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= ARCHIVED WIKISPACES DISCUSSION BELOW =
{{WSArchiveLink}}
'''All discussion below is archived from the Wikispaces export in its original unaltered form.'''
== Calculated 5–limit commas ==
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== No! :( ==
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.
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
List 1:
----
<code><pre>
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
</pre></code>
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


== "Commas" ? ==
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
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
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.
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What do you think is a *large comma*? Greater than 50 cent?


- '''xenwolf''' January 29, 2015, 03:41:53 AM UTC-0800
List 2 repeats 2 numbers: a fraction part and a cents value. The fraction part must be counterparted with a power of two.
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As good a definition as any.


- '''genewardsmith''' January 29, 2015, 04:19:52 PM UTC-0800
There may also be redundant squares, cubes, etc. of the fractions.
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Hope we can get a complete comma list from it!
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...


<ul><li>sagittal.pdf ("Xenharmonikon article") from the Sagittal homepage http://sagittal.org/</li></ul>
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.


- '''Gedankenwelt''' January 29, 2015, 05:58:43 PM UTC-0800
[[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 11:54, 18 September 2018 (UTC)
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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:
 
<ul><li>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).</li><li>We could exclude everything that is larger than a "normal" diesis (&gt; ~56.8 cents), since iirc this the definition with the largest cents value they gave.</li><li>We could define an upper bound for large dieses, and exclude everything above. It probably wouldn't hurt to ask them about their opinion.</li><li>We could simply use a more or less arbitrary cents value (like 120 or 133.33 cents), and exclude everything above.</li></ul>
 
- '''Gedankenwelt''' January 30, 2015, 05:49:12 AM UTC-0800
----
I was just thinking a little bit about <em>extraterrestrial music</em>. 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
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Would people be OK with a cutoff of 100 cents?
 
- '''genewardsmith''' January 30, 2015, 08:52:10 AM UTC-0800
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@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&gt; (~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&gt;/2 ~= 1.8075 cents (half pythagorean schisma = half Mercator's comma)
 
schisma
 
[317 -200&gt;/2 ~= 4.4999 cents
 
kleisma
 
[-19 12&gt;/2 ~= 11.7300 cents (half a pythagorean comma)
 
comma
 
[27 -17&gt;/2 ~= 33.3825 cents (half a pythagorean large-diesis = half a pythagrean limma minus half a pythagorean comma)
 
small-diesis
 
[8 -5&gt;/2 ~= 45.1125 cents (half a pythagorean limma = half a pythagorean apotome minus half a pythagorean comma)
 
(medium-)diesis
 
[-11 7&gt;/2 ~= 56.8425 cents (half a pythagorean apotome)
 
large-diesis
 
[-30 19&gt;/2 ~= 68.5725 cents (half a pythagorean apotome plus half a pythagorean comma)
 
small-semitone
 
[-49 31&gt;/2 ~= 80.3025 cents


limma
== Links in general ==


[-3 -2&gt;/2 ~= 101.9550 cents
I'd like to limit "sophisticated linking" in favor of usability. I'm not very happy with links of type <code><nowiki>[[A|B]]</nowiki></code> even if A→B or B→A. If there is a good chance that both are equally (or comparable) common, I suggest to use <code><nowiki>[[A]]</nowiki></code> and <code><nowiki>[[B]]</nowiki></code>. And sometimes the numbers are easier to remember than names, especially if names are newly invented, so for me [[64/63]] is easier to remember than [[septimal comma]], and in this special case the ratio is unique which the name isn't. Maybe we should try to discuss this with more people. Here are two cases where the problem is noticeable, and I would like to know more about the ideas others have about their way of linking: 1) [https://en.xen.wiki/index.php?title=Small_comma&type=revision&diff=53496&oldid=53464 FloraC "links on names instead of ratios..."] and 2) [https://en.xen.wiki/index.php?title=Symbiosma&diff=next&oldid=53473 Xenwolf: "improved apotome link..."]. Maybe we should start an own project for this? --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 15:37, 27 November 2020 (UTC)


large-semitone
: Hmm... that leaves me with questions about commas like the Quartisma, the Nexuma and the Symbiosma...  These were all recently named, but the ratios of these commas have too many digits...  Also, how unique are the names of these commas? --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 15:43, 27 November 2020 (UTC)


[62 -39&gt;/2 ~= 111.8775 cents
:: I think names and ratios of these three commas will be obscure to most people, so there will be no best option. This probably applies to commas in general (except [[81/80]]=syntonic or meantone comma and [[pythagorean comma]] whose ratio I can remember even worse than my phone number). BTW: in cases like [[7/5]] and [[10/7]], who knows (without cheating!) which of them is Euler's and which Huygens' tritone? --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 15:58, 27 November 2020 (UTC)


apotome
::: Yes, it's an issue for commas with non-unique names. Shall we stick to a ''simple link priciple'', where we always link directly to the page in these tables? [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 19:27, 27 November 2020 (UTC)


