Tripod Notation

A Magic Cauldron of Musical Possibilities

Graham Breed

Originally the tripod was an ancient cooking utensil and was often used tocontain offerings. Later it became a symbol of power and reign. Through-out the three dynasties of Xia, Shang and Zhou, there were nine tripodstaken as a symbol of the imperial power. Whoever owned them would bethe one to seize the throne.

Travel Round Jiangsu

Contents

1 Preliminaries 31.1 Steps and Leaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Nine-Limit Harmony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Magic Temperament . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 A Red, Red Rose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Tricycle Notation 82.1 Pitch Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 The Magic Tricycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 The Tripod Staff 113.1 Tripod Pitches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Semitoe Accidentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Magic Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.4 Crossing the River . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.5 Inch Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.6 Marvel Spellings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Yan Tan Tethera Note Names 194.1 Counting Sheep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Shorter Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Coda 215.1 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.4 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2

1 Preliminaries

千里之行始於足下1

Chinese proverb

1.1 Steps and Leaps

CONTRAPUNTALISTS of old made a distinc-tion between “steps” and “leaps”. Steps

are the building blocks of melody and leapsare the building blocks of harmony. With con-ventional notations, a step moves you fromone scale step to the next and a leap missesout at least one scale step. Hence the distinc-tion between steps and leaps is built into thenotation.

Tripod notations are based on the ideathat an octave is divided into three feet. Aleaps takes you from one foot to another anda step takes you to another note on the samefoot.

A consequence of this is that the ninthdegree of a chord is not on the same foot asa tonic. By the melodic definition, an octaveto a ninth is a second and so a step. By thiswacky definition, the seventh is on the samefoot as the tonic and the ninth is on the next

foot, the same as the third. That may soundstrange but it’s the way the notation works.

The fifth degree of a chord is on the footbelow the octave. So, a single leap takes usfrom the fifth to the octave. This interval is aperfect fourth. It follows that a perfect fourthis a single leap, giving us a concept of “singleleap” that is distinct from that of “third”. If atriad is built of two consecutive, single leapsit follows that a suspended fourth chord is akind of triad.

Of course, an interval between a third anda fourth is a second. If the third and fourthare on the same foot they must be separatedby a step. So the “seconds” of conventionalnotations may be either “steps” or “leaps” intripod notations. This distinction means thattripod notations carry different informationto that of conventional notations.

1.2 The Lattice

YOU CAN SEE a simple lattice or Tonnetz inFigure 1.1. This is a simple structure

that may cause an undue amount of excite-ment. It’s made up of triangles that representmajor and minor triads. Each basic conso-nance – perfect fifths and major and minorthirds – runs in a different direction. Youcan use it to follow traditional chord progres-sions.

Where exact tuning is the issue it’s im-portant remember that, in general, two noteswith the same name that sit on different partsof the lattice are not really the same pitch.

The exception, where they are the same, iswhat I call meantone temperament. Most ofthis exposition is not about meantone tem-perament so the lattice comes first and thetraditional note names are only convenienttags.

To help distinguish notes with the samename, there are indications at the bottomof which foot the note is on. Whenever yousee a pair of notes in Figure 1.1 that seemto be the same, each falls on a different foot.Because of this you can use the traditionalnote name along with the foot indication to

1A Journey of five hundred kilometres begins with a step.

3

1 Preliminaries

Figure 1.1: A section of the lattice with meantone names

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uniquely identify a pitch. Eventually youwill still find duplicates but you get a lot fur-ther than you would with only the meantonenames.

The full lattice would run off the pageat the left and right edges (transposition byfourths and fifths). To go further up anddown (transposition by thirds and sixths) youneed to use double sharps and flats. I leftthem off the diagram to make it simpler.

In the following lattice you can see thatthe key of C major requires two different tun-ings of the note D for each chord to follow its

theoretical tuning. However, each of thesenotes is on a different foot. If you want a Dminor chord, you always take D from foot 1.And if you want a G major chord, you alwaystake D from foot 2. Harmony may take yououtside this set of notes. When that happens,you need to re-define the feet (which foot youstart with is arbitrary).

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1.3 Nine-Limit Harmony

THERE ARE MANY reasons for exploring mi-crotonality. Tripod notation is primarily

concerned with the search for new harmonies.Specifically, it’s intended for nine-limit har-mony. I’ll call well tuned, traditional majorand minor chords Didymic. These are thesimple triangles that make up the lattice.

