Porcupine notation
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[[Mike Battaglia]] posted the following description of a [[Porcupine]] notation to the Xenharmonic Alliance Facebook Group: ==//<span class="messageBody">This is Paul's (and Herman's?) porcupine naming system which names the notes like this:</span>//== //<span class="messageBody"> A B C D E F G H A</span>// //<span class="messageBody"> LLLLLLLs - porcupine[8]</span>// //<span class="messageBody"> You can see that A-H consist of a chain of notes which are all 10/9 apart; B is a 10/9 generator above A, C is a 10/9 generator above B and so on. You don't have to use all those notes, though. If you only use notes A-G, you get this scale:</span>// //<span class="messageBody"> A B C D E F G A</span>// //<span class="messageBody"> ssssssL - porcupine[7]</span>// //<span class="messageBody"> Or, in terms of modes that you're familiar with</span>// //<span class="messageBody"> G H A B C D E F G</span>// //<span class="messageBody"> LsLLLLLL - porcupine[8]</span>// //<span class="messageBody"> G A B C D E F G</span>// //<span class="messageBody"> Lssssss</span>// //<span class="messageBody"> Therefore, you can see that picking this 8-note naming system for porcupine[8] gives you a 7-note naming for porcupine[7] automatically, by default.</span>// //<span class="messageBody"> Let's say we want to play around in porcupine[7]. What we need now is an accidental to represent the porcupine[7] chroma. Since the chroma for porcupine[7] is about a quarter tone, I'll use ^/v for that. (My reasons for not using #/b will become clear below). Therefore, observe</span>// //<span class="messageBody"> G A B C D E F G - porcupine[7] Lssssss</span>// //<span class="messageBody"> G A B Cv D E F^ G - JI major scale as a MODMOS of porcupine[7]</span>// //<span class="messageBody"> G A B C D E Fv G - awesome otonal major scale everyone should use</span>// //<span class="messageBody"> G Av B C D E F G - porcupine[7] sLsssss</span>// //<span class="messageBody"> G B D - major</span>// //<span class="messageBody"> G Bv D - minor</span>// //<span class="messageBody"> OK, great. But now let's say we want to actually use those 8 notes, and throw H back in as the note that's one 10/9 generator above "G." Now what? Well, we need a chroma for that too then, one which represents L-s in porcupine[8]. Since this chroma is more like the size of a "half step," I'll use #/b for it.</span>// //<span class="messageBody"> So then we get</span>// //<span class="messageBody"> G H A B C D E F G - porcupine[8] LsLLLLLL</span>// //<span class="messageBody"> G Hb A B C D E F G - porcupine[8] sLLLLLLL</span>// //<span class="messageBody"> G H A# B C D E F G - porcupine[8] LLsLLLLL</span>// //<span class="messageBody"> etc. OK, you get it.</span>// //<span class="messageBody"> The interesting thing is that H is the same note as Av, and Hb is the same note as G^, and A# is the same note as Bv, and so on. In other words, you end up with different options as to how you want to enharmonically spell notes. So you can use porcupine[7] if you want, and use the A-G nominals, and pick an accidental for your chroma, and everything will work fine. And then if you want to use porcupine[8] you can just add "H" in there as one generator above "G", and pick a porcupine[8] chroma, and everything will still work fine.</span>// //<span class="messageBody"> You can even switch between the two like Igs was suggesting, and you'll find that every pitch which could ever possibly appear in the porcupine tuning system can be indexed by either one of the two notational systems. You can even use the porcupine[7] chroma over porcupine[8] names if you want and so on (but be careful notationally, it's easy to run into trouble - in the scale G H Av B C D E F G, H and Av are the same thing!).</span>// //<span class="messageBody"> I think this is a pretty natural way to do things if you want to use both porcupine[7] and porcupine[8].