Notating 5-limit pental with added 7th harmonics

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Notating 5-limit pental with added 7th harmonics

Post by רועיסיני »

Hi! I have retuned a song to 5-limit pental, which is the temperament that tempers out 847288609443/838860800000, or the difference between 5 syntonic commas and a pythagorean limma, and I also used pure septimal intervals in the accompanimant, so I thought I'd use Sagittal notation to notate it because it's standard and versatile. I haven't used Sagittal before, but I have a good understanding of mathematics and microtonality.
The fifth has complexity 5 in pental, so for the 5-limit notes I can just use :/|: and :/ /|:, but I am not sure what's the best way to notate all the septimal notes. For 7, 35 and 7/5 there are already Athenian symbols - :|):, :/|): and :|(: respectively, so they are not a problem either, but I got stuck when I tried to find symbols for 25/7 and 175.
Preferably, I'd like a symbol for 413343/409600 = [-14 10 -2 1⟩ = (81/80)^2 / (64/63) and one for 729/700 = [6 -2 -2 -1⟩ = (81/80)^2 * (64/63). For the former, the table in viewtopic.php?f=6&t=252 suggests :|~::.::.: so I thought I could use :|~: without the marks, but the latter maps there to :)||(::.: from which I can't remove the mark because :)||(: is the apotome complement of :/ /|:.
The tempered 8505/8192, 729/700 and 25/24 turn out to be about 61, 66 and 71 cents in my tuning, which are very close to 13, 14 and 15 steps of 255edo (which also supports pental), but I haven't found a Sagittal mapping for 255edo to see how it solves this.
What's the most logical solution for this in your opinion?
Thanks,
Roee Sinai.
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Re: Notating 5-limit pental with added 7th harmonics

Post by Dave Keenan »

Welcome to the forum Roee. That is quite a puzzle.

No sagittal notation has previously been devised for 255edo. After much investigation, the only viable solution I can find is to repurpose the left scroll :)|: , giving it a 7-limit (or perhaps 11-limit) definition, which I haven't worked out. The left scroll's default definition, as the 19-schisma, vanishes in 255edo, so it's not unreasonable to repurpose it here. That gives us this 255edo notation with consistent flag arithmetic:

:)|:	:|(:	:)|(:	:/|:	:)/|:	:|):	:|\:	://|:	:)//|:	:/|):	:/|\:	:(|):	:(|\:
1	2	3	4   5	6	7	8	9	10	11	12	13

Notating the rank-2 temperament pental seems like it could be much simpler. It consists of 5 parallel chains of fifths (about 0.8 ¢ narrow), 1/5-octave apart. We know how to notate a chain of fifths. We only need nominals, flats and sharps for that (although a Pythagorean-comma symbol can be useful too). We mainly need Sagittals to distinguish the 5 chains from each other. All we need for that is :\\!: :\!: :h: :/|: ://|:
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Re: Notating 5-limit pental with added 7th harmonics

Post by רועיסיני »

I agree that pental itself is quite simple, the problem is the pure 7th harmonics, that don't come themselves from the pental chain.
In the meantime I have actually devised a notation for 255 myself that is almost entirely consistent with flag arithmetic and uses only symbols from table 1 in the PDF https://sagittal.org/sagittal.pdf:
:~|:   :|(:	:~|(:   :/|:	:|~:   :|):	:|\:   ://|:	:(|(:   :/|):	:/|\:   :(|):	:(|\:
1   2	3   4   5   6	7   8	9   10	11  12	13
:~|: - 17K
:|(: - 7/5K
:~|(: - Sum of flags
:/|: - 5C
:|~: - 23C
:|): - 7C
:|\: - 55C
:/ /|: - 25S
:(|(: - 17/11S
:/|): - 35M
:/|\: - 11M
:(|): - 11L
:(|\: - 35L
The only problem here is that "adding" :(|: to :|(: increases it by 7 steps while "adding" it to :|): or :|\: only increases them by 6 steps, but it is somewhat mitigated by not having the :(|: symbol itself.
This suggests :(|(: for the 175 small diesis 525/512, which is part of the Athenian symbol set, and therefore it suggests the use of :~|(: for the 25/7 comma (kleisma?).
Both of these decisions are consistent with viewing the sequence shown in page 10 there as a seties of commas in steps, if we say that Didymus's comma is 4 steps and Archytas's comma is 5 steps (and we say that :/|\: and :(|): are the same step), and has a nice symmetry to it - just like :|): looks like the apotome complement of itself, and if we add and subtract a syntonic comma we get :/|): and :|(:, where the former looks like the apotome complement of the latter, if we continue the sequence we now get the pairs of :(|(::~||(: and :~|(::(||(: where the double shaft symbol from each pair looks like the single shaft from the other.
The only problem I see with this suggestion is that in pental, the 25/7 comma, which equals to 7/22.8, or about 8.8 cents if we use pure sevenths, turns out to be about a cent smaller than the 7/5 kleisma, so the :~|: flag actually decreases the size of the accidental, although I don't think I'll need to use both :|(: and :~|(: on the same note.
What do you think?
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Re: Notating 5-limit pental with added 7th harmonics