The numbers here refer to harmonics. Theroot, or fundamental, of a tone is one. Har-monic number two is an octave above it,and harmonic number three is a fifth higher

than harmonic number two. Harmonic num-ber four is two octaves above the fundamen-tal. Multiplying the harmonic by two alwaysraises the pitch by an octave. Because ofthis, the interesting new pitches correspondto odd numbered harmonics. The harmoniclimit2 is the largest odd number you need todefine an otonality3 by harmonics.

The otonality that defines Didymic (or five-limit) harmony is a major triad. The roothas a harmonic number of four (two octaves

2As Harry Partch called it; see Genesis of a Music3ibid

4

1 Preliminaries

above the theoretical fundamental). The fifthhas a harmonic number of six (an octaveabove three). The third has a harmonic num-ber of five. Different inversions give differentharmonic numbers but the five-limit is pre-served. A minor chord has the same intervalsas a major chord but in a different order, soit still belongs to the five limit.

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This diagram shows a nine-limit otonal-ity on the lattice. The first three notes—four,five, and six–describe a major triad. The nineis a fifth above the six or a whole tone andan octave above the root. Then we have alast note, with a harmonic number of four-teen, which is twice seven. Seven is an oddnumber larger than five, so this pitch is new.

The lattice is defined for Didymic har-mony, so the fourteen doesn’t really fit onit. It’s placed according to marvel tempera-ment. You can think of this as setting theinterval between harmonics nine and four-teen as being two Didymic major thirds. Thatmeans the interval between the harmonic of

number seven (an octave below the fourteen)and the harmonic of number nine is a kindof major third. Because the true Didymicmajor third is slightly smaller than a thirdof an octave, this new nine-limit major thirdis slightly larger to balance it out. Hence it’ssometimes called a supramajor third. Marveltemperament allows two major thirds and asupramajor third to add up to an octave.

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If all this talk of harmonic numbers is toomuch for you, take a look at the diagramabove. Here, a nine-limit otonality is writtenas an extension of an F major chord. The sev-enth degree, or harmonic number fourteen,is best written as D sharp. Strictly this is anaugmented sixth rather than a seventh, butit happens to be a better fit in meantone tem-perament (which, you may recall, works withconventional note names). A minor seventhis a little to large. The foot indications showthat the D sharp is on the same foot as the F,so in a sense they are a seventh apart.

1.4 Magic Temperament

ALL THIS TALK of meantone temperamentand “conventional note names” may

have given you a warning that we’re aboutto step outside this bubble of conventionalityand enter a new class of temperaments. Thatis indeed the case, and the new temperamentclass is called magic.

Magic and meantone both work withscales of nineteen notes. As with nineteennote equal temperament, E sharp and F flatend up as the same pitch. So, whereas twelvenote equal temperament divides the wholetone into two equal parts, nineteen note equaltemperament splits the semitone in half in-stead. The other diatonic semitone, betweenB and C, is split the same way, to make Bsharp and C flat the same pitch.

The term “diatonic semitone” is itself partof the meantone world. I call the correspond-ing interval in magic temperament a toe. Thetoe can be split into two equal parts in anymagic temperament, not only nineteen toneequal temperament. Hence a toe consists oftwo equal semitoes.

Figure 1.2 uses the nineteen note magicscale to place meantone names on a magiclattice. Sometimes, two different magicpitches will have the same meantone name.However, if the meantone names are the sameand the two pitches are on the same foot,then they really are the same pitch in magictemperament. (Or, at least, the two pitchesare separated by a whole number of octaves,if you interpret the lattice as distinguishing

5

1 Preliminaries

Figure 1.2: A section of the lattice with magic equivalences

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pitches an octave apart.)These equivalences of magic temperament

mean that the nine-limit otonality can be writ-ten in a different way to plain mavel temper-ament, and this is shown below. The mar-vel equivalence still applies, so the fourteenhasn’t moved, but there’s an additional notewith a harmonic number of seven in the dia-gram.

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Ignoring the fourteen, then, this chord ismade up of two triads. Firstly, the usualmajor triad includes the pitches with har-monic numbers of four, five and six. Thetop half of the chord—harmonics six, sevenand nine—form a kind of minor triad. As theinterval between harmonics seven and nineis a supramajor third, the interval between

harmonics six and seven must be smallerthan a minor third to provide balance, andso is a subminor third. This chord, formed byplacing a perfect fifth and a subminor thirdabove the root, is then a subminor triad. Thenine-limit otonality is formed by a major triadon the root and a subminor triad on the fifth.In both cases, the three notes of the triad areon different feet.

