Kite Guitar Exercises and Techniques by Kite Giedraitis

Exercises and techniques for the Kite Guitar by Kite Giedraitis, assumes the downmajor tuning. Highest or top string means thinnest string, lowest or bottom string means thickest string.

Before You Get Your Kite Guitar

You've ordered one, what to do while you wait for it? Learning your way around the major 3rds tuning is half the battle, and you can start right away! Take your 12-equal guitar and tune it in major 3rds as Ralph Patt did. Literally try to play everything you know in this tuning. When your Kite guitar arrives, you can jump in and play all the same shapes and patterns, with only slight adjustments for the increased number of frets.

Exercises and Techniques for Players

There are plenty of guitar exercises to make your fingers more agile or strong. They all apply to the Kite Guitar. These exercises help you navigate 41-equal better. They are for a 6-string guitar, but can be adapted to a 7- or 8-string. For most of these, once you master them, there's no need to practice them further, because you'll naturally reinforce these skills in the course of singing and playing actual songs.

Vocal Exercises

The best way to internalize 41-equal is to sing 41-equal! Get in the habit of singing what you play and playing what you sing.

Beginning exercises:

• Start off by just matching pitch with various guitar notes.
• Play a simple melody and sing along with the guitar.
• Play and sing this melody again, but sing each note first and play it afterwards, to check yourself.
• Play a chord and sing it as an arpeggio.
• Play this chord again, but omit one note, and sing the missing note. Play the note to check yourself.
• Make up your own exercises!

Advanced exercises:

• Play and sing a fretwise (trientonic) melody (steps of one fret).
• Play and sing a microtonal melody (steps of a half-fret).
• Play and sing a melody that uses the mid 2nd and/or the mid 3rd.
• Play and sing a zigzag fretwise melody: P1 vm2 P1 ^m2 P1 vM2 P1 ^M2 P1.
• Play and sing a zigzag microtonal melody: P1 ^1 P1 vm2 P1 m2 P1 ^m2 P1 ~2 P1 vM2 P1 M2 P1.

As before, start by singing along with the guitar, then try singing first and checking yourself later with the guitar.

Parallel Thirds

For all your favorite scales, play ascending and descending lines harmonized in 3rds. If the two notes lie on the same string, use the scale's innate shiftiness to move one of the notes a half-fret up or down. The shifty notes are bolded:

downmajor scale in descending 3rds

high voice	vE	D	C	vB	vA	G	F	vE
low voice	C	vB	vA	G	F	vE	vD	C

Upminor with a raised 7th at the end, a sort of "macro-shiftiness":

upminor scale in descending 3rds

high voice	^Eb	D	C	^Bb	^Ab	G	^F	^Eb	D	vB	C
low voice	C	^Bb	^Ab	G	F	^Eb	D	C	vB	G	G

Equi-minor is reminiscent of Maqam Bayati:

equi-minor scale in descending 3rds

high voice	^Eb	^^Db	C	^Bb	^^Ab	G	^F	^Eb
low voice	C	^Bb	^Ab	G	^F	^Eb	^^Db	C

For pentatonic scales, play parallel "penta-3rds", which span 3 notes of the pentatonic scale. For downmajor, these are mostly 4ths.

dscending downmajor penta-3rds

high voice	vE	vD	C	vA	G	vE	vD	C
low voice	C	vA	G	vE	D	C	vA	G

Downminor penta-4ths are mostly 5ths:

descending downminor penta-4ths

high voice	vEb	C	vBb	G	vF	vEb	C
low voice	G	F	vEb	C	vBb	G	G

The harmonic and subharmonic pentatonic scales aren't shifty. They have a pleasing variety of intervals.

descending harmajor penta-3rds

high voice	vE	D	C	vBb	G	vE	D	C
low voice	C	vBb	G	vE	D	C	vBb	G

Major and minor modes of the subharmonic pentatonic scale:

descending subharmajor penta-3rds

high voice	^E	D	C	^Bb	G	^E	D	C
low voice	C	^Bb	G	^E	D	C	^Bb	G

descending subharminor penta-3rds

high voice	^Eb	C	^A	G	F	^Eb	C
low voice	^A	G	F	^Eb	C	^A	G

This decatonic scale has deca-4ths that are mostly downmajor 3rds.

twin downminor scale in descending deca-4ths

high voice	vE	vEb	vvD	C	vB	vBb	^Ab	G	Gb	vF	vE	vEb	vvD	C
low voice	C	vB	vBb	^Ab	G	Gb	F	vE	vEb	vvD	C	vB	vBb	G

Half-fret Bends

The fact that each 41-equal note only occurs on every other string makes certain scales awkward to play, for example scales with pythagorean 3rds or neutral 3rds. But for 5-limit or 7-limit scales of low odd-limit, it usually works out that you're forced to move to the next string just about when you would want to anyway. For example, the downmajor scale is P1 -- M2 - vM3 - P4 -- P5 - vM6 -- vM7 - P8. The double dashes indicate where you have to move up a string.

There are however two problematic scenarios:

• You're playing the 4th of the scale and you want to hammer on or slide up to the 5th.
• You run out of strings. You're playing the 4th on the top string, and you want to go up to the 5th. (But see also unison leaps below.)

The solution to both issues is to move 3 frets up from the 4th to the down-5th and do a half-fret bend. It's a good idea to practice doing accurate half-fret bends. Here are some exercises that involve playing an off-perfect interval and bending it into tune. No need to practice them all, just find a few that you like.