</span>// <span class="messageBody">~ ~ ~</span> <span class="commentBody">[[Paul Erlich]] commented: //I didn't come up with it but I like it because each of the four consonant triads in porcupine-7 gets a conventional spellings for its quality. D-F-A and E-G-B are minor, and F-A-C and G-B-D are major. Also the minor seventh chords D-F-A-C and E-G-B-D.//</span> <span class="commentBody">//~ ~ ~//</span> <span class="commentBody">Mike added:</span> //<span class="commentBody">BTW with this notation system, one question is - how many lines do you use in a staff? I guess you'd have to have "staff switches" in the middle of a score if you want to switch into a porcupine[8]-oriented notation or what have you. ... Actually, forget staff changes, why not clef changes? 5 lines can accommodate 1 octave of both porcupine[7] and porcupine[8], so there you go.</span>// //<span class="commentBody">~ ~ ~</span>// ==<span class="messageBody">William Lynch's Approach </span>== William Lynch generally likes to reinvent the wheel. He proposed a system of notation for 15 EDO based on porcupine[8] as the fundamental scale. //I think it's worthwhile to explore these scales from scratch with no previous associations. I chose this old-english based alphabet as the nominals of porcupine[8]. They work for porcupine[7] as well if you leave out "ð" from the scale.// //Porcupine 8 alphabet: b c d ð// //e f þ æ// //Porcupine 7 alphabet:// //b c d e f þ æ// //------------// Some diagrams by [[Andrew Heathwaite]] to illustrate: [[image:porcupine spectrum with letter names.png width="499" height="462"]] [[image:porcupine circles 22edo.png width="499" height="462"]] <span class="messageBody">[[image:porcupine generator chain.png]]</span> ---- Andrew added: //Yikes, so #v (sharp-down) is 40/39 and b^^ (flat-double-up) is 65/64 -- and in a 13-limit Porcupine musical setting, unless you're in 37edo, they're not the same, and they're both necessary if you want to spell your scales logically!// //Let's<span class="text_exposed_show"> say you want an 8:9:10:11:12:13:14 chord with letter names A B C D E F G. If you start it on Av, it's:</span>// //<span class="text_exposed_show"> Av B C D E Fb^ G</span>// //<span class="text_exposed_show"> ...with the flat-up turning a 5/3 into 13/8 by subtracting 40/39.</span>// //<span class="text_exposed_show"> But if I start it on A natural, I get:</span>// //<span class="text_exposed_show"> A B^ C^ D^ E^ Fb^^ G^</span>// //<span class="text_exposed_show"> ...with the flat-double-up turning an 8/5 into 13/8 by adding 65/64!</span>// //<span class="text_exposed_show"> To summarize:</span>// //<span class="text_exposed_show"> #v is the chroma in Porcupine[15].</span>// //<span class="text_exposed_show"> In a 13-limit setting, it's 40/39.</span>// //<span class="text_exposed_show"> In 15edo it's tempered out.</span>// //<span class="text_exposed_show"> In 22edo, it's one degree, which would mean, for example, A#v=A^; so we don't need it to describe all the pitches.</span>// //<span class="text_exposed_show"> In 37edo, it's one degree, but A#v != A^, because ^ is two degrees.</span>// //<span class="text_exposed_show"> b^^ is the chroma in Porcupine[22].</span>// //<span class="text_exposed_show"> In a 13-limit setting, it's 65/64.</span>// //<span class="text_exposed_show"> In 15edo, it's equivalent to ^ and # and therefore totally unnecessary.</span>// //<span class="text_exposed_show"> In 22edo, it's tempered out.</span>// //<span class="text_exposed_show"> In 37edo, it's one degree, which means that #v and bvv are in fact the same, and only one accidental would be needed.</span>// //<span class="text_exposed_show"> In other Porcupinefish systems (like 59edo, for instance), it's smaller than #v, and therefore needs a distinct and preferably smaller symbol.</span>// //<span class="text_exposed_show"> That's my proposal for expanding the Porcupine notation [[https://www.facebook.com/MikeBattagliaMusic|Mike Battaglia]] has described into the 13-limit. Does this make sense to people?