If you find the major and subminor tri-ads on the full lattice, you should see thatthe subminor triad is formed by lowering thethird of a minor triad by the same amountthat a minor third is smaller than a majorthird. The difference is the interval I called asemitoe before. So we can say that a minorthird is a semitoe smaller than a major third,a subminor third is a semitoe smaller thana minor third, and a supramajor third is asemitoe larger than a major third.

Raising a supramajor third by anothersemitoe gives a perfect fourth. To see this,find the F sharp at the bottom left-hand cor-ner of Figure 1.2. The foot indications at thebottom show that it’s on foot 2. Directly toits right is a C sharp on foot 1. So, F sharpon foot 2 is a perfect fourth above C sharp onfoot 1. The column to the right of the C sharp,which contains the notes a single leap abovefoot 1, is also foot 2. Follow that column upto see that the major third above C sharp is

6

1 Preliminaries

E sharp, which is identical to F flat on thisfoot. The supramajor third above C sharp isequivalent to F. And the semitoe above that isF sharp, which we already saw is the perfectfourth above C sharp when it’s on foot 2.

Symmetry then tells us that the semitoebelow the subminor third must be the wholetone that you find between a pure, perfectfourth and fifth. Magic temperament has twodifferent sizes of whole tone in the nine-limitthat both end up as major seconds in mean-

tone. This is the larger one, which is called amajor tone. In magic temperament, a majortone is a toe smaller than a minor third.

The smaller whole tone, or minor tone, isa step; that is, both notes are on the samefoot. On the left hand side of the lattice, youcan see that the interval between F sharpan G sharp when on the same foot is threesemitoes. Hence the minor tone in magictemperament is three semitoes.

1.5 A Red, Red Rose

AS THIS EXPOSITION is about new ways ofwriting music, it makes sense to have

some music to write. I’ll use A Red, Red Rose,a song collected by Robert Burns, as a com-mon example. You can see the vocal lineand chords as I like them4 below. There aremore verses and they follow much the samepattern.

The chords are all diatonic and, as we’llsee later, all work together in a generalDidymic system. That means it’s still easy toread in tripod notations. Nine-limit harmonywill tend to have more inherent complexity,and that complexity adds to the complexityof the notation. So this example is a littlesimplistic. But, still, it’s real music.

like

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4Maybe this isn’t the tune that’s commonly used for the song. I don’t care, and in fact I deliberately avoidedlooking at the song in any form of notation so that I can’t be accused of copying a particular arrangement.So there.

7

2 Tricycle Notation

Now you may try to subtract itBut it just won’t go awayThree times one? (What is it? One, two, three!)And that’s the magic number

De La Soul, The Magic Number

Figure 2.1: A Red, Red Rose with tricycle “fingerings”

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BEFORE DEALING WITH the true tripod no-tation, I’ll take a diversion and introduce

a compromise that keeps the traditional notenames and adds a three-fold division of theoctave. The result is that you have, conceptu-ally, three different meantone scales. Thingslike this are sometimes called “bike chains”which leads to the name of the notation sys-tem: tricycle notation.

If we’re looking at a tricycle rather than atripod, the three things we divide the octaveinto must be wheels rather than feet. That’sonly a difference in naming. The tripod’s feetare really the same as the tricycle’s wheelsand you could even call tricycle notation akind of tripod notation.

The structure of tricycle notation follows

directly from the lattice as in Figure 1.1.Notes in the same column are in the samewheel, and the annotations at the bottom tellyou which wheel is which. All we need todo is write the music as normal and add thesame annotations to the score as are on thelattice. The easiest way to do this with stan-dard software is to treat the wheel indicationsas fingerings.1 Even with a crazily extendedmetaphor the wheels (or feet) are not fingers,so I’ll call the annotations “fingerings” withthe scare quotes always present.

You can see A Red, Red Rose marked upthis way in Figure 2.1. The pattern of thechords is very simple, because the notesare always on wheels 1, 2 and 3 countingup. That’s because of the conservative way

1I’m not sure it’s the best way. Maybe one day I’ll explain some variants. I think using “fingerings” is easy tounderstand as well as implement, though, so it works well enough in this context.

8

2 Tricycle Notation

I voiced them. Think of the chords as threeindividual voices, and each moves by steps(in the tripod sense) without crossing. Soeach chord-voice stays on the same wheel.The vocal line covers a wider range and thewheels are mostly determined by the chords.

The “fingerings” add a lot of clutter, so inFigure 2.2 I only included them for ambigu-ous notes. Which is to say only for D because

D is the ambiguous note in a C major scale.Every other note always sounds on the samewheel. As that’s the wheel you’d expect for Cmajor there’s no need to keep reminding thereader of it.