1a) Play a note on the 6th fret and simultaneously play the next higher string open. This is an up-unison. Bend the 6th fret note up a half-fret to make it a unison.

1b) Same as 1a, but played up the neck. Put your 1st finger up the neck far enough that 6 frets is not too big a stretch. Put your 4th finger 1 string lower and 6 frets higher. Bend the lower (4th finger) note up.

1c) Same as 1b, but your 4th finger is 7 frets up. Bend the lower (1st finger) note up. You'll probably have to pull towards the treble side of the fretboard, instead of the usual push towards the bass side.

2a) Play a note on the 5th fret and simultaneously play the next lower string open. This is a down-5th. Bend the 5th fret note up a half-fret to make a perfect 5th. Again, pull don't push.

2b) Same as 2a, but played up the neck. Put your 1st finger on any fret. Put your 4th finger 1 string higher and 5 frets higher. Bend the higher (4th finger) note up. Pull don't push.

2c) Same as 2b, but your 4th finger is 6 frets up. Bend the lower (1st finger) note up.

3a) Play a note on the 7th fret and simultaneously play the open string two strings lower. This is a down-8ve. Bend the 7th fret note up a half-fret to make a perfect 8ve.

3b) Same as 3a, but played up the neck. Put your 1st finger on any fret. Put your 4th finger 2 strings and 7 frets higher. Bend the higher (4th finger) note up.

3c) Same as 2b, but your 4th finger is 8 frets up. Bend the lower (1st finger) note up.

4a) Play a note on the 4th fret and simultaneously play the open string two strings higher. This is an up-4th. Bend the 4th fret note up a half-fret to make a perfect 4th.

4b) Same as 4a, but played up the neck. Put your 1st finger on any fret. Put your 3rd finger 2 strings lower and 4 frets higher. Bend the lower (3rd finger) note up.

4c) Same as 4b, but your 3rd finger is 5 frets higher. Bend the higher (1st finger) note up.

Exactly how far you have to push the string sideways depends on your location on the neck. The most amount of travel is needed halfway up the neck, around the 5th dot (the mid double dot). Closer to the nut or the bridge, you'll need less travel. Therefore be sure to include some of the "b" and "c" exercises.

Half-fret bends can be notated in tablature by putting an up before the fret number. For example, exercise 1a can be written x x ^6 0 x x. This is consistent with an up meaning one edostep of 41. (In Musescore, to add an up, select Staff/Part Properties, then Advanced Style Properties, then Show Fingerings In Tablature. Then ups can be entered as fingerings.)

In relative tab, exercises 1a, 1b & 1c use the unison at (-1,+^6). Exercises 2a, 2b & 2c use the 5th at (+1,+^5). Exercises 3a, 3b & 3c use the 8ve at (+2,+^7). Exercises 4a, 4b & 4c use the 4th at (+2,-^5).

The Circle of 5ths

The most common intervals for root movements are 4ths and 5ths, so it's good to practice moving by these intervals. This exercise walks you through the entire circle of 41 frets.

First play the circle as a bass line:

• Play a low Ab, 6th string 1st dot.
• Move up a 4th to Db. In relative tab, the move is (+1,+2). This puts you on the 5th string.
• Move up a 4th the same way to Gb. This puts you on the 4th string.
• Move down a 5th by (-2,+1) to return to the 6th string.
• Continue cycling through the lowest 3 strings, 6th --> 5th --> 4th --> 6th, until you reach the 4th dot (mid single).
• Move down a 5th by leaping down 3 dots, which is (0,-12) or (0,-3+0).
• Continue as before, cycling through the lowest 3 strings and steadily moving up.
• Whenever you reach the mid single dot (or overshoot it by 1 fret), leap down as before.

After 5 leaps, you should return to where you started.

To sum up, there are three valid moves: (+1,+2), (-2,+1) and (0,-3+0). This exercise moves up by 4ths and down by 5ths, i.e. fourthward in the circle. There is a complimentary exercise that moves fifthwards. Starting at Ab, leap up 3 dots to Eb. Then slowly walk down, cycling through the strings 6th --> 4th --> 5th --> 6th. Once you reach the 1st dot, leap up again. The moves are (-1,-2), (+2,-1) and (0,+3+0).

Once you get the general idea, test yourself by counting the notes out loud as you go. If you say (or sing) "one" for Ab, "two" for the next note, etc., you should return to Ab just as you say "42", which is after all The Answer to the Ultimate Question of Life, the Universe, and Everything!

Get to the point where you can play this 42-note bass line in under 15 seconds. The final step is to play an actual chord over each of these bass notes. It can be a v7 chord or an ^m7 chord, or really any chord you want to practice. Use a close voicing for root-4 chords, a hi3 voicing for root-5 chords, and a hi35 voicing for root-6 chords. (See hi-lo notation.)