[-106 67&gt;/2 ~= 115.4925 cents
:::: Would it be too confusing if we link to the existing lemma (in changing columns)? Or should we add a "further reading" column which would help us to place links (like ''[[Garischismic temperaments|satin comma]]'') that are only loosely coupled with the comma itself? --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 19:41, 27 November 2020 (UTC)


schisma-plus-apotome
::::: A further-reading column for links sounds rather forced. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 08:16, 28 November 2020 (UTC)


[317 -200&gt;/2 + [-11 7&gt;  <h1 id="toc0"> [295 -186&gt;/2 ~</h1>
:::::: Okay, then we'll just let it go. Maybe we or someone else will come up with a really good solution for it later. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 12:56, 28 November 2020 (UTC)
118.1849 cents


kleisma-plus-apotome
== Proposal to change a few comma names ==


[-19 12&gt;/2 + [-11 7&gt;  <h1 id="toc1"> [-41 26&gt;/2 ~</h1>
* <s>Countriton → countertriton or cotriton</s>
125.4150 cents
: <s>"Countriton" sounds like a result of haplology. I, for one, strongly prefer full prefixes. Either countertriton or cotriton would work. </s>
* Moctdel → motel
: "Moctdel" is hard to pronounce. I kinda get that "del" part comes from the grendel temperament (→ Mirkwai clan #Grendel) but I don't see how it's related to moct (millioctave) by any means.  
Ideas? [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 14:10, 2 September 2022 (UTC)


etc up to double-apotome (with limma-plus-apotome also called whole-tone).
Update: Nevermind "countriton". It seems it's formed by co + untriton. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 09:56, 30 November 2022 (UTC)


- '''d.keenan''' November 02, 2016, 06:12:37 PM UTC-0700
== A new way to name commas in color notation? ==
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User Gud2228 recently added Yo-28, Gu-31 and Gu-59 to this page. These are generalizations of the Wa-N format for 3-limit commas. The format is obviously Yo-N = [x, 0, N> and Gu-N = [x, 0, -N>. It can be generalized to other primes, with e.g. Tho-10 meaning [-37 0 0 0 0 10> (Laquinbitho 2nd, about 5¢).


== Go comma ==
I have mixed feelings about this usage. One the one hand, it's great to see people experimenting and improvising with color notation. That's what happens with actual languages after all. And it is a logical extension of existing usage. But still I'd rather not elevate this usage to an official status. IMO it doesn't add enough utility to justify increasing the complexity of the notation.
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
So if Yo-28 isn't worth doing, why is a 3-limit name like Wa-29 OK? Because the alternative to Wa-29 is Quadsawa, which obscures the number 29. But Yo-28's alternative is Quadla-sepquadyo, in which "sepquad" clearly indicates the number 28.
----
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
So I suggest the name Yo-28 be moved to the first column, next to "oquatonic comma", and the 2nd column have the usual color name, Quadla-sepquadyo. Likewise for Gu-31 and Gu-59. -[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 00:46, 11 March 2023 (UTC)
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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
== Color names for 3-limit commas ==
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Color names are mostly a straightforward mapping of the numbers in the monzo to syllables, except that it obscures the 2-count and 3-count. You have to deduce those from the degree, the magnitude, etc. Mostly this works quite well. The degree lets you estimate the size in cents, which is usually more useful information than the number of threes in the ratio.


== Precision of cent values ==
But it doesn't work well with 3-limit commas. The most important fact about such commas is the 3-count! That tells you which edo it implies, which immediately tells you what you get when you temper out the comma. That's why I devised the "wa format", such as Wa-41, or its short form, w-41.
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
But here and on the other two comma pages, the short forms have been changed from e.g. "w-41" to "s<sup>6</sup>w5". IMO this isn't a good idea.  
----
Pythagorean Integer Cents comma is funny. But it's actually 8.9998269225164511026534525240378976445421538045454906193629101364253293539449946526860196588218981646410059857998... cents


- '''PiotrGrochowski''' August 13, 2016, 09:07:54 AM UTC-0700
One could argue that it's somewhat useful to know that this comma turns out to be a very small 5th. IMO that information belongs on the page dedicated to that comma, where there's room to delve into such details. It doesn't belong in the table of commas.  
----
Pythagorean Integer Cents comma is funny. But it's actually 8.9998269225164511026534525240378976445421538045454906193629101364253293539449946526860196588218981646410059857998... cents


- '''PiotrGrochowski''' August 13, 2016, 09:07:54 AM UTC-0700
Also, I don't like seeing the short form removed. It's good to know. It's very useful for equations such as Ly-2 + w-17 = sy2.
----
--[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 05:42, 18 December 2023 (UTC)
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: 43046721 5 44.96629998136541 19683 5 68.71870607634492 81 5 21.5062895967165 3486784401 25 66.4725895780748 1594323 25 47.21241647963268 6561 25 43.012579193433 3 25 70.67242686427875 129140163 125 25.70612688291476 531441 125 64.51886879015092 243 125 49.166137267562604 1 125 41.05885840550059 43046721 625 86.025158386866 19683 625 27.65984767084646 81 625 62.565148002217086 387420489 3125 96.37855374719493 1594323 3125 6.153558074133514 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 padded 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.
 
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.
 
[[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 11:54, 18 September 2018 (UTC)
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