Kite's ups and downs notation: Difference between revisions
Wikispaces>TallKite **Imported revision 584790759 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-04 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-06-04 09:04:08 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>584791671</tt>.<br> | ||
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">="Ups and Downs" | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=__"Ups and Downs" Notation__= | ||
Ups and Downs is a notation system developed by [[KiteGiedraitis|Kite]] that works very well with almost all EDOs and rank 2 tunings. It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol "^" and the down symbol "v". There's also the optional mid symbol "~" which undoes ups and downs (see the Cancelling section). | Ups and Downs is a notation system developed by [[KiteGiedraitis|Kite]] that works very well with almost all EDOs and rank 2 tunings. It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol "^" and the down symbol "v". There's also the optional mid symbol "~" which undoes ups and downs (see the Cancelling section). | ||
To understand the ups and downs notation, let's start with an EDO that doesn't need it. 19-EDO is easy to notate because 7 fifths adds up to one EDO-step. So C# is right next to C, and | To understand the ups and downs notation, let's start with an EDO that doesn't need it. 19-EDO is easy to notate because 7 fifths reduced by 4 octaves adds up to one EDO-step. So C# is right next to C, and the keyboard runs C C# Db D D# Eb E etc. Conventional notation works perfectly with 19-EDO as long as you remember that C# and Db are different notes. | ||
In contrast, 22-EDO is hard to notate because 7 fifths are __three__ EDO-steps, and the usual chain of fifths Eb-Bb-F-C-G-D-A-E-B-F#-C# etc. creates the scale C Db B# C# D Eb Fb D# E F. That's very confusing because B#-Db looks ascending on the page but sounds descending. Also a 4:5:6 chord is written C-D#-G, and the major 3rd becomes an aug 2nd. Some people forgo the chain of fifths for a maximally even scale like C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C. But that's confusing because G-D and A-E are dim 5ths. And if your piece is in G or A, that's really bad. A notation system should work in every key! | In contrast, 22-EDO is hard to notate because 7 fifths are __three__ EDO-steps, and the usual chain of fifths Eb-Bb-F-C-G-D-A-E-B-F#-C# etc. creates the scale C Db B# C# D Eb Fb D# E F. That's very confusing because B#-Db looks ascending on the page but sounds descending. Also a 4:5:6 chord is written C-D#-G, and the major 3rd becomes an aug 2nd. Some people forgo the chain of fifths for a maximally even scale like C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C. But that's confusing because G-D and A-E are dim 5ths. And if your piece is in G or A, that's really bad. A notation system should work in every key! | ||
The solution is to use the sharp symbol to mean "raised by 7 fifths", and to use the up symbol "^" to mean "sharpened by one EDO-step". 22-EDO can be written C-Db-Db^-Dv-D-Eb-Eb^-Ev-E-F etc. The notes are pronounced "D-flat-up, D-down", etc. Now the notes run in order. There's a pattern that's not too hard to pick up on, if you remember that there's 3 ups to a sharp. | The solution is to use the sharp symbol to mean "raised by 7 fifths", and to use the up symbol "^" to mean "sharpened by one EDO-step". 22-EDO can be written C - Db - Db^ - Dv - D - Eb - Eb^ - Ev - E - F etc. The notes are pronounced "D-flat-up, D-down", etc. Now the notes run in order. There's a pattern that's not too hard to pick up on, if you remember that there's 3 ups to a sharp. | ||
The names change depending on the key, just like in conventional notation where F# in D major becomes Gb in Db major. So | The names change depending on the key, just like in conventional notation where F# in D major becomes Gb in Db major. So the B scale is B - C - C^ - C#v - C# - D - D^ - D#v - D# - E etc. | ||
The advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible. | The advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible. | ||
Relative notation for 22-EDO is P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - d5 - ^d5 - vP5 - P5 etc. That's pronounced "upminor 2nd, downmajor 3rd", etc. The ups and downs are leading in relative notation but trailing in absolute notation. You can apply this pattern to any key, with certain keys requiring double-sharps or even triple-sharps. The mid notes always form a chain of fifths. | |||
You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See [[Kite's color notation]] for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations. | You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See [[Kite's color notation]] for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations. | ||
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__**Interval arithmetic**__ | __**Interval arithmetic**__ | ||
In ups and downs notation, as in conventional notation, the chain of fifths runs: | In ups and downs notation, as in conventional notation, the chain of fifths runs: | ||
Ebb-Bbb-Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B#-Fx-Cx etc. | Ebb - Bbb - Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# - Fx - Cx etc. | ||
This chain can be expressed in relative notation: | This chain can be expressed in relative notation: | ||
d2-d6-d3-d7-d4-d1-d5-m2-m6-m3-m7-P4-P1-P5-M2-M6-M3-M7-A4-A1-A5-A2-A6-A3-A7 etc. | d2 - d6 - d3 - d7 - d4 - d1 - d5 - m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 - A4 - A1 - A5 - A2 - A6 - A3 - A7 etc. | ||
To name the interval between any two notes, superimpose one chain onto the other, with P1 lining up with the lower note. For example C-E = M3 because M3 means "raised by 4 fifths" and E is 4 fifths away from C. Likewise, C + M3 = E. | To name the interval between any two notes, superimpose one chain onto the other, with P1 lining up with the lower note. For example C-E = M3 because M3 means "raised by 4 fifths" and E is 4 fifths away from C. Likewise, C + M3 = E. | ||
C - G - D - A - E | C - G - D - A - E | ||
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To add any two intervals, superimpose two copies of the relative chain. m3 + M2 = P4: | To add any two intervals, superimpose two copies of the relative chain. m3 + M2 = P4: | ||
m3-m7-P4-P1 | m3 - m7 - P4 - P1 | ||
P1-P5-M2 | P1 - P5 - M2 | ||
Line up the lower P1 with m3 and look for what lies above M2. | Line up the lower P1 with m3 and look for what lies above M2. | ||
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Here's our fifths: C-G, Db-Ab, Db^-Ab^, Dv-Av, D-A, etc. Most fifths *look* like fifths and are easy to find. So do the 4ths. Our 4\22 maj 2nds are C-D, Db-Eb, Db^-Eb^, Dv-Ev, D-E, Eb-F, good until we reach Eb^-Gb, which is a major 2nd that is spelled as a downminor 3rd. Here's this scale's chain of 5ths: | Here's our fifths: C-G, Db-Ab, Db^-Ab^, Dv-Av, D-A, etc. Most fifths *look* like fifths and are easy to find. So do the 4ths. Our 4\22 maj 2nds are C-D, Db-Eb, Db^-Eb^, Dv-Ev, D-E, Eb-F, good until we reach Eb^-Gb, which is a major 2nd that is spelled as a downminor 3rd. Here's this scale's chain of 5ths: | ||
Gb^ Db^ Ab^ Eb^ Bb^ Gb Db Ab Eb Bb F C G D A E B Gv Dv Av Ev Bv | Gb^ - Db^ - Ab^ - Eb^ - Bb^ - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - Gv - Dv - Av - Ev - Bv | ||
The problem is, there are a few places where the sequence of 7 letters breaks, and we actually have runs of 5 letters. This is the essentially pentatonic-friendly nature of 22-EDO asserting itself. By which is meant, 22-EDO pentatonically is like 19-EDO heptatonically, in that ups and downs are not necessary. Here's 22-EDO in pentatonic notation: | The problem is, there are a few places where the sequence of 7 letters breaks, and we actually have runs of 5 letters. This is the essentially pentatonic-friendly nature of 22-EDO asserting itself. By which is meant, 22-EDO pentatonically is like 19-EDO heptatonically, in that ups and downs are not necessary. Here's 22-EDO in pentatonic notation: | ||
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minor keys: C, C^, C#v, C#, D, D^, Eb^, Ev, E, F, F^, F#v, F#, G, G^, G#v, G#, A, Bb, Bb^, Bv, B | minor keys: C, C^, C#v, C#, D, D^, Eb^, Ev, E, F, F^, F#v, F#, G, G^, G#v, G#, A, Bb, Bb^, Bv, B | ||