If you're really obsessed with music theory, rather than counting to 42, say the actual note names:

Fourthwards:

• Ab Db Gb=vF#
• vB vE vA vD vG vC vF vBb=^^A
• ^^D ^^G ^^C ^^F ^^Bb=vvB
• vvE vvA vvD vvG vvC=^B
• ^E ^A ^D ^G ^C ^F ^Bb ^Eb=D#
• G# C# F# B E A D G C F Bb Eb Ab

Fifthwards:

• Ab Eb Bb F C G D A E B F# C# G# D#=^Eb
• ^Bb ^F ^C ^G ^D ^A ^E ^B=vvC
• vvG vvD vvA vvE vvB=^^Bb
• ^^F ^^C ^^G ^^D ^^A=vBb
• vF vC vG vD vA vE vB vF#=Gb
• Db Ab

Because 41 is a prime number, any one of 41-equal's intervals will generate a similar circle (except the octave of course).

Moving by Plain Minor 2nds

When translating from 12-equal to 41-equal, one may need to move by a major 2nd in two equal or nearly equal root movements. For example, the 12-equal chords might be IV7 -- #IV7 -- V7 or VIm -- bVIm -- V. Assuming the roots translate to IV, V and vVI, the former will involve splitting a plain major 2nd and the latter, a downmajor 2nd.

To split a plain major 2nd, one moves by two frets then jumps to the final chord, e.g. IVv7 -- #IVv7 -- Vv7. Or one jumps to a root two frets from the final chord first, then moves two frets, as in IVv7 -- bVv7 -- Vv7. Either way, this jump is a plain minor 2nd, which is 3 half-frets. Jump up 1 string and back 5 frets, which in relative tab is (+1,-5). Splitting an upmajor 2nd is even easier, since each of the two moves is a simple two-fret move.

But splitting a downmajor 2nd in half is trickier. One must jump twice, and one of the jumps must backtrack by 8 frets. If one jump is (+1,-5), the other must be (-1,+8). If descending, they must be (-1,+5) and (+1,-8). This is not very intuitive, and it's worthwhile to practice rapidly executing these two jumps. Since the jumps can come in either order, practice both routes. Start with playing single notes, then play entire chords on each root.

• Play a note not too close to the nut or heel and not on the highest or lowest string, for example D on the 5th string, 8th fret
• Jump up (+1,-5) (-1,+8) then jump back down the same way i.e. (+1,-8) (-1,+5), making for example D Eb vE Eb D
• Jump up (-1,+8) (+1,-5) then jump back down the same way
• Repeat both exercises, but descending from D, making D C# ^C C# D

Practicing 3 consecutive jumps will train you to split a vm3 into three equal moves (e.g. vbVIIv7 -- vVIv7 -- bVIv7 -- Vv7).

• Play a note not too close to the nut and not on the highest or lowest string, for example D on the 5th string, 8th fret
• Jump up (+1,-5) (-1,+8) (+1,-5) then jump back down the same way = D Eb vE vF vE Eb D
• Descending: jump down (-1,+5) (+1,-8) (-1,+5) then jump back up the same way = D C# ^C ^B ^C C# D

These two exercises will cover 90-95% of the cases. But for notes near the nut, the backtracking jump must come first.

• Play a note near the nut, for example, B on the 5th string, 3rd fret
• Jump up (-1,+8) (+1,-5) (+1,-5) then back down the same way = B C vC# vD vC# C B

Ascending from a note on the 1st string requires two backtracking jumps.

• Play a note on the 1st string, for example E on the 6th fret
• Jump up (-1,+8) (+1,-5) (-1,+8) then back down the same way = E F Gb vG Gb F E

Descending from a note on the lowest string also requires two backtracking jumps. Note that this exercise is impossible if starting on the lowest string too close to the nut, or the 1st string too close to the heel.

Multiple ascending backtracking jumps will walk you through a complex zone and put you in the next higher rainbow zone. In general it's better to stay in one rainbow zone. But sometimes you may want to move to a higher range, and this maneuver avoids a large leap (see the next exercise).

Moving exclusively by plain minor 2nds and upminor 2nds aka aug unisons = (0,+2) can imitate the sound of 12-equal quite closely. The exact order of the steps doesn't matter too much, just do whatever is comfortable. Try traversing these intervals:

• The 4th = 3 m2's + 2 ^m2's
• The 5th = 4 m2 + 3 ^m2
• The 8ve = 7 m2's + 5 ^m2's = the sum of the 4th and 5th

To change zones, try traversing the 5th (+1,-5) (0,+2) (-1,+8) (0,+2) (+1,-5) (0,+2) (-1,+8).

Leaping by 5ths, 8ves and Unisons

Being able to leap up or down by 3 dots can be useful when chording (see the "I Will Survive" translation). Again, start with a bass line, and add chords over them later.

• Play a low Ab (6th string 1st dot), leap up 3 dots to Eb, and leap back down.
• Move up one fret to vvA, leap up to vvE and back down.
• Move up to A, leap up and down, move up to ^^A, leap, etc.

Try to get a steady rhythm going. Keep going up fret by fret until you run out of room, then move down fret by fret. A complimentary exercise starts at Eb on the mid single dot (4th dot), leaps down to Ab and back up. Then go up one fret and continue.

Leaping up an octave is useful when switching from chording to soloing. Start at the low Ab, leap up (+1,+3+2) to Ab and back down. Go up 1 fret and continue. A complimentary exercise starts high, leaps down and leaps back up. Then go up 1 fret and continue.

Leaping up to the unison is useful when you are soloing and you run out of strings. For example, you're playing the 4th of the scale on the top string and not too far up the neck, and you want to go up to the 5th. Start on Eb (1st string 1st dot), leap up to the same Eb on the 3rd string, and leap back down. The leap is (-2,+3+1). Move up a fret and repeat. Keep moving up a fret until you run out of frets.