Post by Dave Keenan »

You clearly have a very good understanding of the system, so you should feel free to use whatever seems best to you. But the main problem I see with your proposal to set Archytas' comma to 5\255 is that the best approximation of the 7th harmonic is given by setting Archytas' comma and hence the right arc or tai symbol :|): equal to 6\255. This makes the interval G:F:!): only 0.6 ¢ wider than 7/4.

We could use the 255edo mapping of the prime 7 to extend the pental mapping to the 7-limit. This would be a different (and far more complex) mapping from the mapping described as 7-limit pental on the Xenharmonic Wiki. This would allow the 7th harmonic to be notated based on the pental chains even though it will be tuned pure.

Here are all the (octave-equivalent) degrees of 255 laid out on a pental plane centered on degree 0. Fifths go down the page. 1/5-octaves across. From the root as zero (yellow), the 5th harmonic is 82 (cyan) and the 7th harmonic is 206 (pink).

253	49	100	151	202
147	198	249	45	96
41	92	143	194	245
190	241	37	88	139
84	135	186	237	33
233	29	80	131	182
127	178	229	25	76
21	72	123	174	225
170	221	17	68	119
64	115	166	217	13
213	9	60	111	162
107	158	209	5	56
1	52	103	154	205
150	201	252	48	99
44	95	146	197	248
193	244	40	91	142
87	138	189	240	36
236	32	83	134	185
130	181	232	28	79
24	75	126	177	228
173	224	20	71	122
67	118	169	220	16
216	12	63	114	165
110	161	212	8	59
4	55	106	157	208
153	204	0	51	102
47	98	149	200	251
196	247	43	94	145
90	141	192	243	39
239	35	86	137	188
133	184	235	31	82
27	78	129	180	231
176	227	23	74	125
70	121	172	223	19
219	15	66	117	168
113	164	215	11	62
7	58	109	160	211
156	207	3	54	105
50	101	152	203	254
199	250	46	97	148
93	144	195	246	42
242	38	89	140	191
136	187	238	34	85
30	81	132	183	234
179	230	26	77	128
73	124	175	226	22
222	18	69	120	171
116	167	218	14	65
10	61	112	163	214
159	210	6	57	108
53	104	155	206	2
(202	253	49	100	151)

Of course this is actually toroidal in 255. It wraps around exactly horizontally, and wraps vertically with an offset, as indicated by the redundant parenthesised last row as compared to the first row.

So the mapping from generators to primes is, in octave-equivalent terms:

0   0   2   1
0   1   5  25

Taking octaves into account it is:

5   5 -13 -69
0   1   5  25

We can refer to this temperament here as "complex 7-limit pental".
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Re: Notating 5-limit pental with added 7th harmonics

Post by Dave Keenan »

If I had instead shown a redundant parenthesised top row:

(104	155	206	2	53)

We would obtain a different complex 7-limit pental mapping. In octave equivalent form:

0   0   2   0
0   1   5 -26

Octave specific:

5   5 -13  80
0   1   5 -26

Since this is arguably equally complex, we could refer to these as the +25g and -26g 7-limit extensions of pental temperament. "g" for "generator". Of course they are equivalent in 255edo, and since this is just a notational convenience for what are actually pure 7th harmonics, we can avail ourselves of both mappings.
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Re: Notating 5-limit pental with added 7th harmonics

Post by Dave Keenan »

Here's notating the easy parts of 255edo with uncontroversial sagittals, based on the complex 7-limit mappings. These sagittals all happen to correspond to even numbers of steps:

D:!(:	E:|):	G:!):	A:|(:	C:\!):
A:!(:	B:|):	D:!):	E:|(:	G:\!):
E:!(:	F:#::|):	A:!):	B:|(:	D:\!):
B:!(:	C:#::|):	E:!):	F:#::|(:	A:\!):
F:#::!(:	G:#::|):	B:!):	C:#::|(:	E:\!):
C:#::!(:	D:#::|):	F:#::!):	G:#::|(:	B:\!):
G:#::!(:	A:#::|):	C:#::!):	D:#::|(:	F:#::\!):
D:#::!(:	E:#::|):	G:#::!):	A:#::|(:	C:#::\!):
A:#::!(:	B:#::|):	D:#::!):	E:#::|(:	G:#::\!):
E:#::!(:		A:#::!):	B:#::|(:	D:#::\!):
B:#::!(:		E:#::!):		A:#::\!):
		B:#::!):		E:#::\!):
				B:#::\!):
			F:b:://|:	
F:b::/|:			C:b:://|:	
C:b::/|:		F:b:	G:b:://|:	
G:b::/|:		C:b:	D:b:://|:	F:b::\!:
D:b::/|:	F:b::\\!:	G:b:	A:b:://|:	C:b::\!:
A:b::/|:	C:b::\\!:	D:b:	E:b:://|:	G:b::\!:
E:b::/|:	G:b::\\!:	A:b:	B:b:://|:	D:b::\!:
B:b::/|:	D:b::\\!:	E:b:	F://|:	A:b::\!:
F:/|:	A:b::\\!:	B:b:	C://|:	E:b::\!:
C:/|:	E:b::\\!:	F	G://|:	B:b::\!:
G:/|:	B:b::\\!:	C	D://|:	F:\!:
D:/|:	F:\\!:	G	A://|:	C:\!:
A:/|:	C:\\!:	D	E://|:	G:\!:
E:/|:	G:\\!:	A	B://|:	D:\!:
B:/|:	D:\\!:	E	F:#:://|:	A:\!:
F:#::/|:	A:\\!:	B	C:#:://|:	E:\!:
C:#::/|:	E:\\!:	F:#:	G:#:://|:	B:\!:
G:#::/|:	B:\\!:	C:#:	D:#:://|:	F:#::\!:
D:#::/|:	F:#::\\!:	G:#:	A:#:://|:	C:#::\!:
A:#::/|:	C:#::\\!:	D:#:	E:#:://|:	G:#::\!:
E:#::/|:	G:#::\\!:	A:#:	B:#:://|:	D:#::\!:
B:#::/|:	D:#::\\!:	E:#:		A:#::\!:
	A:#::\\!:	B:#:		E:#::\!:
	E:#::\\!:			B:#::\!:
	B:#::\\!:			
F:b::/|):				
C:b::/|):		F:b::|):		
G:b::/|):		C:b::|):		F:b::|(:
D:b::/|):	F:b::!(:	G:b::|):		C:b::|(:
A:b::/|):	C:b::!(:	D:b::|):	F:b::!):	G:b::|(:
E:b::/|):	G:b::!(:	A:b::|):	C:b::!):	D:b::|(:
B:b::/|):	D:b::!(:	E:b::|):	G:b::!):	A:b::|(:
F:/|):	A:b::!(:	B:b::|):	D:b::!):	E:b::|(:
C:/|):	E:b::!(:	F:|):	A:b::!):	B:b::|(:
G:/|):	B:b::!(:	C:|):	E:b::!):	F:|(:
D:/|):	F:!(:	G:|):	B:b::!):	C:|(:
A:/|):	C:!(:	D:|):	F:!):	G:|(:
E:/|):	G:!(:	A:|):	C:!):	D:|(:

The fact that both the 5-comma and 7-comma are an even number of steps, means that it doesn't matter how many you add or subtract, they will never give an odd number of steps. The only way you can change from even to odd is moving by factors of 3, i.e. by fifths. Will you really be moving so far from the tonic along the chain of fifths that you will need to notate more than the notes shown above?

If so, you can just use double-sharps and double-flats. Or we can designate a symbol for 3\255 as representing the Pythagorean comma so we don't have to rely on sharps and flats for more than 12 notes in a chain of fifths, say Eb to G#. :|~: is a good symbol for the Pythagorean-comma (3-comma)here because :,::|~: is the Olympian symbol for the 25×49-comma, 33075/32768, which corresponds to 3\255. I confirmed that by finding the dot product of the 255edo map 255 404 592 716] with the vector for the 25×49 comma [-15 3 2 2.

