, please merge Comma with Unnoticeable commas back so that we can have a closer comparison, and complete blank fraction spaces up to like 25 digits.
PiotrGrochowski (talk) 11:54, 18 September 2018 (UTC)