Note that the D of “that’s” (bar seven) istruly ambiguous. There isn’t a D in the chordit sounds over to fix it to either wheel. I’ve setit to be a major sixth above the F.

Figure 2.2: A Red, Red Rose with tricycle “fingerings” for ambiguous notes

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2.1 Pitch Drift

THERE ARE SOME chord sequences thatwork in meantone temperament but

don’t end up on the same note as they startedin a general Didymic tuning. They’re calledcomma pumps because they cause the pitchto rise or fall by a small interval or comma.The popular I IV ii V I comma pump is shownbelow. In tricycle terms, it begins with C onwheel 1 and ends with it on wheel 3. The driftis unavoidable if you common notes in adja-cent chords stay on the same wheel (whichmeans they keep the same pitch).

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The same comma pump, repeated threetimes, is shown later. The notes at the end

should have drifted by three commas. In fact,they end up on the same wheels that theystarted on. This shows a strange feature oftricycle notation, and another reason for itsname: its cyclical. Although notes that differby a single comma are distinguished, pitchesthree commas apart are written identically.

If you followed the example through andworked out the pitch of each note in a statictuning system that preserves the commas,you’d see that the pitch drift is still unavoid-able. The tricycle notation determines thepitch of each note relative to those aroundit. It’s only when you compare notes fromdifferent contexts that their pitches relativeto each other become ambiguous.

This hiding of cumulative pitch drift canbe a problem if you want a strict, static tun-ing. But it also has its advantages. If you

9

2 Tricycle Notation

want the musicians to preserve the Didymicor marvel tuning of each chord as well as theycan but to remove pitch drift, the notationmatches what they should be doing. Unlikemost extended notations, where an acciden-

tal symbol is added to show comma shiftsrelative to a fixed scale, the notation doesn’tget more complex the further the theoreticalpitch drifts from its starting point.

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THE EXAMPLE above shows a magic commapump in tricycle notation. In a general

marvel tuning, it wouldn’t end up where itstarted if the pitches of common notes in ad-jacent chords were preserved. In any magictuning it works without pitch drift.

The root of each chord other than the lastis a Didymic major third above the one be-fore. Each chord is the same type—a Didymicmajor triad with an added harmonic seventh(written as an augmented sixth). That makesit easy to follow, and it gives an idea of whatnine-limit harmony will look like in the nota-tion. The last two chords are a perfect fifthapart, so this is an authentic cadence.

Writing it in meantone requires approxi-

mations from nineteen note equal tempera-ment. That’s made explicit in the bar withtwo dash-tied chords, like this:

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The first chord has an E sharp and thesecond an F flat. Both have the same pitchon a nineteen note scale and so, with the“fingerings”, are the same pitch in magic tem-perament. The dashed slur line is to indicatean approximation, and so a tie in magic tem-perament.

10

3 The Tripod Staff

I’ve devised a system to totally revolutionize music . . . I’vedecimalized it. Instead of the octave, it’s the decatave. AndI’ve invented two new notes: H and J.

Holly, Red Dwarf II:1 (Grant & Naylor)

TRIPOD NOTATION is written on a three linestaff. Each line corresponds to a differ-

ent foot. That means that the whole staffcovers an octave. You can write a note onthe line, above the line, or below the line. Allthree pitches are on the same foot.

The pitches of lines within a staff are sep-arated by a major third. Between one staffand another—that is, between registers—isa river. The river adds an extra semitoe tonotes that cross it. That means that adjacentlines on different staves are separated by asupramajor third.

A note on the line and a note in an ad-jacent space are a toe apart. You can writethe semitoe between them using accidentals.That gives you a string of semitoes on each

foot. Think of it as a tripod with three feet,each of which has three toes. You can writenotes on or between the toes.

Because there’s a space both above andbelow each line, there are two spaces betweenpairs of lines. A note written at the top of thisspace belongs to the foot above, and at thebottom of the space it belongs to the footbelow. The pitches in the same space areseparated by an interval slightly larger thana semitoe.

This summary assumes magic tempera-ment, and tripod notation is primarily de-signed to work with magic, but it can stillbe used for music in any Didymic or marveltuning.

3.1 Tripod Pitches

THREE FEET TO an octave and three noteson each foot gives you nine notes within

the octave without accidentals. I give themnumbers from one to nine in order of ascend-ing pitch, with the river between notes nineand one as you ascend. I chose them so thatthe pitches are roughly evenly spaced whichhelps to preserve contours. You can call themthe tripod nominals or the tripod scale.