Major keys are almost entirely natural, down, flat or upflat. The one exception is F^ major, needed because Gb major would use Cb. Likewise, minor keys are mostly natural, up, sharp or downsharp. Exceptions: Ev minor for D# minor, and Bv minor for A# minor, to avoid E#. In addition, three minor keys are named to match their relative major. This isn't as strict a rule, and the other names may be used as alternatives. Thus Bb minor and Bb^ minor are preferred over A^ minor and A#v minor, to match their relative majors Db major and Db^ major. Also Eb^ minor is preferred over D#v minor, to match its relative major Gb^ major. These two keys<span style="line-height: 1.5;"> break the rule for naming black keys because they have a Cb^.There is unfortunately no way to notate these keys and follow the rule!</span> | Major keys are almost entirely natural, down, flat or upflat. The one exception is F^ major, needed because Gb major would use Cb. Likewise, minor keys are mostly natural, up, sharp or downsharp. Exceptions: Ev minor for D# minor, and Bv minor for A# minor, to avoid E#. In addition, three minor keys are named to match their relative major. This isn't as strict a rule, and the other names may be used as alternatives. Thus Bb minor and Bb^ minor are preferred over A^ minor and A#v minor, to match their relative majors Db major and Db^ major. Also Eb^ minor is preferred over D#v minor, to match its relative major Gb^ major. These two keys<span style="line-height: 1.5;"> break the rule for naming black keys because they have a Cb^. There is unfortunately no way to notate these keys and follow the rule!</span> | ||
<span style="line-height: 1.5;">Key signatures: </span> | <span style="line-height: 1.5;">Key signatures: </span> | ||
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__**Other EDOs**__ | ==__**Other EDOs**__== | ||
EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest: | EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest: | ||
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"perfect" EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢ | "perfect" EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢ | ||
fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢ | fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢ | ||
This is in addition to the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy. | This is in addition to the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy. | ||
[[image:The fifth of EDOs 5-53.png width="800" height="1035"]] | |||
The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same "generation" occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. The colored regions of the tree are what I call **kites**. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others. | The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same "generation" occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. The colored regions of the tree are what I call **kites**. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others. | ||
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Eb + m3 --> E# + M3 = G## --> Gbb | Eb + m3 --> E# + M3 = G## --> Gbb | ||
The second special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. (42edo, 49edo, etc. have a fifth wider than 4\7.) In these five edos, there are zero keys per sharp/flat, and all intervals are perfect. That's because the scale that is produced by a chain of fifths is exactly the same scale as produced by a chain of 2nds, 3rds, 4ths, etc. Since any of these intervals is a potential generator, and since the generator is perfect by definition, they must all be perfect. | The second special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. (42edo, 49edo, etc. have a fifth wider than 4\7.) In these five edos, there are zero keys per sharp/flat, and all intervals are perfect. That's because the scale that is produced by a chain of fifths is exactly the same scale as produced by a chain of 2nds, 3rds, 4ths, etc. Since any of these intervals is a potential generator, and since the generator is perfect by definition, they must all be perfect. There are no major or minor intervals. | ||
The chain of fifths in "perfect" EDOs (3/2 maps to 4\7): | The chain of fifths in "perfect" EDOs (3/2 maps to 4\7): | ||
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21edo: 1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8 | 21edo: 1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8 | ||
21edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C | 21edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C | ||
Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. The sharp symbol actually indicates raising by zero EDOsteps, and F = F#. One could simply redefine the sharp and flat symbols to mean up and down in perfect EDOs, perhaps to make one's notation software easier to use. But this would be confusing because B - F# isn't a perfect fifth because it's | Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. The sharp symbol actually indicates raising by zero EDOsteps, and F = F#. One could simply redefine the sharp and flat symbols to mean up and down in perfect EDOs, perhaps to make one's notation software easier to use. But this would be confusing because B - F# isn't a perfect fifth because it's actually . | ||
The 3rd special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo, 18edo and 23edo. Heptatonic fifth-based notation is impossible in these cases. The minor 2nd, which is the sum of five 4ths minus two 8ves, becomes a descending interval. Thus the major 3rd is wider than the perfect 4th, etc. Such EDOs are dealt with below. | The 3rd special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo, 18edo and 23edo. Heptatonic fifth-based notation is impossible in these cases. The minor 2nd, which is the sum of five 4ths minus two 8ves, becomes a descending interval. Thus the major 3rd is wider than the perfect 4th, etc. Such EDOs are dealt with below. | ||
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0-7-13-20 = Cv Evv Gv Bvv is "Cv.vM7", "C down, downmajor seven". | 0-7-13-20 = Cv Evv Gv Bvv is "Cv.vM7", "C down, downmajor seven". | ||
Sus chords: as | Sus chords: as in conventional notation, "sus" means the 3rd is replaced by the named note, a 2nd or 4th. "Sus4" implies a perfect 4th, and other 4ths are specified explicitly as sus^4 for an up-fourth, etc. Some larger edos would have susv4, susvv4, etc. "Sus2" implies a major 2nd. In most edos, this M2 is always a perfect 4th below the perfect 5th, implying an approximate 8:9:12 chord. See the fourthwards EDOs below for an exception. | ||
"Aug" and "dim" chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. An A3,P5 chord is A3 = "aug three chord" | "Aug" and "dim" chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. An A3,P5 chord is A3 = "aug three chord" (not "aug chord", because that refers to the conventional aug chord M3,A5). Likewise d3,P5 is a "dim three chord", and m3,d5 is a "dim" chord. | ||
0-3-13 = C Dv G = Csusv2 | 0-3-13 = C Dv G = Csusv2 | ||
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0-7-13-18-26-32 = C Ev G Bb D F^ = C11(v3,^11) | 0-7-13-18-26-32 = C Ev G Bb D F^ = C11(v3,^11) | ||
You can write out chord progressions using | You can write out chord progressions using ups/downs notation to name the roots. Here's the first 4 chords of Paul Erlich's 22edo composition "Tibia": | ||
G.vM7(no5) = "G downmajor seven, no five" | G.vM7(no5) = "G downmajor seven, no five" | ||
Eb^.v(add9) = "E flat up, downmajor, add nine" | Eb^.v(add9) = "E flat up, downmajor, add nine" | ||
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#VI or vVII | #VI or vVII | ||
VII or vI | VII or vI | ||
These are pronounced "down-two", "up-flat-three", "down-sharp-four", etc. Periods are never needed after the root in relative notation because ups and downs are always leading, never trailing. Here's the "Tibia" chords again: | |||
IvM7(no5) = "one downmajor seven, no five" | IvM7(no5) = "one downmajor seven, no five" | ||
^bVIv(add9) = "up-flat six downmajor, add nine" | ^bVIv(add9) = "up-flat six downmajor, add nine" | ||
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||= ||= || || || etc. ||= ||= || | ||= ||= || || || etc. ||= ||= || | ||
The 22-tone keyboard, with alternate tunings for the black keys ("^3" means triple-up): | |||
||= keyspan from C ||= genspan from C ||= note ||= genspan from C ||= note || | ||= keyspan from C ||= genspan from C ||= note ||= genspan from C ||= note || | ||
||= 0 ||= 0 ||= C ||= ||= || | ||= 0 ||= 0 ||= C ||= ||= || | ||
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==__Generators other than a fifth__== | ==__Generators other than a fifth__== | ||
Porcupine in 22-tone is generated by a 2nd = 3\22 | The main reason to use ups and downs is to allow fifth-generated heptatonic notation in frameworks and EDOs that aren't fully compatible with such a notation, i.e. those not on the sides of the 4\7 kite. The main reason to use a generator other than a fifth is to use a notation more compatible with one's chosen framework or EDO. Thus there is little reason to use ups and downs in such a situation. | ||
An example of a rank-2 tuning with a non-fifth generator is porcupine. Porcupine in 22-tone is generated by a 2nd = 3\22. | |||
Genchain: A# B# C# D# E# F# G# A B C D E F G Ab Bb Cb Db Eb Fb Gb | Genchain: A# B# C# D# E# F# G# A B C D E F G Ab Bb Cb Db Eb Fb Gb | ||
