Kite Guitar Scales: Difference between revisions
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This is the practical guide to Kite Guitar scales. See also [[Kite Giedraitis's Categorizations of 41edo Scales]], which is more theoretical. | This is the practical guide to Kite Guitar scales. See also [[Kite Giedraitis's Categorizations of 41edo Scales]], which is more theoretical. | ||
There are many possible 41edo scales. Those discussed here are those with at least 5 notes, and which | There are many possible 41edo scales. Those discussed here are those with at least 5 notes, and which have a plain perfect 5th from the tonic. Scales that are awkward to play on the Kite guitar are avoided. An '''awkward''' scale has a step which requires a jump of more than four frets. Thus plain minor 2nds and 3rds are avoided. A scale naturally hops from one string to the next as it goes up or down. Unlike other guitars, the Kite guitar doesn't let one hop freely. For example, the 3-limit scale fragment P1 M2 M3 P4 requires 3 hops, 2 upward and 1 downward. Any scale which doesn't have exactly three upward hops per octave will be awkward, because the downward hop will always be at least 6 frets, and usually 7 or more. Almost every scale with a low prime limit and/or a low odd limit is not awkward. | ||
[[MOS scale|MOS (moment of symmetry) scales]] have only two step sizes, with the less frequent steps evenly distributed throughout the scale. MOS scales are an important part of microtonal scale theory. But almost every 41-edo MOS scale with a perfect 5th is awkward. The only exception is scales from the [[Magic|Laquinyo]] temperament, which have a small step of only one fret. They have either a very lopsided L/s ratio or more than 12 notes. They are discussed further in the Nineteen-tone section. | [[MOS scale|MOS (moment of symmetry) scales]] have only two step sizes, with the less frequent steps evenly distributed throughout the scale. MOS scales are an important part of microtonal scale theory. But almost every 41-edo MOS scale with a perfect 5th is awkward. The only exception is scales from the [[Magic|Laquinyo]] temperament, which have a small step of only one fret. They have either a very lopsided L/s ratio or more than 12 notes. They are discussed further in the Nineteen-tone section. | ||
Every scale can be thought of as a chord, e.g. the 12edo major pentatonic scale is a 6add9 pentad. Many pentads and heptads have an [[Essential tempering commas|innate comma]] which 41edo does not temper out. Thus many Kite Guitar scales | Every scale can be thought of as a chord, e.g. the 12edo major pentatonic scale is a 6add9 pentad. Many pentads and heptads have an [[Essential tempering commas|innate comma]] which 41edo does not temper out. Thus many Kite Guitar scales have '''dual''' notes, meaning a note may vary by 1 edostep, in order to avoid a wolf 5th. In the tables below, a note that may be either a M2 or a vM2 is indicated by (v)M2. A scale with a dual note or two is called a '''fluid''' scale. In general, majorish scales have a dual 2nd and minorish scales have a dual 4th. But it depends on the chord progression. For example, Iv - IVv - Vv7 - Iv requires a major scale with a dual 4th. Step sizes can also be dual, leading to near-MOS scales (see below). | ||
== The Format == | == The Format == | ||
The modes of a scale are grouped together. Not every mode is shown. Often two scales are modes only because of the | The modes of a scale are grouped together. Not every mode is shown. Often two scales are modes only because of the dual notes, e.g. downmajor and upminor. Two modes of a scale will use the same prime subgroup, so modes are grouped by subgroup. Subgroups are explained on the theoretical scales page: [[Kite Giedraitis's Categorizations of 41edo Scales]]. | ||
Each scale has steps of various sizes, shown as a series of edosteps. A dash separates the P1-P5 section of the scale from the P5-P8 section. An odd number always hops a string, an even number never does. This chart translates the edostep sizes into 41-edo notation: | Each scale has steps of various sizes, shown as a series of edosteps. A dash separates the P1-P5 section of the scale from the P5-P8 section. An odd number always hops a string, an even number never does. Thus every non-awkward scale has exactly three odd-numbered steps. This chart translates the edostep sizes into 41-edo notation: | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
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|^m3 | |^m3 | ||