To see how nine-limit harmony comes outwith these nominals, look at the diagram be-low. The ninth degree of the chord is a semi-tone above the seventh tripod nominal, soI used a double shafted arrow to raise thepitch accordingly. The same arrows are inFigure 3.1 along with arrows pointing downto lower a pitch by a semitoe.

Because a toe equally divides a semitoe,and adjacent scale degrees are often a toe

apart, there are some pitches that can bewritten on two different scale degrees. I showboth alternatives on the lattice. Having thesechoices means you can adjust the spellingof a chord to preserve the pitch contour andhelp avoid collisions.

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The foot indications at the bottom are re-dundant now. Notes one to three are alwayson foot 1 and so on. They may help you to

11

3 The Tripod Staff

Figure 3.1: A section of the lattice with tripod scale degrees

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reconcile the tripod pitches with the mean-tone pitches in Figure 1.1. I’ve matched uptripod and tricycle notations so that C is onthe first line, and so equal to tripod note two.

There are simple rules that tell you how towrite an interval with tripod pitches. For ex-ample, a major third is three scale steps andleaps between adjacent feet. A minor thirdis a semitoe smaller than the major third.Where an interval crosses the river, you ad-just by removing a semitoe. Hence the majorthird above note nine is a semitoe below notethree.

The lattice continues above and below thesection shown in the example. However, thepitches really do run out on either side, if

you only use single and double semitoe shifts.This reflects the fact that perfect fifths arenot so simple an interval in magic tempera-ment as they are in meantone. Music thatuses long chains of fifths is not likely to workwell in tripod notation, but music that useslong chains of major thirds may work betterthan in standard, meantone-based notation.

I only showed full toe shifts where they de-scribe a pitch that can’t be written any otherway. All of the tripod nominals can be writ-ten in terms of a double semitoe shift from atleast one other scale degree. You aren’t likelyto use those equivalences very often and itwould clutter up the lattice to show them all.

3.2 Semitoe Accidentals

I CHOSE MY accidentals from the Sagittalsystem of Dave Keenan and George Secor.1

A magic semitoe lies between a usual semi-tone and quartertone, and so could be eithera single or double shafted arrow in Sagittal. Ichose one of the smallest and most importantdouble shafted arrows.

You can see A Red, Red Rose in tripod

notation with Sagittal accidentals in Figure3.2. The big numerals show the register. Cleffour is an octave above clef three and you cangeneralize this in an obvious way. Clef fourcorresponds to the bottom of the usual trebleclef. There’s a ledger line that no note can beplaced on to mark the river.

1http://sagittal.org/

12

3 The Tripod Staff

Figure 3.2: A Red, Red Rose in magic tripod notation

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3.3 Magic Scales

YOU CAN SEE some important magic scalesin Figure 3.3 in both ascending and de-

scending forms. First off, the Huating scalehas seven notes, and is in that sense com-parable to the traditional diatonic. But itspitches are spread unevenly. Its step sizesare either a semitoe (smaller than a semitone)or a minor third.

The nineteen notes of Pengcheng are al-most evenly spaced. It’s too big to functionas a mode but good as a set of notes to tunean instrument to. If you write it in tricyclenotation with magic equivalences, each con-ventional name always refers to a pitch on thesame foot. In tripod notation, it can be writ-ten with all the tripod nominals, all pitchesa single semitone either higher or lower thana nominal, and one additional note betweenthe last and first tripod nominals.

The table below shows the melodic struc-ture of Pengcheng. An “s” is a step of a semi-toe and an “S” is slightly larger. You can

see that each “S” takes you from one foot toanother. Figure 3.3 shows the two forms ofPengcheng in tripod notation, with upwardshifts used in the ascending scale and down-ward shifts in the descending scale.

Haizhou has three more notes (one oneach foot) and includes pairs of pitches thatwould be approximated the same way inmeantone. That means you can use it towrite diatonic music where a conventionalnote name is associated with more than onepitch. It always requires at least one full toeshift, to fill in the gap between the lowest andhighest tripod nominals. It can be writtenwith all the nominals and all possible single-semitoe shifts along with this single full toeshift. Figure 3.3 shows the two ways this canbe done, with a pitch a full toe from either thefirst or last tripod nominal, for the ascendingand descending scales. A Red, Red Rose is asubset of Haizhou in Figure 3.2, but doesn’trequire the full toe shift.