Scale fragment: D D# Eb E | Scale fragment: D D# Eb E | ||
Keyboard: D * * E * * F * * G * * * A * * B * * C * * D | Keyboard: D * * E * * F * * G * * * A * * B * * C * * D | ||
Because | Because sharp = 1 key, ups and downs aren't needed. | ||
The porcupine genchain: | |||
||= genspan from D ||= 22-tone keyspan from D ||= note || | |||
||= -18 ||= 12 ||= G### || | |||
||= -17 ||= 15 ||= Ax || | |||
||= -16 ||= 18 ||= Bx || | |||
||= -15 ||= 21 ||= Cx || | |||
||= -14 ||= 2 ||= Dx || | |||
||= -13 ||= 5 ||= Ex || | |||
||= -12 ||= 8 ||= Fx || | |||
||= -11 ||= 11 ||= Gx || | |||
||= -10 ||= 14 ||= A# || | |||
||= -9 ||= 17 ||= B# || | |||
||= -8 ||= 20 ||= C# || | |||
||= -7 ||= 1 ||= D# || | |||
||= -6 ||= 4 ||= E# || | |||
||= -5 ||= 7 ||= F# || | |||
||= -4 ||= 10 ||= G# || | |||
||= -3 ||= 13 ||= A || | |||
||= -2 ||= 16 ||= B || | |||
||= -1 ||= 19 ||= C || | |||
||= 0 ||= 0 ||= D || | |||
||= 1 ||= 3 ||= E || | |||
||= 2 ||= 6 ||= F || | |||
||= 3 ||= 9 ||= G || | |||
||= 4 ||= 12 ||= Ab || | |||
||= 5 ||= 15 ||= Bb || | |||
||= 6 ||= 18 ||= Cb || | |||
||= 7 ||= 21 ||= Db || | |||
||= 8 ||= 2 ||= Eb || | |||
||= 9 ||= 5 ||= Fb || | |||
||= 10 ||= 8 ||= Gb || | |||
||= 11 ||= 11 ||= Abb || | |||
||= 12 ||= 14 ||= Bbb || | |||
||= 13 ||= 17 ||= Cbb || | |||
||= 14 ||= 20 ||= Dbb || | |||
||= 15 ||= 1 ||= Ebb || | |||
||= 16 ||= 4 ||= Fbb || | |||
||= 17 ||= 7 ||= Gbb || | |||
||= 18 ||= 10 ||= Abbb || | |||
||= ||= etc. ||= || | |||
The 22-tone | The 22-tone keyboard, tuned to a rank-2 porcupine tuning, with alternate tunings for the black keys: | ||
||= keyspan from D ||= genspan from D ||= note ||= genspan from D ||= note || | ||= keyspan from D ||= genspan from D ||= note ||= genspan from D ||= note || | ||
||= 0 ||= 0 ||= D ||= ||= || | ||= 0 ||= 0 ||= D ||= ||= || | ||
||= 1 ||= -7 ||= D# ||= +15 ||= Ebb || | ||= 1 ||= -7 ||= D# ||= +15 ||= Ebb || | ||
||= 2 ||= - | ||= 2 ||= -14 ||= Dx ||= +8 ||= Eb || | ||
||= 3 ||= | ||= 3 ||= +1 ||= E ||= ||= || | ||
||= 4 ||= | ||= 4 ||= -6 ||= E# ||= +16 ||= Fbb || | ||
||= 5 ||= - | ||= 5 ||= -13 ||= Ex ||= +9 ||= Fb || | ||
||= 6 ||= | ||= 6 ||= +2 ||= F ||= ||= || | ||
||= 7 ||= - | ||= 7 ||= -5 ||= F# ||= +17 ||= Gbb || | ||
||= 8 ||= | ||= 8 ||= -12 ||= Fx ||= +10 ||= Gb || | ||
||= 9 ||= | ||= 9 ||= +3 ||= G ||= ||= || | ||
||= 10 ||= - | ||= 10 ||= -4 ||= G# ||= +18 ||= Abbb || | ||
||= 11 ||= -11 ||= | ||= 11 ||= -11 ||= Gx ||= +11 ||= Abb || | ||
||= 12 ||= - | ||= 12 ||= -18 ||= G### ||= +4 ||= Ab || | ||
||= 13 ||= | ||= 13 ||= -3 ||= A ||= ||= || | ||
||= 14 ||= - | ||= 14 ||= -10 ||= Bx ||= +12 ||= Bbb || | ||
||= 15 ||= - | ||= 15 ||= -17 ||= B# ||= +5 ||= Bb || | ||
||= 16 ||= - | ||= 16 ||= -2 ||= B ||= ||= || | ||
||= 17 ||= | ||= 17 ||= -9 ||= Cx ||= +13 ||= Cbb || | ||
||= 18 ||= - | ||= 18 ||= -16 ||= C# ||= +6 ||= Cb || | ||
||= 19 ||= - | ||= 19 ||= -1 ||= C ||= ||= || | ||
||= 20 ||= - | ||= 20 ||= -8 ||= Dx ||= +14 ||= Dbb || | ||
||= 21 ||= | ||= 21 ||= -15 ||= D# ||= +7 ||= Db || | ||
||= 22 ||= 0 ||= | ||= 22 ||= 0 ||= D ||= ||= ||</pre></div> | ||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Ups and Downs Notation</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation"></a><!-- ws:end:WikiTextHeadingRule:0 -->&quot;Ups and Downs&quot; Notation</h1> | <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Ups and Downs Notation</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x&quot;Ups and Downs&quot; Notation"></a><!-- ws:end:WikiTextHeadingRule:0 --><u>&quot;Ups and Downs&quot; Notation</u></h1> | ||
<br /> | <br /> | ||
Ups and Downs is a notation system developed by <a class="wiki_link" href="/KiteGiedraitis">Kite</a> that works very well with almost all EDOs and rank 2 tunings. It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol &quot;^&quot; and the down symbol &quot;v&quot;. There's also the optional mid symbol &quot;~&quot; which undoes ups and downs (see the Cancelling section).<br /> | Ups and Downs is a notation system developed by <a class="wiki_link" href="/KiteGiedraitis">Kite</a> that works very well with almost all EDOs and rank 2 tunings. It only adds 3 symbols to standard notation, so it's very easy to learn. The name comes from the up symbol &quot;^&quot; and the down symbol &quot;v&quot;. There's also the optional mid symbol &quot;~&quot; which undoes ups and downs (see the Cancelling section).<br /> | ||
<br /> | <br /> | ||
To understand the ups and downs notation, let's start with an EDO that doesn't need it. 19-EDO is easy to notate because 7 fifths adds up to one EDO-step. So C# is right next to C, and | To understand the ups and downs notation, let's start with an EDO that doesn't need it. 19-EDO is easy to notate because 7 fifths reduced by 4 octaves adds up to one EDO-step. So C# is right next to C, and the keyboard runs C C# Db D D# Eb E etc. Conventional notation works perfectly with 19-EDO as long as you remember that C# and Db are different notes.<br /> | ||
<br /> | <br /> | ||
In contrast, 22-EDO is hard to notate because 7 fifths are <u>three</u> EDO-steps, and the usual chain of fifths Eb-Bb-F-C-G-D-A-E-B-F#-C# etc. creates the scale C Db B# C# D Eb Fb D# E F. That's very confusing because B#-Db looks ascending on the page but sounds descending. Also a 4:5:6 chord is written C-D#-G, and the major 3rd becomes an aug 2nd. Some people forgo the chain of fifths for a maximally even scale like C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C. But that's confusing because G-D and A-E are dim 5ths. And if your piece is in G or A, that's really bad. A notation system should work in every key!<br /> | In contrast, 22-EDO is hard to notate because 7 fifths are <u>three</u> EDO-steps, and the usual chain of fifths Eb-Bb-F-C-G-D-A-E-B-F#-C# etc. creates the scale C Db B# C# D Eb Fb D# E F. That's very confusing because B#-Db looks ascending on the page but sounds descending. Also a 4:5:6 chord is written C-D#-G, and the major 3rd becomes an aug 2nd. Some people forgo the chain of fifths for a maximally even scale like C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C. But that's confusing because G-D and A-E are dim 5ths. And if your piece is in G or A, that's really bad. A notation system should work in every key!<br /> | ||
<br /> | <br /> | ||
The solution is to use the sharp symbol to mean &quot;raised by 7 fifths&quot;, and to use the up symbol &quot;^&quot; to mean &quot;sharpened by one EDO-step&quot;. 22-EDO can be written C-Db-Db^-Dv-D-Eb-Eb^-Ev-E-F etc. The notes are pronounced &quot;D-flat-up, D-down&quot;, etc. Now the notes run in order. There's a pattern that's not too hard to pick up on, if you remember that there's 3 ups to a sharp.<br /> | The solution is to use the sharp symbol to mean &quot;raised by 7 fifths&quot;, and to use the up symbol &quot;^&quot; to mean &quot;sharpened by one EDO-step&quot;. 22-EDO can be written C - Db - Db^ - Dv - D - Eb - Eb^ - Ev - E - F etc. The notes are pronounced &quot;D-flat-up, D-down&quot;, etc. Now the notes run in order. There's a pattern that's not too hard to pick up on, if you remember that there's 3 ups to a sharp.<br /> | ||
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The names change depending on the key, just like in conventional notation where F# in D major becomes Gb in Db major. So | The names change depending on the key, just like in conventional notation where F# in D major becomes Gb in Db major. So the B scale is B - C - C^ - C#v - C# - D - D^ - D#v - D# - E etc.<br /> | ||
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The advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible.<br /> | The advantage to this notation is that you always know where your fifth is. And hence your 4th, and your major 9th, hence the maj 2nd and the min 7th too. You have convenient landmarks to find your way around, built into the notation. The notation is a map of unfamiliar territory, and we want this map to be as easy to read as possible.<br /> | ||
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Relative notation for 22-EDO is P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - d5 - ^d5 - vP5 - P5 etc. That's pronounced &quot;upminor 2nd, downmajor 3rd&quot;, etc. The ups and downs are leading in relative notation but trailing in absolute notation. You can apply this pattern to any key, with certain keys requiring double-sharps or even triple-sharps. The mid notes always form a chain of fifths.<br /> | |||
<br /> | <br /> | ||
You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See <a class="wiki_link" href="/Kite%27s%20color%20notation">Kite's color notation</a> for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations.<br /> | You can loosely relate the ups and downs to JI: major = red or fifthward white, downmajor = yellow, upminor = green, minor = blue or fourthwards white. Or simply up = green, down = yellow, and mid = white, blue or red. (See <a class="wiki_link" href="/Kite%27s%20color%20notation">Kite's color notation</a> for an explanation of the colors.) These correlations are for 22-EDO only, other EDOs have other correlations.<br /> | ||