I note that the small wing-shaped mark on the left replaces the straight mark on the right that you have seen in older material.

We can't use the standard symbol for the Pythagorean comma :'::/|: , because the 5-schisma :': is negative in this temperament. i.e. the Pythagorean comma is smaller than Didymus' comma.
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Re: Notating 5-limit pental with added 7th harmonics

Post by Dave Keenan »

This is incomplete, but hopefully it's enough so you understand what I have in mind. This would not be a general 255edo notation, but purely a pental-plus-7th-harmonics notation.

D:!(:	E:|):	G:!):	A:|(:	C:\!):
A:!(:	B:|):	D:!):	E:|(:	G:\!):
E:!(:	F:#::|):	A:!):	B:|(:	D:\!):
B:!(:	C:#::|):	E:!):	F:#::|(:	A:\!):
F:#::!(:	G:#::|):	B:!):	C:#::|(:	E:\!):
C:#::!(:	E:b::|~::|):	F:#::!):	G:#::|(:	B:\!):
G:#::!(:	B:b::|~::|):	C:#::!):		F:#::\!):
E:b::|~::!(:	F:|~::|):	G:#::!):		C:#::\!):
B:b::|~::!(:	C:|~::|):	E:b::!~:		G:#::\!):
F:|~::!(:	G:|~::|):	B:b::!~:		
C:|~::!(:	D:|~::|):	F:!~:		
G:|~::!(:	A:|~::|):	C:!~:		
D:|~::!(:	F:!~::\\!:	G:!~:		
A:!~::/|:	C:!~::\\!:	D:!~:		
E:!~::/|:	G:!~::\\!:	A:!~:		
B:!~::/|:	D:!~::\\!:	E:!~:		
F:#::!~::/|:	A:!~::\\!:	B:!~:		
C:#::!~::/|:	E:!~::\\!:	F:#::!~:		
G:#::!~::/|:	B:!~::\\!:	C:#::!~:	E:b:://|:	
E:b::/|:	F:#::!~::\\!:	G:#::!~:	B:b:://|:	
B:b::/|:	C:#::!~::\\!:	E:b:	F://|:	
F:/|:	G:#::!~::\\!:	B:b:	C://|:	E:b::\!:
C:/|:	E:b::\\!:	F	G://|:	B:b::\!:
G:/|:	B:b::\\!:	C	D://|:	F:\!:
D:/|:	F:\\!:	G	A://|:	C:\!:
A:/|:	C:\\!:	D	E://|:	G:\!:
E:/|:	G:\\!:	A	B://|:	D:\!:
B:/|:	D:\\!:	E	F:#:://|:	A:\!:
F:#::/|:	A:\\!:	B	C:#:://|:	E:\!:
C:#::/|:	E:\\!:	F:#:	G:#:://|:	B:\!:
G:#::/|:	B:\\!:	C:#:		F:#::\!:
	F:#::\\!:	G:#:		C:#::\!:
	C:#::\\!:	E:b::|~:		G:#::\!:
	G:#::\\!:	B:b::|~:		
		F:|~:		
		C:|~:		
		G:|~:		
		D:|~:		
		A:|~:		
		E:|~:		
		B:|~:		
		F:#::|~:		
		C:#::|~:		
E:b::/|):		G:#::|~:		
B:b::/|):		E:b::|):		
F:/|):		B:b::|):		E:b::|(:
C:/|):	E:b::!(:	F:|):		B:b::|(:
G:/|):	B:b::!(:	C:|):	E:b::!):	F:|(:
D:/|):	F:!(:	G:|):	B:b::!):	C:|(:
A:/|):	C:!(:	D:|):	F:!):	G:|(:
E:/|):	G:!(:	A:|):	C:!):	D:|(:
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Re: Notating 5-limit pental with added 7th harmonics

Post by רועיסיני »