1 1⇑ 2 2⇑ 3 3⇑ 4 4⇑ 5 5⇑ 6s s s s s S s s s s

6 6⇑ 7 7⇑ 8 8⇑ 9 9⇑ 1⇓ 1s S s s s s s S s

13

3 The Tripod Staff

Figure 3.3: Some magic scales in tripod notation

“Huating” scales of seven notes

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WHEN A PART doesn’t sit squarely on ei-ther side of the river, you can extend

that staff to cover two octaves. That’s what Idid in the example above. It shows the samemagic comma pump as in Section 2.2. Extralines are added for notes above clef four, andthere’s no need for a ledger line to mark the

river that’s now in the middle of the staff. Youcan see where the river is because the linesare more widely spaced on either side of it.

As we’ve got two octaves to play with now,why not duplicate notes in octaves to put theroots in the bass?

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3 The Tripod Staff

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3.5 Inch Shifts

THE INTERVAL between two notes written atdifferent positions in the same space is

slightly larger than a toe. I call the amountby which it exceeds a toe an inch.

You can describe more pitches by writingshifts of an inch, as in Figure 3.4. The single-shaft arrows show raising and lowering by aninch. A lot more pitches can be written thisway than with only single and double semitoeshifts. You still have to combine inches andsemitoes to cover the Haizhou scale in Figure3.3 though.2

Here’s a Didymic comma pump writtenwith inch shifts.

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There’s another new symbol for a semitoeand and inch combining in the same direc-tion. You may end up using it more oftenthan the single inch shifts because it allowsyou to reach notes that lie on the edges offeet, which are the most likely notes to re-quire re-spelling. They also help you writenotes on either side of the river in the correctpitch order.

3.6 Marvel Spellings

SO FAR I’ve described the pitches in termsof magic temperament, where each toe

can be divided into two equal semitoes. Liketricycle notation, the tripod staff can be usedto specify pitches in any marvel temperament.This can work because there are two ways ofspelling many intervals.

For example, here are some major triads.

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The first two have the same pattern ofscale steps if you ignore the river. This is the

2This can be called a wart of the notation: an ugly feature that can’t easily be removed.

15

3 The Tripod Staff

Figure 3.5: A section of the lattice with simple, marvel spellings

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better pattern for a major triad because itcan be written using the nominals. Follow-ing it gives you the pitches shown in Figure3.5. That isn’t a very useful slice of the lat-tice. Sometimes you want to spell a chorddifferently, like the other two chords in theexample. The final chord is the real problem:it can’t be re-written correctly because lower-ing the top note by a scale degree will leave iton the wrong foot.

One way to get more correct chords is touse inch shifts. Pitches an inch apart areon the same scale degree but different feet.You can then use the much fatter lattice inFigure 3.6 and have a bit of freedom to moveby fifths. Some problematic major and mi-nor triads (and a full nine-limit otonality) canthen be written as follows (accidentals don’tpersist).

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This uses a different symbol for the inchshifts to Section 3.5. The symbol used therewas for commas of Didymus (the differencebetween a major and minor tone). It isn’tconsistent with the spelling rules here so Ichose a different Sagittal symbol that is.

A Red, Red Rose requires some inch

spellings for general didymic harmony, andyou can see it in Figure 3.6.

Another way to get more notes with mar-vel spelling is to use two different symbols forthe different sized semitones. Some problemchords are then written like this:

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The symbol for alternative semitoe shiftsis the inverse of the one for standard shiftsand curved. Because it turns a major thirdinto a minor third, the downward shift has acommon function with flats. For this reason Iuse sharp and flat signs on the lattice in Fig-ure 3.7. This semitone/semitoe equivalenceis close but remember that semitones withpopular tunings are generally larger thansemitoes so it’s best not to use sharp andflat symbols in tripod notation.

Figure 3.7 also uses inch shifts, definedas Didymic commas. To make them distinctfrom the inches in Figure 3.6 I used “har-poons” that match the Sagittal symbols. Thisallows us to write a large number of pitches,with a fair amount of complexity. You canadd whatever symbols you like from AthenianSagittal but you could always use the same

16

3 The Tripod Staff

Figure 3.6: A section of the lattice with marvel spellings using inch shifts and A Red, RedRose using them

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symbols on a conventional staff and it wouldbe easier to read.

Some music still ends up simpler in tri-pod notation with the twin spellings of semi-toes. The semitoe shifts can both be writtenwith two shafts so if you miss the distinctionyou still have notation for magic tempera-ment. Two semitoes in the same directioncan always be three shaft symbols. Inchescombining with semitoes give twin-headedtwo shaft symbols which makes them looklarger than semitoes but smaller than toes,which is correct. There are two different sym-bols for inch shifts but none of the contexts

I’ve outlined here require you to use both ofthem, and they’re both single shaft symbolsthat look like hooks. I don’t know what to doabout semitoe shifts that combine in oppo-site directions—probably best to ignore them.Any nine-limit chord can be written correctlywith only one set of magic nominals. If noteschange their spelling from one chord to thenext you can find a way to indicate this.