</span>// //<span class="commentBody">BTW with this notation system, one question is - how many lines do you use in a staff? I guess you'd have to have "staff switches" in the middle of a score if you want to switch into a porcupine[8]-oriented notation or what have you.</span>// =<span class="commentBody">Kite Giedraitis's approach </span>= <span class="commentBody"> Porcupine divides the perfect 4th into 3 equal steps, thus its [[pergen]] is (P8, P4/3). </span>[[xenharmonic/Ups and Downs Notation|Ups and downs ]][[xenharmonic/Ups and Downs Notation|notation]]<span class="commentBody"> can be used. The generator is vM2. The enharmonic is v</span><span style="vertical-align: super;">3</span><span class="commentBody">A1, thus C^</span><span style="vertical-align: super;">3</span> equals C#. The alternate generator is G - E = ^^m2. The genchain is <span class="commentBody">...^1 - M2 - vM3 - ^4 - P5 - vM6 - ^m7 - __**P1**__ - vM2 - ^m3 - P4 - v5 - ^m6 - m7 - v8...</span> <span class="commentBody">In C, this would be </span> <span class="commentBody">...C^ - D - Ev - F^ - G - Av - Bb^ - __**C**__ - Dv - Eb^ - F - Gv - Ab^ - Bb - Cv...</span> <span class="commentBody">Unlike the other proposals here, the notes without ups and downs still form the familiar chain of 5ths ...Eb Bb F C G D A E B F# C#..., and interval arithmetic remains unchanged. Staff notation is as usual, with the addition of up and down accidentals before certain notes.</span> <span class="commentBody">Porcupine[8] in A is A Bv C</span>^ D Ev F^ G Av A. Porcupine[7] would omit Av. <span class="commentBody">Example comma pump, with brackets indicating an enharmonic equivalence:</span> <span class="commentBody">C.v -- Av.^m -- Dv.v -- [Bvv=Bb^].^m -- Eb^.v -- Ab^.v -- C.v</span> <span class="commentBody">See the ups and downs page for an explanation of the chord names.</span> This is the rank-2 notation,<span class="commentBody"> for a generator of indeterminate cents</span>. The edo notation uses ups and downs to represent one EDOstep. If an edo tempers out porcupine, it must be sharp-3, sharp-6, sharp-9, etc. (See the scale tree on the ups and downs page.) The notation is identical for sharp-3 edos<span class="commentBody"> (15, 22, 29, etc.). For sharp-6 edos (e.g. 30 or 72), simply double all ups and downs. The generator is vvM2. The genchain is P1 - vvM2 - ^^m3 - P4 - vv5... or C - Dvv - Eb^^ - F - Gvv... Sharp-9 edos are rarely used, but the generator would be v</span><span style="font-size: 14.4px; vertical-align: super;">3</span><span class="commentBody">M2. </span>
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<html><head><title>Porcupine Notation</title></head><body><a class="wiki_link" href="/Mike%20Battaglia">Mike Battaglia</a> posted the following description of a <a class="wiki_link" href="/Porcupine">Porcupine</a> notation to the Xenharmonic Alliance Facebook Group:<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-This is Paul's (and Herman's?) porcupine naming system which names the notes like this:"></a><!-- ws:end:WikiTextHeadingRule:0 --><em><span class="messageBody">This is Paul's (and Herman's?) porcupine naming system which names the notes like this:</span></em></h2> <br /> <em><span class="messageBody"> A B C D E F G H A</span></em><br /> <em><span class="messageBody"> LLLLLLLs - porcupine[8]</span></em><br /> <br /> <em><span class="messageBody"> You can see that A-H consist of a chain of notes which are all 10/9 apart; B is a 10/9 generator above A, C is a 10/9 generator above B and so on. You don't have to use all those notes, though. If you only use notes A-G, you get this scale:</span></em><br /> <br /> <em><span class="messageBody"> A B C D E F G A</span></em><br /> <em><span class="messageBody"> ssssssL - porcupine[7]</span></em><br /> <br /> <em><span class="messageBody"> Or, in terms of modes that you're familiar with</span></em><br /> <br /> <em><span class="messageBody"> G H A B C D E F G</span></em><br /> <em><span class="messageBody"> LsLLLLLL - porcupine[8]</span></em><br /> <br /> <em><span class="messageBody"> G A B C D E F G</span></em><br /> <em><span class="messageBody"> Lssssss</span></em><br /> <br /> <em><span class="messageBody"> Therefore, you can see that picking this 8-note naming system for porcupine[8] gives you a 7-note naming for porcupine[7] automatically, by default.