A complimentary exercise is for when you are on the lowest string and high up the neck, and you want to go further down. Start at the highest fret of the 6th string, leap down by (+2,-3-1), and leap back up. Move down a fret and repeat.

Combination Exercises

These are not for the faint of heart!

If you have enough frets, you can combine any leaping exercise with the circle of 5ths exercise. For example, start at Ab, leap up a 5th or an 8ve, leap back, move up a 4th to Db, leap up and down, move on to Gb, etc. Or start on Ab, leap up an 8ve, move up a 4th to Db, leap down, move up a 4th to Gb, etc.

You can leap up an octave using a half-fret bend by a move of (+2,+^7). Finally, you can combine this with the circle of 5ths exercise by repeatedly leaping up, leaping down, and moving up a 4th.

You can add a vocal exercise to all this by singing what you play.

Alternate Fingering Techniques

Cross-fret barre

When looking for a fingering, often a cross-fret barre (aka diagonal barre) is the solution. The closer fret spacing of the Kite guitar makes this much easier than it is on the 12-equal guitar. For example, 4 2 3 1 might be fingered as 3 1 2 1, and 2 4 4 3 1 as 1 3 4 2 1.

Harmonics

The 3rd harmonic is written in guitar tablature as <12>, meaning the 12th-fret harmonic. The 2nd harmonic falls midway between the 20th and 21st frets, and is written as <^20> or <v21>. The latter is preferable because one must place one's finger just behind the 21st fret, exactly as if the tab were 21. In other words, a guitarist is used to seeing "21" and placing their finger in the proper spot for <v21>. Furthermore ^20 without the angle brackets means bend the 20th-fret note up a half-fret. While ^20 and <^20> do look different, ^20 and <v21> are easier to distinguish.

The 5th harmonic falls between the 6th and 7th fret, but closer to the 7th. It's written here as <6.6> for greater accuracy, but in tablature it will usually be rounded off to <v7>. (In Musescore, to add an up, select Staff/Part Properties, then Advanced Style Properties, then Show Fingerings In Tablature. Then ups can be entered as fingerings.)

Here are all the places harmonics 1-12 occur, excluding those above the 41st fret. Be sure to pluck on an anti-node!

Harmonics 1-12 on the Kite guitar

harmonic -->	1	2	3	4	5	6	7	8	9	10	11		12
fret	open	<v21>	<12>	<v9>	<6.6>	<5.4>	<v5>	<4>	<v4>	<3.1>	<2.8>	<17.9>	<2.6>
			<v33>	<41>	<15.1>		<10>	<13.9>	<v8>	<v11>	<5.9>	<23.3>	<15.9>
					<27.1>		<v17>	<29>	<v18>	<35.6>	<9.4>	<29.9>	<25.9>
							<25>		<24>		<13.4>	<38.4>
							<37>
combo						2nd + 3rd				2nd + 5th			3rd + 4th

The last row of the table indicates combo-harmonics. These let you play a harmonic by node-ing twice. For example, the 6th harmonic can be played by node-ing both the 2nd and 3rd harmonic simultaneously, e.g. <v21> with <12>, or <v21> with <v33>.

These harmonics, along with open strings, provide alternative fingerings for notes on the 13th, 14th and 15th frets:

13th fret	14th fret	15th fret
13 x x = x x 0	x 14 = <v21> x	x x x 15 = <v9> x x x
x x x 13 = <12> x x x		x x x x x 15 = <15.1> x x x x x

In the last equation, the harmonic is ~6¢ sharp of the 41-equal note.

Quarter-fret Bends, Sixth-fret Bends, etc.

When soloing over an ^m7 chord, a sustained 4th creates an innate-comma pentad. The effect is subtle but noticeable, and once you hear it, it's hard to unhear. The comma can be tamed by splitting the difference. Play the plain 4th, then bend it up a quarter-fret to a half-up 4th. The bend needn't be exact. Unlike bending the down-5th a half-fret up to the 5th, the end result isn't to lock into a specific ratio. In fact, rather than play a static half-up 4th, a moving bend that starts at the plain 4th and goes up past the half-up 4th and then back down sounds better. This is called a fuzzy 4th, specifically an upfuzzy 4th. But over a vm7 chord, we want a downfuzzy 4th. Either play the down 4th and bend it up, or play the plain 4th and bend it down (harder, see below).

Any chord that has two notes an upmajor or downmajor 2nd apart will create a fuzzy note. In practice, this 2nd may be voiced as a 7th or 9th.

fuzzy notes

if the chord has both...		soloist must bend	example chords
plain	upped or downed
root	minor 7th	the perfect 4th	^7	v7	^m7	vm7	^d^7	vdv7	^9	v9
4th	minor 3rd	the minor 7th	^m,4	vm,4
5th	major 6th	the major 2nd	^6	v6	^m6	vm6
9th	major 3rd	the major 6th	^,9	v,9	^M9	vM9	^9	v9

Both the top and bottom rows of the table apply to the dominant 9th chord, thus it has two fuzzy notes.