Fortunately, A Red, Red Rose doesn’t re-quire any of this complexity because the har-mony is so simple. You can see it in Figure3.7.

17

3 The Tripod Staff

Figure 3.7: A section of the lattice with twin marvel spellings and A Red, Red Rose usingthem

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18

4 Yan Tan Tethera Note Names

I can’t sing like that, not I! I can sing genuine music, like Fand G; but not anything so much out of the order of nateras that.

Sammy Blore, Two on a Tower II (Thomas Hardy)

4.1 Counting Sheep

THERE ARE MANY systems of special num-bers associated with counting sheep in

and around the north of England. The storygoes that, after the Romans left, the Anglo-Saxon invaders settled mainly in the valleysof England and left hill farming to the nativeBritons. For centuries there were speakers oflanguages similar to Welsh living in Englandand not troubling the historical record. A tra-dition of using variations of British numbersto count sheep outlived these languages, andeven after it stopped being used for practi-cal purposes it survived as a way of amusingchildren.

Whether this story is true or not doesn’tconcern us here. As these Yan Tan Tetheranumbers have a certain musical quality tothem I’ll use them to name the tripod nomi-nals. The results are in the table on the right.They’re taken, for various reasons it would betoo tedious to go into, from the “DerbyshireDales” list given by Wikipedia.1 You thenraise a note by a semitoe by adding a “jig”to the name and lower it by adding a “bum”.These suffixes follow from the words “bum-fit” and “jiggit” used for the numbers fifteenand twenty in some variants. Double shiftsbecome “bubum” and “jijig”.

I think these names are more suitablethan English letter names or numbers be-cause they can’t be confused with the tradi-tional note names, or any of the ways num-bers are used in music (including, of course,the foot annotations in this exposition).

Full Name Short Terse Trad.1⇓⇓ Yanna-bubum Yabub Ybb1⇓ Yannabum Yab Yb1 Yan Yan Y B1⇑ Yannajig Yaj Yj2⇓ Tannabum Tab Tb2 Tan Tan T C2⇑ Tannajig Taj Tj3⇓ Tethabum Eb Eb3 Tethera Eth E D[3⇑ Tethajig Ej Ej D3⇑⇑ Tetha-jijig Ejij Ejj4⇓⇓ Metha-bubum Mebub Mbb4⇓ Methabum Meb Mb D4 Methera Meth M D]4⇑ Methajig Mej Mj5⇓ Pippabum Pub Pb5 Pip Pip P E5⇑ Pippajig Pij Pj6⇓ Sethabum Seb Sb6 Sethera Seth S F6⇑ Sethajig Sej Sj6⇑⇑ Setha-jijig Sejij Sjj7⇓⇓ Letha-bubum Lebub Lbb7⇓ Lethabum Leb Lb7 Lethera Leth L G[7⇑ Lethajig Lej Lj G8⇓ Hovabum Hob Hb G8 Hovera Hov H G]8⇑ Hovajig Hoj Hj9⇓ Dovabum Dob Db9 Dovera Dov D A9⇑ Dovajig Doj Dj9⇑⇑ Dova-jijig Dojij Djj

1http://en.wikipedia.org/wiki/Yan_Tan_Tethera

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4 Yan Tan Tethera Note Names

Figure 4.1: A section of the lattice with Yan Tan Tethera names

Djj S T Lj/Hb MbGGGG

Sjj

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Ejj

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wwwYj/Tb L

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Sj Tj/Eb

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Ej

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1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

4.2 Shorter Names

THE FULL Yan Tan Tethera names have acertain rhythm to them, but there are

some contexts—like solfege—where they’retoo cumbersome. For this reason I added theshorter names in the “Short” column. Eachnote requiring a single shift is one syllable,making it easier to sing and type. Notes re-quiring a full toe shift are two syllables. Pip-pabum is shortened to “pub” as a specialcase because “pip” and “pib” can easily beconfused.

Sometimes even the short names aren’tshort enough. For that reason I added the

“Terse” column. Each nominal and each shiftare written as a single letter to match tradi-tional names (the last column). I renamed“teth” to “eth” so that each short name of anominal starts with a different letter, makingthese terse names unique.2 There is a prece-dent for this: some systems have “eddero”instead of “tethera”. Maybe you’ll prefer touse “ethera” for the full name as well.

One use for the terse names is to notatelattice points. For an example see Figure4.1.