| Line 1,125: | Line 1,129: | ||
<u><strong>Interval arithmetic</strong></u><br /> | <u><strong>Interval arithmetic</strong></u><br /> | ||
In ups and downs notation, as in conventional notation, the chain of fifths runs:<br /> | In ups and downs notation, as in conventional notation, the chain of fifths runs:<br /> | ||
Ebb-Bbb-Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#-A#-E#-B#-Fx-Cx etc.<br /> | Ebb - Bbb - Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# - Fx - Cx etc.<br /> | ||
This chain can be expressed in relative notation:<br /> | This chain can be expressed in relative notation:<br /> | ||
d2-d6-d3-d7-d4-d1-d5-m2-m6-m3-m7-P4-P1-P5-M2-M6-M3-M7-A4-A1-A5-A2-A6-A3-A7 etc.<br /> | d2 - d6 - d3 - d7 - d4 - d1 - d5 - m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 - A4 - A1 - A5 - A2 - A6 - A3 - A7 etc.<br /> | ||
To name the interval between any two notes, superimpose one chain onto the other, with P1 lining up with the lower note. For example C-E = M3 because M3 means &quot;raised by 4 fifths&quot; and E is 4 fifths away from C. Likewise, C + M3 = E.<br /> | To name the interval between any two notes, superimpose one chain onto the other, with P1 lining up with the lower note. For example C-E = M3 because M3 means &quot;raised by 4 fifths&quot; and E is 4 fifths away from C. Likewise, C + M3 = E.<br /> | ||
C - G - D - A - E<br /> | C - G - D - A - E<br /> | ||
| Line 1,133: | Line 1,137: | ||
<br /> | <br /> | ||
To add any two intervals, superimpose two copies of the relative chain. m3 + M2 = P4:<br /> | To add any two intervals, superimpose two copies of the relative chain. m3 + M2 = P4:<br /> | ||
m3-m7-P4-P1<br /> | m3 - m7 - P4 - P1<br /> | ||
P1-P5-M2<br /> | P1 - P5 - M2<br /> | ||
Line up the lower P1 with m3 and look for what lies above M2.<br /> | Line up the lower P1 with m3 and look for what lies above M2.<br /> | ||
<br /> | <br /> | ||
| Line 1,146: | Line 1,150: | ||
Here's our fifths: C-G, Db-Ab, Db^-Ab^, Dv-Av, D-A, etc. Most fifths *look* like fifths and are easy to find. So do the 4ths. Our 4\22 maj 2nds are C-D, Db-Eb, Db^-Eb^, Dv-Ev, D-E, Eb-F, good until we reach Eb^-Gb, which is a major 2nd that is spelled as a downminor 3rd. Here's this scale's chain of 5ths:<br /> | Here's our fifths: C-G, Db-Ab, Db^-Ab^, Dv-Av, D-A, etc. Most fifths *look* like fifths and are easy to find. So do the 4ths. Our 4\22 maj 2nds are C-D, Db-Eb, Db^-Eb^, Dv-Ev, D-E, Eb-F, good until we reach Eb^-Gb, which is a major 2nd that is spelled as a downminor 3rd. Here's this scale's chain of 5ths:<br /> | ||
<br /> | <br /> | ||
Gb^ Db^ Ab^ Eb^ Bb^ Gb Db Ab Eb Bb F C G D A E B Gv Dv Av Ev Bv<br /> | Gb^ - Db^ - Ab^ - Eb^ - Bb^ - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - Gv - Dv - Av - Ev - Bv<br /> | ||
<br /> | <br /> | ||
The problem is, there are a few places where the sequence of 7 letters breaks, and we actually have runs of 5 letters. This is the essentially pentatonic-friendly nature of 22-EDO asserting itself. By which is meant, 22-EDO pentatonically is like 19-EDO heptatonically, in that ups and downs are not necessary. Here's 22-EDO in pentatonic notation:<br /> | The problem is, there are a few places where the sequence of 7 letters breaks, and we actually have runs of 5 letters. This is the essentially pentatonic-friendly nature of 22-EDO asserting itself. By which is meant, 22-EDO pentatonically is like 19-EDO heptatonically, in that ups and downs are not necessary. Here's 22-EDO in pentatonic notation:<br /> | ||
| Line 1,163: | Line 1,167: | ||
minor keys: C, C^, C#v, C#, D, D^, Eb^, Ev, E, F, F^, F#v, F#, G, G^, G#v, G#, A, Bb, Bb^, Bv, B<br /> | minor keys: C, C^, C#v, C#, D, D^, Eb^, Ev, E, F, F^, F#v, F#, G, G^, G#v, G#, A, Bb, Bb^, Bv, B<br /> | ||
<br /> | <br /> | ||
Major keys are almost entirely natural, down, flat or upflat. The one exception is F^ major, needed because Gb major would use Cb. Likewise, minor keys are mostly natural, up, sharp or downsharp. Exceptions: Ev minor for D# minor, and Bv minor for A# minor, to avoid E#. In addition, three minor keys are named to match their relative major. This isn't as strict a rule, and the other names may be used as alternatives. Thus Bb minor and Bb^ minor are preferred over A^ minor and A#v minor, to match their relative majors Db major and Db^ major. Also Eb^ minor is preferred over D#v minor, to match its relative major Gb^ major. These two keys<span style="line-height: 1.5;"> break the rule for naming black keys because they have a Cb^.There is unfortunately no way to notate these keys and follow the rule!</span> | Major keys are almost entirely natural, down, flat or upflat. The one exception is F^ major, needed because Gb major would use Cb. Likewise, minor keys are mostly natural, up, sharp or downsharp. Exceptions: Ev minor for D# minor, and Bv minor for A# minor, to avoid E#. In addition, three minor keys are named to match their relative major. This isn't as strict a rule, and the other names may be used as alternatives. Thus Bb minor and Bb^ minor are preferred over A^ minor and A#v minor, to match their relative majors Db major and Db^ major. Also Eb^ minor is preferred over D#v minor, to match its relative major Gb^ major. These two keys<span style="line-height: 1.5;"> break the rule for naming black keys because they have a Cb^. There is unfortunately no way to notate these keys and follow the rule!</span><br /> | ||
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<span style="line-height: 1.5;">Key signatures: </span><br /> | <span style="line-height: 1.5;">Key signatures: </span><br /> | ||
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<u><strong>Other EDOs</strong></u>< | <!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x&quot;Ups and Downs&quot; Notation-Other EDOs"></a><!-- ws:end:WikiTextHeadingRule:2 --><u><strong>Other EDOs</strong></u></h2> | ||
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EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:<br /> | EDOs come in 5 categories, based on the size of the fifth. From widest to narrowest:<br /> | ||
&quot;fifth-less&quot; EDOs, with fifths wider than 720¢<br /> | &quot;fifth-less&quot; EDOs, with fifths wider than 720¢<br /> | ||
| Line 1,192: | Line 1,195: | ||
&quot;perfect&quot; EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢<br /> | &quot;perfect&quot; EDOs, with a fifth = four sevenths of an octave = 4\7 = 686¢<br /> | ||
fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢<br /> | fourthwards EDOs aka Mavila EDOs, with a fifth less than 686¢<br /> | ||
<br /> | <br /> | ||
This is in addition to the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy.<br /> | This is in addition to the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO. The fifth is defined as the nearest approximation to 3/2. There is a little leeway to this in certain EDOs like 18 which have two possible fifths with nearly equal accuracy.<br /> | ||
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The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same &quot;generation&quot; occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. The colored regions of the tree are what I call <strong>kites</strong>. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others.<br /> | The above diagram is actually a section of the Stern-Brocot tree. The tree usually has ratios, not octave fractions (i.e. 4/7, not 4\7 as above). Also it's usually arranged vertically with nodes of the same &quot;generation&quot; occurring at the same height. For example, 5\9 and 7\12 are both children of 4\7, and would usually be level with each other. Here the nodes are arranged vertically by denominator, i.e., the EDO itself. The colored regions of the tree are what I call <strong>kites</strong>. The heptatonic kite is blue and the pentatonic kite is orange. Every kite has a head (4\7 for the blue kite), a central spine (8\14, 12\21, etc.), a fifthward side (7\12, 11\19, etc.) and a fourthward side (5\9, 9\16, etc.). Every node not on a spine is part of three kites. It's the head of one kite and on the side of two others.<br /> | ||
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Black and white keys: C * * * * * * * * D * * * * * * * * E * * * F * * * * * * * * G * * * * * * * * A * * * * * * * * B * * * C<br /> | Black and white keys: C * * * * * * * * D * * * * * * * * E * * * F * * * * * * * * G * * * * * * * * A * * * * * * * * B * * * C<br /> | ||
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<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Naming Chords"></a><!-- ws:end:WikiTextHeadingRule:4 --><u>Naming Chords</u></h1> | ||