The next use of quarter-fret bends is less essential. One can hide pitch shifts by sharpening an entire chord by some fraction of a half-fret. Obviously it won't work if a chord uses open strings. Play a progression with a pitch shift, e.g. Iv - vVI^m - vII^m - Vv7 - Iv. The 3rd chord has vD and the 4th chord has D. Bend the entire 3rd chord up a quarter-fret by ear, so that its vD becomes a half-down D. This creates another pitch shift, because the chord now has a half-down A which differs from the previous chord's vA. However, two small 15¢ shifts are better than one large 30¢ one. Alternatively, bend the 2nd chord up a sixth-fret and the 3rd chord up a third-fret, to create three pitch shifts of 10¢ each.

To practice such bends, do one of the half-fret bend exercises in two or three stages.

In 41-equal, 5-over intervals like 5/4 and 5/3 are about 6¢ flat. This issue is even more subtle than the innate-comma pentad, but still noticeable. One can correct this by applying a tenth-fret bend to certain notes of the chord. This sounds hard, but fortunately there are only a few chord shapes to apply this to. One quickly gets in the habit of "leaning on" certain notes in these shapes.

For example, with a downmajor chord in R-5-10 (aka hi3) voicing, bend the 3rd up slightly with your pinkie. Listen closely for interference beats that slow down as you bend up. It may help to play the actual coinciding harmonics first. As you play 4 x 3 x 5 x, play matching artificial harmonics at <v11> x x x <v26> x, and also at x x <v10> x <17> x (see Harmonics above). For a 4 x 3 5 5 x voicing, to bend the 3rd up, you'll need to pull your pinkie down towards the treble side of the fretboard. For a 1st inversion x 4 3 5 x x voicing, push your finger up towards the bass side. It's rather difficult to bend the 3rd in a close 4 4 3 5 x x voicing.

It's also possible to correct the 6¢ sharpness of 5-under intervals by bending a note slightly down. Press the string firmly against the fingerboard and push it towards the bridge. This is harder to do by the nut, because bending down stretches the string behind the fret, and there's very little to stretch there.

Primes 11 and 13

Whereas primes 5 and 7 are tuned slightly flat in 41-equal (5.8¢ and 3.0¢ respectively), primes 11 and 13 are slightly sharp (4.8¢ and 8.3¢ respectively). Thus ratios that have either 5 or 7 on one side and either 11 or 13 on the other are doubly mistuned. Bending up to ratios using 11 or 13 lets us fine-tune them. When primes 5 or 7 are present, best to underbend a bit, to match their flatness. This also makes ratios with primes 11 and 13 more physically accessible. For 13/8, the ^m6 is much easier to reach than the ~6. 11/6, 11/9 and 13/12 also become easier.

11-over and 13-over ratios require just under a half-fret bend. 11-under and 13-under ones require just over a half-fret bend. Both are denoted here with an up, e.g. ^4.

Augmented triads using primes 11 or 13

chord	color name		41-equal name	example	frets	fingering
7:9:11	ru loru-5	r(1or5)	up-downsharp5	C^(v#5) = C ^E vG#	4 5 ^4	1 3 2
8:10:13	yo tho-6 no5	y,3o6no5	down-upsharp5	Cv(^#5) = C vE ^G#	4 4 ^5	1 2 3

Exercises and Techniques for Composers and Arrangers

These are not physical exercises for your fingers, but mental exercises for your mind.

Interesting Root Movements

Given a chord, what chord can you move to that has at least 2 notes in common? Root movement intervals will tend to be not plain. Harmonic chords will tend to be followed by subharmonic chords and vice versa. The ^9 chord can often have its root omitted, becoming a vdv7 or ^m6 chord. These tables list only some of the possibilities.

from the down7 chord

common tones	progression	guitar tab		notes
root (of I chord)	Iv7 -- IVv7	4 - 3 1 5	- 6 6 5 3	only 1 note in common, but too basic to leave out
5th	Iv7 -- Vv7	- 6 6 5 3	4 - 3 1 5	ditto
	Iv7 -- V^9	- 6 6 5 3	4 5 3 2 2	ditto, two nice 1-fret voice movements
root & 5th	Iv7 -- IVv9	4 - - 1 5 4	- 6 6 5 3 4
	Iv7 -- IV^9	4 - - 1 5 4	- 6 7 5 4 4	one tiny half-fret voice movement
	Iv7 -- I^9	4 - 3 1 5	4 5 3 2 2	less satisfying because the root doesn't change
	Iv7 -- vVI^d^7	- 4 4 3 1	5 - 2 3 5	vVI chord can "flip" to a IVv9noR chord
root & 3rd	Iv7 -- vII^9 or #IVvdv7	4 - 3 1 5	(7) 8 6 5 5	leads nicely into the IVv7 chord
3rd & 5th	Iv7 -- vVI^9 or #Ivdv7	- 4 4 3 1	(5) 6 4 3 3	#Ivdv7 leads nicely into the vVI^m7 chord
root & 7th	Iv7 -- vbVI^9	- 4 4 3 1	3 4 2 1 1	leads nicely into the Vv7 chord
	Iv7 -- vbIII^m6	4 - - 1 5 4 -	- 2 - 5 3 2 2	leads nicely into the IVv7 chord
5th & 7th	Iv7 -- vbIII^9 or vbVII^m6	4 - - 1 5 4	0 (2) 3 1 0 0	leads nicely into the Vv7 chord
3rd & 7th	Iv7 -- ^VII^9 or vIIIvdv7	4 - 3 1 5	(3) 4 2 1 1	a weird one