Dave Keenan wrote: Wed Apr 12, 2023 12:57 pm the main problem I see with your proposal to set Archytas' comma to 5\255 is that the best approximation of the 7th harmonic is given by setting Archytas' comma and hence the right arc or tai symbol :|): equal to 6\255
I know that Archytas' comma is 6 steps of 255edo, and indeed in the system I made for 225edo I had it notate this interval, but in the table in page 10 of the PDF it is used for 5 steps and therefore in the sentence that related to that table I treated it as 5 steps. I use 255edo and that table as sepatare independent places from which I can take inspiration or justification for my choice of accidentals for pental with added independent 7th harmonics, which is a rank-3 temperament.
I brought out 255edo only because it had a nice parallel to the different sizes of 8505/8192, 729/700 and 25/24 that occured in my tuning, in order to take inspiration for my symbol for 729/700 from some notation of it, and indeed I have done that (twice including the end of this post). It has other stuff that I don't want to straight up lift for my notation, the most significant of which being that two hemifamity commas (aka 7/5 kleismas) equal the syntonic comma there. If you want a pental EDO that distinguishes all the commas I want to notate differently, 195edo or 320edo may be a good choice.
Dave Keenan wrote: Wed Apr 12, 2023 12:57 pm So the mapping from generators to primes is, in octave-equivalent terms:

0   0   2   1
0   1   5  25
No rank-2 mapping will suffice for me, because I am intending to use this for a rank-3, not a rank-2 temperament, and therefore notating, in this case, both 64/63 and 324/234 by the same accidental will be inaccurate and confusing in my opinion.
Dave Keenan wrote: Wed Apr 12, 2023 2:59 pm Here's notating the easy parts of 255edo with uncontroversial sagittals, based on the complex 7-limit mappings.
What you have made here seems to be a notation for hemipental, which is another 7-limit extension for pental, where (5120/5103)2~81/80 (which is the simplest one 255edo supports), but it doesn't fit my needs, not because I will be moving far from the tonic along the chain of fifths, but because it doesn't distinguish between commas I want to distinguish between, for example, between 5120/5103 and 413343/409600. I also want my notation to be self-explanatory, without additional text (for people who are willing to read about Sagittal or to reconstruct it from the symbols, and can understand the temperament from the seemingly weird intervals used), and using this system people might think I'm using hemipental and not free 7ths, which will be incorrect. One more thing that I think is suboptimal in this approach is that, for example. a 7th harmonic over C://|: turns out to be notated as A:|):, although by Sagittal rules it should be optimally notated as a kind of a B, and I try to avoid these sevenths that look like sixths as much as possible.

In the end, I think I'll use :/|: and ://|: for 5-limit pental, and :|):, :|(:, :/|):, :~|:, and :)//|: for septimal alterations.
:~|: came to my mind because it's :~|(:, from my previous suggestion, without the :|(: flag that suggests it is bigger than the corresponding accidental, and it also has a dedicated size of about 8.73c which is very close to the size of the 25/7 comma/kleisma in pental-with-sevenths, and :)//|: was actually from your original reply with the 255 system, and also because its original size is between the ://|: and :/|): accidentals. This system also has the same symmetry to it as the previous one because :)//|: looks like the apotome complement of :~|:.
Thanks!
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Re: Notating 5-limit pental with added 7th harmonics

Post by Dave Keenan »

I don't think it's necessary (or possible) to distinguish a notation for hemipental (if that's what I was giving) from a notation for pental-plus-pure-7ths. You just have to say precisely what the 7-limit symbols represent in each case. This is basic to Sagittal. It's designed so the meaning of symbols has some flexibility.

I'm glad you've found a notation that you're happy with. If it implies a notation for 255edo, would you please spell out what that is, for future readers.

I note that 255/244 is strictly called the 25/7-kleisma in the systematic comma naming scheme used in the Sagittal documentation. The 25/7-comma would be larger by a Pythagorean comma. The kleisma/comma boundary is at half the size of a Pythagorean comma. See footnote 7 at the bottom of page 8 of https://sagittal.org/sagittal.pdf.

All the best.
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Re: Notating 5-limit pental with added 7th harmonics

Post by רועיסיני »

Dave Keenan wrote: Thu Apr 13, 2023 8:43 am If it implies a notation for 255edo, would you please spell out what that is, for future readers.
It doesn't, unfortunately. I have taken :)//|: from 9 steps in your your original 255edo notation and the only other controvertial Sagittal I've used is for an interval which is mapped to 2 steps of 255, which if you're indeed playing inside 255edo is already taken by :|(:.
Dave Keenan wrote: Thu Apr 13, 2023 8:43 am I note that 255/244 is strictly called the 25/7-kleisma in the systematic comma naming scheme used in the Sagittal documentation. The 25/7-comma would be larger by a Pythagorean comma.
Interesting. The interval I chose to notate for 25/7 is actually smaller than the pythagorean comma by a pythagorean kleisma, so it seems to be neither of those. I chose it because it equals (64/63)/(81/80)2.
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