2You may still prefer to use “teb” for tethabum so that it doesn’t look like E flat.

20

5 Coda

In 256 BC . . . Prince Zhao of Qin State came to fetch thetripods. In fact, he had only seized eight and the ninthsuddenly flew into the Sishui River . . . when the tripod wasalmost pulled out of the water, the celestial dragon in Sishuiriver wouldn’t give it up and bit off the rope so that thetripod sank into the water again and was never seen afterthat.

Travel Round Jiangsu

5.1 Sources

TRAVEL ROUND JIANGSU was published bythe China Forestry Publishing House in

2001. The writers from Xuzhou are listed asLiu Shaogun[sic] (劉韶云), Zhao Hong (趙虹),Zhang Yan (張灩) and Shang Xuguang (單旭光).

The image on the front page is a pro-

cessed version of a digital copy supplied byWang Hongzhen (王洪震). It shows a rubbingof a stone relief, probably Han Dynasty, ofthe story told by the Travel Round Jiangsuquotes. Problems caused by the processingare, of course, my fault.

5.2 Implementation

I WROTE this exposition using Lilypond-book with LATEX. You can download the

full source code from http://x31eq.com/magic/tripod-code.zip which includesstandalone LilyPond files, so you don’t needto get LATEX working. Also MIDI files usingpitch bends!

Early versions (before February 6) used

different Sagittal symbols. I’m afraid they re-ally were wrong, so I made the change, whichinvolves one symbol changing its meaning.This only affects Sections 3.5 and 3.6.

From October 21 the Sagittal URL is up-dated and I fixed a minor error.

This version was compiled on October 21,2009.

5.3 Acknowledgements

THANK YOU to Hudson Lacerda for helpingme get tripod notation working with Mi-

croABC. I didn’t end up using it here butnever mind that. Some details were influ-enced by his comments.

He also mentioned something that ArnoldSchoenberg called a trigram in Style and Idea.There are three spaces between each line,and you can write notes in the middle one us-ing a ledger line. That gives you twelve notesto the octave without a need for accidentals.

I didn’t know about it before I thought of tri-pod notation, so it didn’t influence me, but itdoes look very similar. I should mention it ifonly to stop others doing so.

My gratitude also to the Lilypond teamfor a program that handles tripod notationunreasonably well for software aimed at Com-mon Practice notation. And Dave and Georgefor the Sagittal symbols and the microtonalcommunity in general for providing the back-ground for these ideas to come out of.

21

5 Coda

5.4 Glossary

TERE ARE some terms in this expositionthat are either uncommon or I invented.

Some might be in the Tonalsoft Encyclope-dia.1 Here, anyway, are some brief defini-tions.

comma A small interval. In particular, thecomma of Didymus (also known as thesyntonic comma) that separates the twowhole tones used in nine-limit harmony.

comma pump A chord sequence that leavesyou a comma away from where youstarted.

Didymic Relating to conventional triadic har-mony, and pitches related according toit. It’s defined by an octave, perfect fifthand major third.

five-limit Equivalent to Didymic.

foot One of a threefold division of the octave.Each note of a triad stands on a differ-ent foot.

inch A small interval in magic temperament.Around forty to the octave, dependingon the tuning.

lattice A regular pattern showing relation-ships between pitches.

magic A marvel temperament class with ad-ditional approximations.

marvel A nine-limit temperament class withthe same structure as Didymic har-mony.

meantone A way of tuning that gives eachnote in conventional notation a uniquepitch with certain regularity. A marveltemperament class.

nine-limit harmony An extension of triadicharmony including some new intervals.

nominal A pitch you write without an acci-dental or key signature.

otonality A standard chord used to define asystem of harmony.

Sagittal A comprehensive system of acciden-tals for microtonal music.

semitoe Half a toe. The interval between amajor and minor third.

subminor A semitoe smaller than minor.

supramajor A semitoe larger than major.

river An extra ledger line or space or semitoethat separates the staves or registers oftripod notation.

temperament A way of tuning that approx-imates the ideal intervals of some har-monic system. In return for the approxi-mation, you expect the tempered systemto be simpler than the ideal one.

temperament class A set of temperamentsthat have the same structure but dif-ferent tunings.

toe A standard interval of magic tempera-ment. Equivalent to the semitone ofa major scale, but on the large side.Around ten to the octave.

tripod A crazy system of notation with threefeet.

Tonnetz The German word for lattice that’ssometimes used in English.

tricycle A crazy system of notation using atraditional staff with foot indications.

wheel A foot in tricycle notation.

1http://tonalsoft.com/enc/encyclopedia.aspx

22