</span></em><br /> <br /> <em><span class="messageBody"> Let's say we want to play around in porcupine[7]. What we need now is an accidental to represent the porcupine[7] chroma. Since the chroma for porcupine[7] is about a quarter tone, I'll use ^/v for that. (My reasons for not using #/b will become clear below). Therefore, observe</span></em><br /> <br /> <em><span class="messageBody"> G A B C D E F G - porcupine[7] Lssssss</span></em><br /> <em><span class="messageBody"> G A B Cv D E F^ G - JI major scale as a MODMOS of porcupine[7]</span></em><br /> <em><span class="messageBody"> G A B C D E Fv G - awesome otonal major scale everyone should use</span></em><br /> <em><span class="messageBody"> G Av B C D E F G - porcupine[7] sLsssss</span></em><br /> <em><span class="messageBody"> G B D - major</span></em><br /> <em><span class="messageBody"> G Bv D - minor</span></em><br /> <br /> <em><span class="messageBody"> OK, great. But now let's say we want to actually use those 8 notes, and throw H back in as the note that's one 10/9 generator above "G." Now what? Well, we need a chroma for that too then, one which represents L-s in porcupine[8]. Since this chroma is more like the size of a "half step," I'll use #/b for it.</span></em><br /> <br /> <em><span class="messageBody"> So then we get</span></em><br /> <br /> <em><span class="messageBody"> G H A B C D E F G - porcupine[8] LsLLLLLL</span></em><br /> <em><span class="messageBody"> G Hb A B C D E F G - porcupine[8] sLLLLLLL</span></em><br /> <em><span class="messageBody"> G H A# B C D E F G - porcupine[8] LLsLLLLL</span></em><br /> <br /> <em><span class="messageBody"> etc. OK, you get it.</span></em><br /> <br /> <em><span class="messageBody"> The interesting thing is that H is the same note as Av, and Hb is the same note as G^, and A# is the same note as Bv, and so on. In other words, you end up with different options as to how you want to enharmonically spell notes. So you can use porcupine[7] if you want, and use the A-G nominals, and pick an accidental for your chroma, and everything will work fine. And then if you want to use porcupine[8] you can just add "H" in there as one generator above "G", and pick a porcupine[8] chroma, and everything will still work fine.</span></em><br /> <br /> <em><span class="messageBody"> You can even switch between the two like Igs was suggesting, and you'll find that every pitch which could ever possibly appear in the porcupine tuning system can be indexed by either one of the two notational systems. You can even use the porcupine[7] chroma over porcupine[8] names if you want and so on (but be careful notationally, it's easy to run into trouble - in the scale G H Av B C D E F G, H and Av are the same thing!).</span></em><br /> <br /> <em><span class="messageBody"> I think this is a pretty natural way to do things if you want to use both porcupine[7] and porcupine[8].</span></em><br /> <br /> <span class="messageBody">~ ~ ~</span><br /> <br /> <span class="commentBody"><a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a> commented: <em>I didn't come up with it but I like it because each of the four consonant triads in porcupine-7 gets a conventional spellings for its quality. D-F-A and E-G-B are minor, and F-A-C and G-B-D are major. Also the minor seventh chords D-F-A-C and E-G-B-D.</em></span><br /> <br /> <span class="commentBody"><em>~ ~ ~</em></span><br /> <br /> <span class="commentBody">Mike added:</span><br /> <br /> <em><span class="commentBody">BTW with this notation system, one question is - how many lines do you use in a staff? I guess you'd have to have "staff switches" in the middle of a score if you want to switch into a porcupine[8]-oriented notation or what have you. ... Actually, forget staff changes, why not clef changes? 