from the upminor7 chord

common tones	progression	guitar tab		notes
5th	I^m7 -- Vv7	- 6 5 5 4	4 - 3 1 5	only 1 note in common, but too basic to leave out
root & 5th	I^m7 -- ^VIvm7	- 4 3 3 2	6 4 5 3	weird but cool
root & 3rd	I^m7 -- IV^m7	4 - 3 2 4	- 6 5 5 4
	I^m7 -- ^bVIv7	- 4 3 3 2	4 - 3 1 5	nice
3rd & 7th	I^m7 -- ^bIIIvm7	4 - 3 2 4	- 3 1 2 0	weird but cool
5th & 7th	I^m7 -- V^m7	- 4 3 3 2	2 - 1 0 2
root, 3rd & 5th	I^m7 -- IV^9	4 - 3 2 4 4	- 6 7 5 4 4	one of my favorites, even though 3 common tones

These are similar to neo-Riemannian progressions, but using 41-equal 7-limit tetrads not 12-equal 5-limit triads. A more exact extension of the 12-equal case would allow only two tetrads, the v7 (harmonic) and vdv7 (subharmonic) ones. Interestingly, just as in the 5-limit 12-equal case, you can get from one tetrad to any other in five steps or less (four if the first and last chord have the same quality). Thus one can modulate to any key in 41edo fairly quickly.

Harmonizing Fretwise Melodies

Write a melody with steps of vm2 (one fret), with perhaps an occasional m2 for string-hopping. Write chords under it. See if you can improve on my attempt:

x 2 4 4 3 x -- ^^Ebv

3 x 2 4 4 x -- ^Gv7

x 4 6 6 5 x -- vFv

x 5 4 4 6 x -- ^C^m

x 2 4 4 x 1 -- ^^Ebv7

5 x 4 2 6 x -- ^^Abv7

x 0 2 3 5 x -- ^D^6

3 x 2 4 4 x -- ^Gv

Rotations aka Inversions

In music theory, the word inversion has distinct, but related, meanings when applied to intervals, chords, voices (in counterpoint), and melodies. These exercises are not about chord inversions in the sense of putting the 3rd or 5th in the bass. Rather, they use melodic inversions, i.e. flipping everything upside down. It's similar to what Jacob Collier calls negative harmony. To avoid confusion, we'll call them rotations, for reasons that will become obvious.

Motivating Example

Take the classic V7 - I chord progression and tune it 7-limit:

x x 4 4 3 1

x x 4 6 6 5

I can derive an entirely different, yet vaguely similar chord progression from this one by rotating everything 180 degrees:

x x 8 6 5 5

x x 4 3 3 5

I took something nice, but a bit of a cliche, rotated it, and found something else nice, but fresh and new! And I didn't need any music theory to do that. I found it purely mechanically, without thinking about intervals or chords at all. I just looked at my fingers and did some "spatial math".

If we want, we can apply music theory after the fact. The chords are IV^m6 - I^m, or equivalently ^IIvdv7 - I^m. The original chord progression has a diminished 5th resolving inward to a downmajor 3rd, giving a feeling of tension and resolution. The new one does too. The original chord progression has 3 voices moving by 2nds in parallel motion. The new one does too. In both progressions, one voice stays still, providing oblique motion. Different, yet similar.

Rotating an Interval

Let's start with an easy exercise. Play any interval smaller than an octave. Now move the lower note up an octave for a new interval, the octave inverse.

interval	tab	--->	tab	interval
vM3	x x 5 5 x x		x x x 5 x 6	^m6

Notice how major intervals become minor and vice versa. Augmented becomes diminished and up becomes down. But not everything changes. Perfect stays perfect. 3-limit remains 3-limit, 5-limit remains 5-limit, 7-limit remains 7-limit, and 11-limit remains 11-limit. A highly consonant interval remains at least fairly consonant. A highly dissonant interval won't improve much.

Rotating a Melody

Play a short stepwise melody. Next, play it reversing the direction of each step. Ascending becomes descending and vice versa.

note	tab	--->	tab	note
A	x x x 7 x x		x x x 7 x x	A
B	x x x x 4 x		x x 10 x x x	G
vC#	x x x x 7 x		x x 7 x x x	^F
B	x x x x 4 x		x x 10 x x x	G
A	x x x 7 x x		x x x 7 x x	A

Look at the shape the first melody traces out on the fretboard. It's a long skinny triangle. We move along each of the two long sides, then retrace our steps.

Now look at the new melody. The long skinny triangle has been rotated 180 degrees. We take the same path along this new triangle.

Play any scale, going from the tonic up to the octave. Next, start at the fifth of the scale and go downward to the low fifth. The C major scale becomes C minor, C dorian becomes C mixolydian, and C lydian becomes C phrygian. C locrian rotates to itself! Up becomes down: C upminor becomes C downmajor. Again, the prime limit doesn't change.

In general, when rotating, the old tonic becomes the new fifth, and the old fifth becomes the new tonic. This tends to preserve the original key.

Play an actual melody from a song you know. Consider the interval from the tonic up to the starting note. From the fifth, go down by this interval. Wherever you land is the starting note for the rotated melody. For example, Mary Had A Little Lamb goes vM3 M2 P1 M2 vM3 vM3 vM3. The first note is vM3. P5 minus vM3 is ^m3. So start at ^m3 to get ^m3 P4 P5 P4 ^m3 ^m3 ^m3. Note that a pentatonic melody is still pentatonic after rotation.