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Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos. The quality of an interval is defined by its position on the chain of 5ths (or more generally, the chain of generators). Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc.<br /> | Ups and downs allow us to name any chord easily. First we need an exact definition of major, minor, perfect, etc. that works with all edos. The quality of an interval is defined by its position on the chain of 5ths (or more generally, the chain of generators). Perfect is 0-1 steps away, major/minor are 2-5 steps away, aug/dim are 6-12 steps away, etc.<br /> | ||
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Eb + m3 --&gt; E# + M3 = G## --&gt; Gbb<br /> | Eb + m3 --&gt; E# + M3 = G## --&gt; Gbb<br /> | ||
<br /> | <br /> | ||
The second special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. (42edo, 49edo, etc. have a fifth wider than 4\7.) In these five edos, there are zero keys per sharp/flat, and all intervals are perfect. That's because the scale that is produced by a chain of fifths is exactly the same scale as produced by a chain of 2nds, 3rds, 4ths, etc. Since any of these intervals is a potential generator, and since the generator is perfect by definition, they must all be perfect.<br /> | The second special case is when the edo's fifth equals 4\7, as in 7edo, 14edo, 21edo, 28edo, and 35edo. (42edo, 49edo, etc. have a fifth wider than 4\7.) In these five edos, there are zero keys per sharp/flat, and all intervals are perfect. That's because the scale that is produced by a chain of fifths is exactly the same scale as produced by a chain of 2nds, 3rds, 4ths, etc. Since any of these intervals is a potential generator, and since the generator is perfect by definition, they must all be perfect. There are no major or minor intervals.<br /> | ||
<br /> | <br /> | ||
The chain of fifths in &quot;perfect&quot; EDOs (3/2 maps to 4\7):<br /> | The chain of fifths in &quot;perfect&quot; EDOs (3/2 maps to 4\7):<br /> | ||
| Line 1,277: | Line 1,279: | ||
21edo: 1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8<br /> | 21edo: 1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8<br /> | ||
21edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C<br /> | 21edo: C - C^ - Dv - D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C<br /> | ||
Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. The sharp symbol actually indicates raising by zero EDOsteps, and F = F#. One could simply redefine the sharp and flat symbols to mean up and down in perfect EDOs, perhaps to make one's notation software easier to use. But this would be confusing because B - F# isn't a perfect fifth because it's | Just as ups and downs aren't needed in 19edo, sharps and flats aren't needed in 21edo. The sharp symbol actually indicates raising by zero EDOsteps, and F = F#. One could simply redefine the sharp and flat symbols to mean up and down in perfect EDOs, perhaps to make one's notation software easier to use. But this would be confusing because B - F# isn't a perfect fifth because it's actually .<br /> | ||
<br /> | <br /> | ||
The 3rd special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo, 18edo and 23edo. Heptatonic fifth-based notation is impossible in these cases. The minor 2nd, which is the sum of five 4ths minus two 8ves, becomes a descending interval. Thus the major 3rd is wider than the perfect 4th, etc. Such EDOs are dealt with below.<br /> | The 3rd special case is when the edo's fifth is wider than 3\5, as in 8edo, 13edo, 18edo and 23edo. Heptatonic fifth-based notation is impossible in these cases. The minor 2nd, which is the sum of five 4ths minus two 8ves, becomes a descending interval. Thus the major 3rd is wider than the perfect 4th, etc. Such EDOs are dealt with below.<br /> | ||
| Line 1,283: | Line 1,285: | ||
Chord names are based entirely on the ups/downs interval names, not on JI ratios. This avoids identifying one EDOstep with multiple ratios, as happens in 22edo when 0-7-18 implies 4:5:7 but 0-9-18 implies 9:12:16. 18\22 is neither 7/4 nor 16/9, it's 18\22!<br /> | Chord names are based entirely on the ups/downs interval names, not on JI ratios. This avoids identifying one EDOstep with multiple ratios, as happens in 22edo when 0-7-18 implies 4:5:7 but 0-9-18 implies 9:12:16. 18\22 is neither 7/4 nor 16/9, it's 18\22!<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Naming Chords-22edo chord names"></a><!-- ws:end:WikiTextHeadingRule:6 --><u>22edo chord names</u></h2> | ||
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Let's review the 22edo interval names:<br /> | Let's review the 22edo interval names:<br /> | ||
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0-7-13-20 = Cv Evv Gv Bvv is &quot;Cv.vM7&quot;, &quot;C down, downmajor seven&quot;.<br /> | 0-7-13-20 = Cv Evv Gv Bvv is &quot;Cv.vM7&quot;, &quot;C down, downmajor seven&quot;.<br /> | ||
<br /> | <br /> | ||
Sus chords: as | Sus chords: as in conventional notation, &quot;sus&quot; means the 3rd is replaced by the named note, a 2nd or 4th. &quot;Sus4&quot; implies a perfect 4th, and other 4ths are specified explicitly as sus^4 for an up-fourth, etc. Some larger edos would have susv4, susvv4, etc. &quot;Sus2&quot; implies a major 2nd. In most edos, this M2 is always a perfect 4th below the perfect 5th, implying an approximate 8:9:12 chord. See the fourthwards EDOs below for an exception.<br /> | ||
<br /> | <br /> | ||
&quot;Aug&quot; and &quot;dim&quot; chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. An A3,P5 chord is A3 = &quot;aug three chord&quot; | &quot;Aug&quot; and &quot;dim&quot; chords: many of the larger EDOs have an aug 3rd distinct from the perfect 4th, and a dim 3rd distinct from the major 2nd. An A3,P5 chord is A3 = &quot;aug three chord&quot; (not &quot;aug chord&quot;, because that refers to the conventional aug chord M3,A5). Likewise d3,P5 is a &quot;dim three chord&quot;, and m3,d5 is a &quot;dim&quot; chord.<br /> | ||
<br /> | <br /> | ||
0-3-13 = C Dv G = Csusv2<br /> | 0-3-13 = C Dv G = Csusv2<br /> | ||
| Line 1,370: | Line 1,372: | ||
0-7-13-18-26-32 = C Ev G Bb D F^ = C11(v3,^11)<br /> | 0-7-13-18-26-32 = C Ev G Bb D F^ = C11(v3,^11)<br /> | ||
<br /> | <br /> | ||
You can write out chord progressions using | You can write out chord progressions using ups/downs notation to name the roots. Here's the first 4 chords of Paul Erlich's 22edo composition &quot;Tibia&quot;:<br /> | ||
G.vM7(no5) = &quot;G downmajor seven, no five&quot;<br /> | G.vM7(no5) = &quot;G downmajor seven, no five&quot;<br /> | ||
Eb^.v(add9) = &quot;E flat up, downmajor, add nine&quot;<br /> | Eb^.v(add9) = &quot;E flat up, downmajor, add nine&quot;<br /> | ||
| Line 1,400: | Line 1,402: | ||
#VI or vVII<br /> | #VI or vVII<br /> | ||
VII or vI<br /> | VII or vI<br /> | ||
<br /> | <br /> | ||
These are pronounced &quot;down-two&quot;, &quot;up-flat-three&quot;, &quot;down-sharp-four&quot;, etc. Periods are never needed after the root in relative notation because ups and downs are always leading, never trailing. Here's the &quot;Tibia&quot; chords again:<br /> | |||
<br /> | |||
<br /> | |||
IvM7(no5) = &quot;one downmajor seven, no five&quot;<br /> | IvM7(no5) = &quot;one downmajor seven, no five&quot;<br /> | ||
^bVIv(add9) = &quot;up-flat six downmajor, add nine&quot;<br /> | ^bVIv(add9) = &quot;up-flat six downmajor, add nine&quot;<br /> | ||
| Line 1,409: | Line 1,412: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:4037:&lt;img src=&quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 1035px; width: 800px;&quot; /&gt; --><img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg/570451171/800x1035/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%201.jpg" alt="Tibia in G with ^v, rygb 1.jpg" title="Tibia in G with ^v, rygb 1.jpg" style="height: 1035px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:4037 --><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><!-- ws:end:WikiTextHeadingRule:8 --><!-- ws:start:WikiTextLocalImageRule:4038:&lt;img src=&quot;/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 957px; width: 800px;&quot; /&gt; --><img src="/file/view/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg/570451199/800x957/Tibia%20in%20G%20with%20%5Ev%2C%20rygb%202.jpg" alt="Tibia in G with ^v, rygb 2.jpg" title="Tibia in G with ^v, rygb 2.jpg" style="height: 957px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:4038 --></h2> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><!-- ws:end:WikiTextHeadingRule:10 --> </h2> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><!-- ws:end:WikiTextHeadingRule:10 --> </h2> | ||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><a name="Naming Chords-Chord names in other EDOs"></a><!-- ws:end:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:12:&lt;h2&gt; --><h2 id="toc6"><!-- ws:end:WikiTextHeadingRule:12 --> </h2> | ||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Naming Chords-Chord names in other EDOs"></a><!-- ws:end:WikiTextHeadingRule:14 --><u>Chord names in other EDOs</u></h2> | |||