5 lines can accommodate 1 octave of both porcupine[7] and porcupine[8], so there you go.</span></em><br /> <br /> <em><span class="commentBody">~ ~ ~</span></em><br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-William Lynch's Approach"></a><!-- ws:end:WikiTextHeadingRule:2 --><span class="messageBody">William Lynch's Approach </span></h2> William Lynch generally likes to reinvent the wheel. He proposed a system of notation for 15 EDO based on porcupine[8] as the fundamental scale.<br /> <br /> <em>I think it's worthwhile to explore these scales from scratch with no previous associations. I chose this old-english based alphabet as the nominals of porcupine[8]. They work for porcupine[7] as well if you leave out "ð" from the scale.</em><br /> <br /> <em>Porcupine 8 alphabet: b c d ð</em> <em>e f þ æ</em><br /> <em>Porcupine 7 alphabet:</em> <em>b c d e f þ æ</em><br /> <br /> <em>------------</em><br /> <br /> Some diagrams by <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a> to illustrate:<br /> <br /> <!-- ws:start:WikiTextLocalImageRule:7:<img src="/file/view/porcupine%20spectrum%20with%20letter%20names.png/353430856/499x462/porcupine%20spectrum%20with%20letter%20names.png" alt="" title="" style="height: 462px; width: 499px;" /> --><img src="/file/view/porcupine%20spectrum%20with%20letter%20names.png/353430856/499x462/porcupine%20spectrum%20with%20letter%20names.png" alt="porcupine spectrum with letter names.png" title="porcupine spectrum with letter names.png" style="height: 462px; width: 499px;" /><!-- ws:end:WikiTextLocalImageRule:7 --><br /> <!-- ws:start:WikiTextLocalImageRule:8:<img src="/file/view/porcupine%20circles%2022edo.png/353430910/499x462/porcupine%20circles%2022edo.png" alt="" title="" style="height: 462px; width: 499px;" /> --><img src="/file/view/porcupine%20circles%2022edo.png/353430910/499x462/porcupine%20circles%2022edo.png" alt="porcupine circles 22edo.png" title="porcupine circles 22edo.png" style="height: 462px; width: 499px;" /><!-- ws:end:WikiTextLocalImageRule:8 --><br /> <br /> <span class="messageBody"><!-- ws:start:WikiTextLocalImageRule:9:<img src="/file/view/porcupine%20generator%20chain.png/349915824/porcupine%20generator%20chain.png" alt="" title="" /> --><img src="/file/view/porcupine%20generator%20chain.png/349915824/porcupine%20generator%20chain.png" alt="porcupine generator chain.png" title="porcupine generator chain.png" /><!-- ws:end:WikiTextLocalImageRule:9 --></span><br /> <br /> <br /> <hr /> <br /> <br /> Andrew added:<br /> <br /> <em>Yikes, so #v (sharp-down) is 40/39 and b^^ (flat-double-up) is 65/64 -- and in a 13-limit Porcupine musical setting, unless you're in 37edo, they're not the same, and they're both necessary if you want to spell your scales logically!</em><br /> <br /> <em>Let's<span class="text_exposed_show"> say you want an 8:9:10:11:12:13:14 chord with letter names A B C D E F G. If you start it on Av, it's:</span></em><br /> <br /> <em><span class="text_exposed_show"> Av B C D E Fb^ G</span></em><br /> <br /> <em><span class="text_exposed_show"> ...with the flat-up turning a 5/3 into 13/8 by subtracting 40/39.</span></em><br /> <br /> <em><span class="text_exposed_show"> But if I start it on A natural, I get:</span></em><br /> <br /> <em><span class="text_exposed_show"> A B^ C^ D^ E^ Fb^^ G^</span></em><br /> <br /> <em><span class="text_exposed_show"> ...with the flat-double-up turning an 8/5 into 13/8 by adding 65/64!</span></em><br /> <br /> <em><span class="text_exposed_show"> To summarize:</span></em><br /> <br /> <em><span class="text_exposed_show"> #v is the chroma in Porcupine[15].</span></em><br /> <em><span class="text_exposed_show"> In a 13-limit setting, it's 40/39.</span></em><br /> <em><span class="text_exposed_show"> In 15edo it's tempered out.</span></em><br /> <em><span class="text_exposed_show"> In 22edo, it's one degree, which would mean, for example, A#v=A^; so we don't need it to describe all the pitches.</span></em><br /> <em><span class="text_exposed_show"> In 37edo, it's one degree, but A#v != A^, because ^ is two degrees.