Rotating a Chord

Rotating in place: Play any chord in root position and close voicing as an ascending arpeggio. Think of this arpeggio as a melody. Starting on the highest note, rotate the melody to get a descending arpeggio. You should end up on the lowest note of the original chord. This new arpeggio is your rotated chord. Example: a v7 chord 4 4 3 1 rotates to 4 2 1 1. Again, look at the fretboard shapes. Everything got rotated 180 degrees.

Try this with other chords. Some chords rotate to themselves!

What if we include the octave?

chord	voicing	tab		tab	chord	voicing
Dv	close	x 8 8 7 x x	--->	x 8 7 7 x x	D^m	close
Dv	add-8	x 8 8 7 9 x	--->	x 8 10 9 9 x	G^m	add-low-5

Adding the octave results in a different root. To keep the same root, rather than rotating in place, follow our rule: the root becomes the 5th and vice versa. x 8 8 7 9 x becomes 6 8 7 7 x x, and Dv becomes D^m. The old voicing was close, going up from the root. The new voicing is also close, but now it goes down from the 5th.

If the 5th is diminished, the root-to-fifth rule still works: 4 3 1 becomes 4 2 1, and updim becomes downdim.

What about open voicings? To avoid running out of strings, let's assume a 7-string guitar in a low-7 tuning.

chord	voicing	tab		tab	chord	voicing
D^m	hi-3 add-8	x x 8 x 7 9 8	--->	7 6 8 x 7 x x	Dv	low-3, add-low-5

Not a very nice voicing. You can only take rotations so far. The basic rules of voicing and voice leading still hold. Good chord voicings imitate the harmonic series: larger intervals between the lower voices and smaller intervals between the upper voices. In accompaniments, good bass melodies tend to have big leaps, and good melodies in the upper voices tend to have smaller steps. So a good voicing will often rotate to a bad one, and you'll often want to revoice after rotating.

• major rotates to minor
• up rotates to down
• harmonic chords rotate to subharmonic chords
• stacked chords rotate to stacked chords
• the prime limit doesn't change

7th chords rotate to 6th chords, but every 6th chord has a 7th chord homonym. So 7th chords can rotate to 7th chords, as in our first example 4 4 3 1 --> 4 2 1 1. You can think of this as Cv7 becomes vEb^m6/C, or as Cv7 becomes Cvdv7.

Rotating a Chord Progression

First rotate each individual chord type. Next, play the roots of each chord as a bass line. Voice each root in whatever octave you want. Rotate the bass line like you would rotate any melody. Don't change the starting note, except perhaps by an octave. Finally, play the rotated chords using the roots from the rotated bass line.

chord progression	bass line of roots		new bass line	new roots	new chords
Iv - vVI^m - IVv - Vv	P8 - vM6 - P4 - P5	--->	P1 - ^m3 - P5 - P4	I - ^bIII - V - IV	I^m - ^bIIIv - V^m - IV^m.

Some chord progressions rotate to themselves. Two examples: I^m - IVv and I^m - Vv.

Rotating an Entire Song

We can rotate both a chord progression and an associated melody. Use the root-to-fifth rule for both.

lyrics	original		rotated
	melody	chords	melody	chords
Happy	P5	Iv	P8	I^m
Birth-	vM6		^m7
day	P5		P8
To	P8		P5
You	vM7	Vv	^m6	IV^m
Happy	P5		P8
Birth-	M6		^m7
day	P5		P8
To	M9		P4
You	P8	Iv	P5	I^m

All these rotations work in 12-equal, or any tuning system, but unless your guitar is isomorphic, the fretboard shapes won't simply rotate.

The Big Switcheroo

Like the previous exercise, this creates a new melody or chord progression from an old one. You simply swap up for down, so that upminor becomes downminor, etc. Swap intervals, not notes. The quality (major, minor, perfect, augmented or diminished) is unchanged. Plain and mid intervals are unchanged. (The reason mid is unchanged is that it's simultaneously both dupminor and dudmajor. But the former would become dudminor, and the latter dupmajor. Rather than changing to two things, which doesn't make any sense, it doesn't change at all.)

lyrics	original		swapped
	melody	chords	melody	chords
Happy	P5	Iv	P5	I^
Birth-	vM6		^M6
day	P5		P5
To	P8		P8
You	vM7	Vv	^M7	V^
Happy	P5		P5
Birth-	M6		M6
day	P5		P5
To	M9		M9
You	P8	Iv	P8	I^

This swapping has the effect of interchanging prime 5 with prime 7 (i.e. 5-limit becomes 7-limit-no-fives), and harmonic with subharmonic. Rotations also exchange harmonic with subharmonic, so rotating followed by switching preserves this property.

lyrics	original		first rotated		then swapped
	melody	chords	melody	chords	melody	chords
Happy	P5	Iv	P8	I^m	P8	Ivm
Birth-	vM6		^m7		vm7
day	P5		P8		P8
To	P8		P5		P5
You	vM7	Vv	^m6	IV^m	vm6	IVvm
Happy	P5		P8		P8
Birth-	M6		^m7		vm7
day	P5		P8		P8
To	M9		P4		P4
You	P8	Iv	P5	I^m	P5	Ivm

