Notating 5-limit pental with added 7th harmonics
Notating 5-limit pental with added 7th harmonics
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.
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
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:
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
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
Re: Notating 5-limit pental with added 7th harmonics
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:
- 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?
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
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).
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:
Taking octaves into account it is:
We can refer to this temperament here as "complex 7-limit pental".
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
If I had instead shown a redundant parenthesised top row:
We would obtain a different complex 7-limit pental mapping. In octave equivalent form:
Octave specific:
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.
(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
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:
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.
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 F C C F G G 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 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 C F G C F D F G C A C D F G E G A C D B D E G A F A B D E C E F A B G B C E F D F G 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
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 F G B G B C F E F G C B C E G F G 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 E F G B B C E F F G 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 F A F A B C E C E F G B G B C F F G C C E G G B F C G D A E B F C E G B E F B E C E F B G B C E F D F G B C A C D F G E G A C D
Re: Notating 5-limit pental with added 7th harmonics
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.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 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.
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 12:57 pm So the mapping from generators to primes is, in octave-equivalent terms:
0 0 2 1 0 1 5 25
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.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.
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
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.
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.
Re: Notating 5-limit pental with added 7th harmonics
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 If it implies a notation for 255edo, would you please spell out what that is, for future readers.
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.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.