<br /> | <br /> | ||
15edo: 3 keys per #/b, so ups and downs are needed.<br /> | 15edo: 3 keys per #/b, so ups and downs are needed.<br /> | ||
| Line 1,506: | Line 1,509: | ||
0-12-18 = susv4<br /> | 0-12-18 = susv4<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Naming Chords-Cross-EDO considerations"></a><!-- ws:end:WikiTextHeadingRule:16 --><strong><u>Cross-EDO considerations</u></strong></h2> | ||
<br /> | <br /> | ||
In 22edo, the major chord is 0-8-13 = 0¢-436¢-709¢. In 19edo, it's 0-6-11 = 0¢-379¢-695¢. The two chords sound quite different, because &quot;major 3rd&quot; is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound red and 19edo major chords sound yellow.<br /> | In 22edo, the major chord is 0-8-13 = 0¢-436¢-709¢. In 19edo, it's 0-6-11 = 0¢-379¢-695¢. The two chords sound quite different, because &quot;major 3rd&quot; is defined only in terms of the fifth, not in terms of what JI ratios it approximates. To describe the sound of the chord, color notation can be used. 22edo major chords sound red and 19edo major chords sound yellow.<br /> | ||
| Line 1,517: | Line 1,520: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Naming Chords-EDOs with an inaccurate 3/2"></a><!-- ws:end:WikiTextHeadingRule:18 --><u>EDOs with an inaccurate 3/2</u></h2> | ||
<br /> | <br /> | ||
Not counting the trivial edos 2, 3, 4 and 6, there are only seven such edos. As seen in the above diagram, they are the ones to the left of the central line in the light blue region, plus the ones to the right of the central line in the orange region. The ones on the left edge of the blue region are the fourthward ones like 16edo, and have been dealt with already. 23edo can be notated similarly to 16edo by using a fifth of 13\23 instead of 14\23. That leaves only four edos: 8, 11, 13, and 18.<br /> | Not counting the trivial edos 2, 3, 4 and 6, there are only seven such edos. As seen in the above diagram, they are the ones to the left of the central line in the light blue region, plus the ones to the right of the central line in the orange region. The ones on the left edge of the blue region are the fourthward ones like 16edo, and have been dealt with already. 23edo can be notated similarly to 16edo by using a fifth of 13\23 instead of 14\23. That leaves only four edos: 8, 11, 13, and 18.<br /> | ||
| Line 1,591: | Line 1,594: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:20:&lt;h1&gt; --><h1 id="toc10"><a name="Summary of EDO notation"></a><!-- ws:end:WikiTextHeadingRule:20 --><u><strong>Summary of EDO notation</strong></u></h1> | ||
<br /> | <br /> | ||
Besides the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO, there are five EDO categories, based on the size of the fifth:<br /> | Besides the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO, there are five EDO categories, based on the size of the fifth:<br /> | ||
| Line 2,963: | Line 2,966: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Summary of EDO notation--&quot;Fifth-less&quot; EDOs (8, 11, 13 and 18)"></a><!-- ws:end:WikiTextHeadingRule:22 --><u><strong>&quot;Fifth-less&quot; EDOs (8, 11, 13 and 18)</strong></u></h3> | ||
<br /> | <br /> | ||
<strong><u>8edo</u>:</strong> (generator = 1\8 = perfect 2nd = 150¢)<br /> | <strong><u>8edo</u>:</strong> (generator = 1\8 = perfect 2nd = 150¢)<br /> | ||
| Line 2,993: | Line 2,996: | ||
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb<br /> | E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Summary of EDO notation--Alternate pentatonic notation for EDOs 8, 13 and 18"></a><!-- ws:end:WikiTextHeadingRule:24 --><u><strong>Alternate pentatonic notation for EDOs 8, 13 and 18</strong></u></h3> | ||
<br /> | <br /> | ||
All three EDOs use the same pentatonic fifthwards chain of fifths: ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d etc.<br /> | All three EDOs use the same pentatonic fifthwards chain of fifths: ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d etc.<br /> | ||
| Line 3,014: | Line 3,017: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Summary of EDO notation--Fourthward EDOs (9, 16 and 23)"></a><!-- ws:end:WikiTextHeadingRule:26 --><u>Fourthward EDOs (9, 16 and 23)</u></h3> | ||
<br /> | <br /> | ||
All fourthwards EDOs use the same chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 etc.<br /> | All fourthwards EDOs use the same chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 etc.<br /> | ||
| Line 3,035: | Line 3,038: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Summary of EDO notation--&quot;Perfect&quot; EDOs (7, 14, 21, 28 and 35)"></a><!-- ws:end:WikiTextHeadingRule:28 --><u>&quot;Perfect&quot; EDOs (7, 14, 21, 28 and 35)</u></h3> | ||
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All perfect EDOs use the same chain of fifths: P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.<br /> | All perfect EDOs use the same chain of fifths: P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.<br /> | ||
| Line 3,066: | Line 3,069: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="Summary of EDO notation--Pentatonic EDOs (5, 10, 15, 20, 25 and 30)"></a><!-- ws:end:WikiTextHeadingRule:30 --><u>Pentatonic EDOs (5, 10, 15, 20, 25 and 30)</u></h3> | ||
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All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br /> | All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br /> | ||
| Line 3,101: | Line 3,104: | ||
P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8<br /> | P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="Summary of EDO notation--Alternative pentatonic notation for pentatonic EDOs:"></a><!-- ws:end:WikiTextHeadingRule:32 --><u>Alternative pentatonic notation for pentatonic EDOs:</u></h3> | ||
<br /> | <br /> | ||
Pentatonic fourthwards chain of fifthoids: Ms3 - Ms7 - P4d - P1 - P5d - ms3 - ms7 - d4d etc.<br /> | Pentatonic fourthwards chain of fifthoids: Ms3 - Ms7 - P4d - P1 - P5d - ms3 - ms7 - d4d etc.<br /> | ||
| Line 3,124: | Line 3,127: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:34:&lt;h3&gt; --><h3 id="toc17"><a name="Summary of EDO notation--&quot;Sweet&quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)"></a><!-- ws:end:WikiTextHeadingRule:34 --><u>&quot;Sweet&quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)</u></h3> | ||
<br /> | <br /> | ||
All sweet EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br /> | All sweet EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.<br /> | ||
| Line 3,157: | Line 3,160: | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Summary of EDO notation-Ups and downs solfege"></a><!-- ws:end:WikiTextHeadingRule:36 --><u>Ups and downs solfege</u></h2> | ||
<br /> | <br /> | ||
Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:<br /> | Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:<br /> | ||
| Line 3,187: | Line 3,190: | ||
etc.<br /> | etc.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:38:&lt;h2&gt; --><h2 id="toc19"><a name="Summary of EDO notation-Rank-2 Notation"></a><!-- ws:end:WikiTextHeadingRule:38 --><u>Rank-2 Notation</u></h2> | ||
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Ups and downs can be used to notate rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)<br /> | Ups and downs can be used to notate rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)<br /> | ||
| Line 3,775: | Line 3,778: | ||
<br /> | <br /> | ||
The 22-tone keyboard, with alternate tunings for the black keys (&quot;^3&quot; means triple-up):<br /> | |||
| Line 5,348: | Line 5,351: | ||
There is no variant of D adjacent to C, and there is no 2nd with keyspan 1 or -1. Some other method of notation must be used for these two frameworks.<br /> | There is no variant of D adjacent to C, and there is no 2nd with keyspan 1 or -1. Some other method of notation must be used for these two frameworks.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><!-- ws:end:WikiTextHeadingRule:40 --> </h2> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:42:&lt;h2&gt; --><h2 id="toc21"><a name="Summary of EDO notation-Generators other than a fifth"></a><!-- ws:end:WikiTextHeadingRule:42 --><u>Generators other than a fifth</u></h2> | ||
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Porcupine in 22-tone is generated by a 2nd = 3\22 | The main reason to use ups and downs is to allow fifth-generated heptatonic notation in frameworks and EDOs that aren't fully compatible with such a notation, i.e. those not on the sides of the 4\7 kite. The main reason to use a generator other than a fifth is to use a notation more compatible with one's chosen framework or EDO. Thus there is little reason to use ups and downs in such a situation.<br /> | ||