Modulation via Dim7 Chords

This is analogous to 12-equal's Cdim7 --> B7 --> E (or EM7 or Em7). The idea is to lower one of the 4 notes in the dim7 chord by a semitone to transform it into a dom7 chord, then use a typical V7 - I cadence to arrive at one of 4 new keys. Thus the dim7 chord is sort of a "portal" to other keys. The following table uses a single 12-equal dim7 chord that is spelled 4 different ways for convenience. This starting chord is one of the three possible dim7 chords in 12-equal, thus two more tables would be needed to show all possible modulations. (Briefly, Gdim7 goes to D, F, Ab and B, and Bdim7 goes to F#, A, C and Eb.) The note that is lowered is bolded.

using a dim7 chord to modulate in 12-equal

starting dim7 chord	bolded note is lowered		bolded note is raised
	new V7 chord	new key	new IVm6 chord	new key
Adim7 = A C Eb Gb	Ab7 = Ab C Eb Gb	Db	Ebm6 = Bb C Eb Gb	Bb
B#dim7 = A B# D# F#	B7 = A B D# F#	E	F#m6 = A C# D# F#	C#
D#dim7 = A C D# F#	D7 = A C D F#	G	Am6 = A C E F#	E
F#dim7 = A C Eb F#	F7 = A C Eb F	Bb	Cm6 = A C Eb G	G

The last two columns take this idea further. The bolded note is raised to make a min6 chord that resolves down a 4th to the new tonic. This IVm6 - I cadence is simply the rotation of V7 - I. Note that both cadences take you to the same 4 keys. Also, the chord that results from raising the note can be interpreted as a dom9noR chord, in which case it resolves the same as if the note had been lowered. In other words, instead of being lowered to become the root, the note is raised to become the ninth. For example, Ebm6 can resolve to Bb, but if heard as Ab9noR, it can resolve to Db. Likewise, the Ab7 chord can be interpreted as Ebm6,11no5, and thus can resolve to Bb. (This is perhaps more plausible in 41-equal than in 12-equal.) In all these cadences, the C-Gb dim 5th resolves inward to a 3rd.

Let's extend this idea to 41-equal. A plain dim7 chord is possible, but awkward on the Kite guitar. So we will focus on the ^dim7 and vdim7 chords. Neither of these are symmetrical, so 40 more tables would be needed! How much to raise/lower by? The bolded note has another chord note a tritone above it. The bolded note is either lowered to make that interval a perfect 5th, or raised to make a perfect 4th.

Sometimes another note in the chord needs to shift by a half-fret to make a low-odd-limit chord. When this happens, the shifting note is bolded in the "new chord" column. Without this shift, the new chord is more dissonant. But the down add-7 chord is a very familiar dissonance. Furthermore the vanishing Ruyoyo comma means the 45/32 downaug 4th is really a consonant 7/5 dim 5th. So it's listed in the 2nd row as an alternative to shifting. The upminor add6 chord is included as a possibility because it's a rotation of v,7, and has the same vanishing innate comma. But the other two non-shifting chords, F# minor add-down6 and D add-up7, are less plausible because of their dissonant plain 3rds. Those two are written in (italics).

using an updim7 chord to modulate in 41-equal

starting ^d7 chord	bolded note is lowered		bolded note is raised
	new V chord	new key	new IV chord	new key
A^d7 = A ^C Eb ^Gb	Ab^7 = Ab ^C Eb ^Gb	Db	Eb^m6 = Bb ^C Eb ^Gb	Bb
A^d7 = A ^C vD# F#	Bv,7 = A B vD# F# Bv7 = vA B vD# F#	E	F#vm6 = vA C# vD# F# (F#mv6 = A C# vD# F#)	C#
A^d7 = A ^C Eb F#	D^7 = A ^C D ^F# (D,^7 = A ^C D F#)	G	A^m,6 = A ^C E F# A^m6 = A ^C E ^F#	E
A^d7 = A ^C Eb ^Gb	^Fv7 = A ^C Eb ^F	^Bb	^Cvm6 = A ^C Eb ^G	^G

Note that unlike in 12-equal, the IV-I cadences take you to mostly different keys than the V-I cadences.

Again, there are alternate interpretations for the new chord.

• Ab^7 = Eb^m6,11no5
• Eb^m6 = Ab^9noR
• ^Fv7 = ^Cvm6,11no5
• ^Cvm6 = ^Fv9noR

Here's a similar table for the downdim7 chord. All the non-shifting chords contain mid intervals.

using a downdim7 chord to modulate in 41-equal

starting vd7 chord	bolded note is lowered		bolded note is raised
	new V chord	new key	new IV chord	new key
^Avd7 = ^A C ^Eb Gb	^Abv7 = ^Ab C ^Eb Gb	^Db	^Ebvm6 = ^Bb C ^Eb Gb	^Bb
^Avd7 = ^A C D# vF#	vB^7 = A vB D# vF# (vB^,~7 = ^A vB D# vF#)	vE	vF#^m6 = A vC# D# vF# (vF#~,^6 = ^A vC# D# vF#)	vC#
^Avd7 = ^A C ^Eb vF#	^Dv7 = ^A C ^D F# (^D~,v7 = ^A C ^D vF#)	^G	^Avm6 = ^A C ^E F# (^Avm,~6 = ^A C ^E vF#)	^E
^Avd7 = ^A C ^Eb Gb	F^7 = ^A C ^Eb F	Bb	C^m6 = ^A C ^Eb G	G