</span></em><br /> <br /> <em><span class="text_exposed_show"> b^^ is the chroma in Porcupine[22].</span></em><br /> <em><span class="text_exposed_show"> In a 13-limit setting, it's 65/64.</span></em><br /> <em><span class="text_exposed_show"> In 15edo, it's equivalent to ^ and # and therefore totally unnecessary.</span></em><br /> <em><span class="text_exposed_show"> In 22edo, it's tempered out.</span></em><br /> <em><span class="text_exposed_show"> In 37edo, it's one degree, which means that #v and bvv are in fact the same, and only one accidental would be needed.</span></em><br /> <em><span class="text_exposed_show"> In other Porcupinefish systems (like 59edo, for instance), it's smaller than #v, and therefore needs a distinct and preferably smaller symbol.</span></em><br /> <br /> <em><span class="text_exposed_show"> That's my proposal for expanding the Porcupine notation <a class="wiki_link_ext" href="https://www.facebook.com/MikeBattagliaMusic" rel="nofollow">Mike Battaglia</a> has described into the 13-limit. Does this make sense to people?</span></em><br /> <em><span class="commentBody">BTW with this notation system, one question is - how many lines do you use in a staff? I guess you'd have to have "staff switches" in the middle of a score if you want to switch into a porcupine[8]-oriented notation or what have you.</span></em><br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h1> --><h1 id="toc2"><a name="Kite Giedraitis's approach"></a><!-- ws:end:WikiTextHeadingRule:4 --><span class="commentBody">Kite Giedraitis's approach </span></h1> <br /> <span class="commentBody"> Porcupine divides the perfect 4th into 3 equal steps, thus its <a class="wiki_link" href="/pergen">pergen</a> is (P8, P4/3). </span><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">Ups and downs </a><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">notation</a><span class="commentBody"> can be used. The generator is vM2. The enharmonic is v</span><span style="vertical-align: super;">3</span><span class="commentBody">A1, thus C^</span><span style="vertical-align: super;">3</span> equals C#. The alternate generator is G - E = ^^m2. The genchain is<br /> <span class="commentBody">...^1 - M2 - vM3 - ^4 - P5 - vM6 - ^m7 - <u><strong>P1</strong></u> - vM2 - ^m3 - P4 - v5 - ^m6 - m7 - v8...</span><br /> <span class="commentBody">In C, this would be </span><br /> <span class="commentBody">...C^ - D - Ev - F^ - G - Av - Bb^ - <u><strong>C</strong></u> - Dv - Eb^ - F - Gv - Ab^ - Bb - Cv...</span><br /> <span class="commentBody">Unlike the other proposals here, the notes without ups and downs still form the familiar chain of 5ths ...Eb Bb F C G D A E B F# C#..., and interval arithmetic remains unchanged. Staff notation is as usual, with the addition of up and down accidentals before certain notes.</span><br /> <br /> <span class="commentBody">Porcupine[8] in A is A Bv C</span>^ D Ev F^ G Av A. Porcupine[7] would omit Av.<br /> <br /> <span class="commentBody">Example comma pump, with brackets indicating an enharmonic equivalence:</span><br /> <span class="commentBody">C.v -- Av.^m -- Dv.v -- [Bvv=Bb^].^m -- Eb^.v -- Ab^.v -- C.v</span><br /> <span class="commentBody">See the ups and downs page for an explanation of the chord names.</span><br /> <br /> This is the rank-2 notation,<span class="commentBody"> for a generator of indeterminate cents</span>. The edo notation uses ups and downs to represent one EDOstep. If an edo tempers out porcupine, it must be sharp-3, sharp-6, sharp-9, etc. (See the scale tree on the ups and downs page.) The notation is identical for sharp-3 edos<span class="commentBody"> (15, 22, 29, etc.). For sharp-6 edos (e.g. 30 or 72), simply double all ups and downs. The generator is vvM2. The genchain is P1 - vvM2 - ^^m3 - P4 - vv5... or C - Dvv - Eb^^ - F - Gvv... Sharp-9 edos are rarely used, but the generator would be v</span><span style="font-size: 14.4px; vertical-align: super;">3</span><span class="commentBody">M2. </span></body></html>