<br /> | |||
An example of a rank-2 tuning with a non-fifth generator is porcupine. Porcupine in 22-tone is generated by a 2nd = 3\22.<br /> | |||
Genchain: A# B# C# D# E# F# G# A B C D E F G Ab Bb Cb Db Eb Fb Gb<br /> | Genchain: A# B# C# D# E# F# G# A B C D E F G Ab Bb Cb Db Eb Fb Gb<br /> | ||
Scale fragment: D D# Eb E<br /> | Scale fragment: D D# Eb E<br /> | ||
Keyboard: D * * E * * F * * G * * * A * * B * * C * * D<br /> | Keyboard: D * * E * * F * * G * * * A * * B * * C * * D<br /> | ||
Because | Because sharp = 1 key, ups and downs aren't needed.<br /> | ||
<br /> | <br /> | ||
The porcupine genchain:<br /> | |||
The | |||
| Line 5,367: | Line 5,371: | ||
<td style="text-align: center;">22-tone keyspan from D<br /> | <td style="text-align: center;">22-tone keyspan from D<br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;">note<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">- | <td style="text-align: center;">-18<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">12<br /> | ||
</td> | </td> | ||
<td> | <td style="text-align: center;">G###<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">- | <td style="text-align: center;">-17<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">15<br /> | ||
</td> | </td> | ||
<td> | <td style="text-align: center;">Ax<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;">-11<br /> | <td style="text-align: center;">-16<br /> | ||
</td> | |||
<td style="text-align: center;">18<br /> | |||
</td> | |||
<td style="text-align: center;">Bx<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-15<br /> | |||
</td> | |||
<td style="text-align: center;">21<br /> | |||
</td> | |||
<td style="text-align: center;">Cx<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-14<br /> | |||
</td> | |||
<td style="text-align: center;">2<br /> | |||
</td> | |||
<td style="text-align: center;">Dx<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-13<br /> | |||
</td> | |||
<td style="text-align: center;">5<br /> | |||
</td> | |||
<td style="text-align: center;">Ex<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-12<br /> | |||
</td> | |||
<td style="text-align: center;">8<br /> | |||
</td> | |||
<td style="text-align: center;">Fx<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">-11<br /> | |||
</td> | </td> | ||
<td style="text-align: center;">11<br /> | <td style="text-align: center;">11<br /> | ||
</td> | </td> | ||
<td>Gx<br /> | <td style="text-align: center;">Gx<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,399: | Line 5,443: | ||
<td style="text-align: center;">14<br /> | <td style="text-align: center;">14<br /> | ||
</td> | </td> | ||
<td>A#<br /> | <td style="text-align: center;">A#<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,407: | Line 5,451: | ||
<td style="text-align: center;">17<br /> | <td style="text-align: center;">17<br /> | ||
</td> | </td> | ||
<td>B#<br /> | <td style="text-align: center;">B#<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,415: | Line 5,459: | ||
<td style="text-align: center;">20<br /> | <td style="text-align: center;">20<br /> | ||
</td> | </td> | ||
<td>C#<br /> | <td style="text-align: center;">C#<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,423: | Line 5,467: | ||
<td style="text-align: center;">1<br /> | <td style="text-align: center;">1<br /> | ||
</td> | </td> | ||
<td>D#<br /> | <td style="text-align: center;">D#<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,431: | Line 5,475: | ||
<td style="text-align: center;">4<br /> | <td style="text-align: center;">4<br /> | ||
</td> | </td> | ||
<td>E#<br /> | <td style="text-align: center;">E#<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,439: | Line 5,483: | ||
<td style="text-align: center;">7<br /> | <td style="text-align: center;">7<br /> | ||
</td> | </td> | ||
<td>F#<br /> | <td style="text-align: center;">F#<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,447: | Line 5,491: | ||
<td style="text-align: center;">10<br /> | <td style="text-align: center;">10<br /> | ||
</td> | </td> | ||
<td>G#<br /> | <td style="text-align: center;">G#<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,455: | Line 5,499: | ||
<td style="text-align: center;">13<br /> | <td style="text-align: center;">13<br /> | ||
</td> | </td> | ||
<td>A<br /> | <td style="text-align: center;">A<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,463: | Line 5,507: | ||
<td style="text-align: center;">16<br /> | <td style="text-align: center;">16<br /> | ||
</td> | </td> | ||
<td>B<br /> | <td style="text-align: center;">B<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,471: | Line 5,515: | ||
<td style="text-align: center;">19<br /> | <td style="text-align: center;">19<br /> | ||
</td> | </td> | ||
<td>C<br /> | <td style="text-align: center;">C<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,479: | Line 5,523: | ||
<td style="text-align: center;">0<br /> | <td style="text-align: center;">0<br /> | ||
</td> | </td> | ||
<td>D<br /> | <td style="text-align: center;">D<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,487: | Line 5,531: | ||
<td style="text-align: center;">3<br /> | <td style="text-align: center;">3<br /> | ||
</td> | </td> | ||
<td>E<br /> | <td style="text-align: center;">E<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,495: | Line 5,539: | ||
<td style="text-align: center;">6<br /> | <td style="text-align: center;">6<br /> | ||
</td> | </td> | ||
<td>F<br /> | <td style="text-align: center;">F<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,503: | Line 5,547: | ||
<td style="text-align: center;">9<br /> | <td style="text-align: center;">9<br /> | ||
</td> | </td> | ||
<td>G<br /> | <td style="text-align: center;">G<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,511: | Line 5,555: | ||
<td style="text-align: center;">12<br /> | <td style="text-align: center;">12<br /> | ||
</td> | </td> | ||
<td>Ab<br /> | <td style="text-align: center;">Ab<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,519: | Line 5,563: | ||
<td style="text-align: center;">15<br /> | <td style="text-align: center;">15<br /> | ||
</td> | </td> | ||
<td>Bb<br /> | <td style="text-align: center;">Bb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,527: | Line 5,571: | ||
<td style="text-align: center;">18<br /> | <td style="text-align: center;">18<br /> | ||
</td> | </td> | ||
<td>Cb<br /> | <td style="text-align: center;">Cb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,535: | Line 5,579: | ||
<td style="text-align: center;">21<br /> | <td style="text-align: center;">21<br /> | ||
</td> | </td> | ||
<td>Db<br /> | <td style="text-align: center;">Db<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,543: | Line 5,587: | ||
<td style="text-align: center;">2<br /> | <td style="text-align: center;">2<br /> | ||
</td> | </td> | ||
<td>Eb<br /> | <td style="text-align: center;">Eb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,551: | Line 5,595: | ||
<td style="text-align: center;">5<br /> | <td style="text-align: center;">5<br /> | ||
</td> | </td> | ||
<td>Fb<br /> | <td style="text-align: center;">Fb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,559: | Line 5,603: | ||
<td style="text-align: center;">8<br /> | <td style="text-align: center;">8<br /> | ||
</td> | </td> | ||
<td>Gb<br /> | <td style="text-align: center;">Gb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,567: | Line 5,611: | ||
<td style="text-align: center;">11<br /> | <td style="text-align: center;">11<br /> | ||
</td> | </td> | ||
<td>Abb<br /> | <td style="text-align: center;">Abb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,575: | Line 5,619: | ||
<td style="text-align: center;">14<br /> | <td style="text-align: center;">14<br /> | ||
</td> | </td> | ||
<td>Bbb<br /> | <td style="text-align: center;">Bbb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,583: | Line 5,627: | ||
<td style="text-align: center;">17<br /> | <td style="text-align: center;">17<br /> | ||
</td> | </td> | ||
<td>Cbb<br /> | <td style="text-align: center;">Cbb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,591: | Line 5,635: | ||
<td style="text-align: center;">20<br /> | <td style="text-align: center;">20<br /> | ||
</td> | </td> | ||
<td>Dbb<br /> | <td style="text-align: center;">Dbb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,599: | Line 5,643: | ||
<td style="text-align: center;">1<br /> | <td style="text-align: center;">1<br /> | ||
</td> | </td> | ||
<td>Ebb<br /> | <td style="text-align: center;">Ebb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,607: | Line 5,651: | ||
<td style="text-align: center;">4<br /> | <td style="text-align: center;">4<br /> | ||
</td> | </td> | ||
<td>Fbb<br /> | <td style="text-align: center;">Fbb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,615: | Line 5,659: | ||
<td style="text-align: center;">7<br /> | <td style="text-align: center;">7<br /> | ||
</td> | </td> | ||
<td>Gbb<br /> | <td style="text-align: center;">Gbb<br /> | ||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">18<br /> | |||
</td> | |||
<td style="text-align: center;">10<br /> | |||
</td> | |||
<td style="text-align: center;">Abbb<br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,623: | Line 5,675: | ||
<td style="text-align: center;">etc.<br /> | <td style="text-align: center;">etc.<br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,629: | Line 5,681: | ||
<br /> | <br /> | ||
The 22-tone keyboard, tuned to a rank-2 porcupine tuning, with alternate tunings for the black keys:<br /> | |||
| Line 5,672: | Line 5,724: | ||
<td style="text-align: center;">2<br /> | <td style="text-align: center;">2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">- | <td style="text-align: center;">-14<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Dx<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+ | <td style="text-align: center;">+8<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Eb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,684: | Line 5,736: | ||
<td style="text-align: center;">3<br /> | <td style="text-align: center;">3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">+1<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">E<br /> | <td style="text-align: center;">E<br /> | ||
| Line 5,696: | Line 5,748: | ||
<td style="text-align: center;">4<br /> | <td style="text-align: center;">4<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">-6<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">E#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+16<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Fbb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,708: | Line 5,760: | ||
<td style="text-align: center;">5<br /> | <td style="text-align: center;">5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">- | <td style="text-align: center;">-13<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Ex<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+ | <td style="text-align: center;">+9<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Fb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,720: | Line 5,772: | ||
<td style="text-align: center;">6<br /> | <td style="text-align: center;">6<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">+2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">F<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,732: | Line 5,784: | ||
<td style="text-align: center;">7<br /> | <td style="text-align: center;">7<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">- | <td style="text-align: center;">-5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">F#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+ | <td style="text-align: center;">+17<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Gbb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,744: | Line 5,796: | ||
<td style="text-align: center;">8<br /> | <td style="text-align: center;">8<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">-12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Fx<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+10<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Gb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,756: | Line 5,808: | ||
<td style="text-align: center;">9<br /> | <td style="text-align: center;">9<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">+3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">G<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 5,768: | Line 5,820: | ||
<td style="text-align: center;">10<br /> | <td style="text-align: center;">10<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">- | <td style="text-align: center;">-4<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">G#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+ | <td style="text-align: center;">+18<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Abbb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,782: | Line 5,834: | ||
<td style="text-align: center;">-11<br /> | <td style="text-align: center;">-11<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Gx<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+11<br /> | <td style="text-align: center;">+11<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Abb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,792: | Line 5,844: | ||
<td style="text-align: center;">12<br /> | <td style="text-align: center;">12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">- | <td style="text-align: center;">-18<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">G###<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+ | <td style="text-align: center;">+4<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Ab<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,804: | Line 5,856: | ||
<td style="text-align: center;">13<br /> | <td style="text-align: center;">13<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">-3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">A<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 5,816: | Line 5,868: | ||
<td style="text-align: center;">14<br /> | <td style="text-align: center;">14<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">- | <td style="text-align: center;">-10<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Bx<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+ | <td style="text-align: center;">+12<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Bbb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,828: | Line 5,880: | ||
<td style="text-align: center;">15<br /> | <td style="text-align: center;">15<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">- | <td style="text-align: center;">-17<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">B#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+ | <td style="text-align: center;">+5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Bb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,840: | Line 5,892: | ||
<td style="text-align: center;">16<br /> | <td style="text-align: center;">16<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">- | <td style="text-align: center;">-2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">B<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,852: | Line 5,904: | ||
<td style="text-align: center;">17<br /> | <td style="text-align: center;">17<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">-9<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Cx<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+13<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Cbb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,864: | Line 5,916: | ||
<td style="text-align: center;">18<br /> | <td style="text-align: center;">18<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">- | <td style="text-align: center;">-16<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">C#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+ | <td style="text-align: center;">+6<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Cb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,876: | Line 5,928: | ||
<td style="text-align: center;">19<br /> | <td style="text-align: center;">19<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">- | <td style="text-align: center;">-1<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">C<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,888: | Line 5,940: | ||
<td style="text-align: center;">20<br /> | <td style="text-align: center;">20<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">- | <td style="text-align: center;">-8<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Dx<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">+ | <td style="text-align: center;">+14<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">Dbb<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,900: | Line 5,952: | ||
<td style="text-align: center;">21<br /> | <td style="text-align: center;">21<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">-15<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">D#<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">+7<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">Db<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 5,914: | Line 5,966: | ||
<td style="text-align: center;">0<br /> | <td style="text-align: center;">0<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"> | <td style="text-align: center;">D<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 5,923: | Line 5,975: | ||
</table> | </table> | ||
</body></html></pre></div> | |||