|} | |} | ||
The edosteps that are affected by | The edosteps that are affected by dual-ness are underlined. For example, the downmajor scale is P1 (v)M2 vM3 P4 P5 vM6 vM7 P8. The edosteps are listed as if the 2nd were plain: <u>76</u>47-674. Downing the 2nd makes <u>67</u>47-674. Almost always, the two underlined step sizes differ by only one, and upping or downing merely swaps the two numbers. Therefore the dual-ness rarely affects the moves (see below). The only exceptions are the dorian, locrian and dodecatonic scales, all of which are problematic anyway. | ||
The step sizes column shows the sizes used. Two modes of a scale will have the same step sizes, so modes are also grouped by step sizes. The largest-to-smallest ratio L/s indicates how even the scale is. For example, the downminor heptatonic scale has a very large L/s ratio of 8/2 = 4, giving it a lopsided feel. But the downminor ''pentatonic'' scale has a very small L/s ratio of only 9/7 = 1.29, giving it an even [[5-edo|equipentatonic]] feel. | The step sizes column shows the sizes used. Two modes of a scale will have the same step sizes, so modes are also grouped by step sizes. The largest-to-smallest ratio L/s indicates how even the scale is. For example, the downminor heptatonic scale has a very large L/s ratio of 8/2 = 4, giving it a lopsided feel. But the downminor ''pentatonic'' scale has a very small L/s ratio of only 9/7 = 1.29, giving it an even [[5-edo|equipentatonic]] feel. | ||
The step count column analyzes the scale by the usual MOS notation of how many large and small steps there are. Some scales also have m for medium, and even XL for extra large and xs for extra small. Most scales are not actually MOS, but a | The step count column analyzes the scale by the usual MOS notation of how many large and small steps there are. Some scales also have m for medium, and even XL for extra large and xs for extra small. Most scales are not actually MOS, but a near-MOS. For example, the first two pentatonic scales are 2L 1m 2s, where L=11, m=7 and s=6. The single m step can be thought of as a fluid version of the s step, making a 2L 3s near-MOS scale. Near-MOS scales are listed as alternates, e.g. "2L 1m 2s or 2L 3s". | ||
Harmonic and subharmonic scales are contiguous segments of the harmonic and subharmonic series respectively. They are never | Harmonic and subharmonic scales are contiguous segments of the harmonic and subharmonic series respectively. They are never fluid. Harmonic and subharmonic may be abbreviated as har- and subhar-, e.g. harmajor pentatonic. Pentatonic scales use (sub)harmonics 5-10, and heptatonic scales use (sub)harmonics 7-14. In harmonic scales, the step sizes get smaller as you ascend. In subharmonic scales, they get larger. In general, given a choice between an Ls sequence and an sL sequence, the first is often more otonal, and more consonant. For example, P1-M2-vM3 vs. P1-vM2-vM3, or P1-vm3-P4 vs. P1-^M2-P4, or even P1-vM3-P5 vs. P1-^m3-P5. (One exception: P4-d5-P5 is more otonal that P4-A4-P5. But P1-^m2-M2 is better than P1-m2-M2.) Likewise for the choice between LLs and LsL and sLL, or between Lss and sLs and ssL, the first is generally more consonant. | ||
Scales are loosely named similarly to how chords are named. Adding up or down to a scale name affects the 3rd, 6th and 7th. However, there are usually | Scales are loosely named similarly to how chords are named. Adding up or down to a scale name affects the 3rd, 6th and 7th. However, there are usually dual notes not implied by the name. Harmonic and subharmonic scales are named after the tonic triad, minus the up or down. | ||
Some scales are listed as chains of 5ths. For example, the downmajor scale is P1 (v)M2 vM3 P4 P5 vM6 vM7 P8. There are two chains: P4-P1-P5-M2 and vM2-vM6-vM3-vM7. This is condensed to P415M2 vM2637. Here the two chains overlap on a | Some scales are listed as chains of 5ths. For example, the downmajor scale is P1 (v)M2 vM3 P4 P5 vM6 vM7 P8. There are two chains: P4-P1-P5-M2 and vM2-vM6-vM3-vM7. This is condensed to P415M2 vM2637. Here the two chains overlap on a dual note. However, the near-equidistant heptatonic scales do not, and have a wolf 5th. | ||
The moves column is perhaps the most practical information in the table. It says how many frets to move up or down as you ascend the scale. Positive numbers refer to forward moves that move up the fretboard on a single string. Negative numbers refer to backwards moves that move up a string, then down the fretboard. The moves are not listed in order of size. Rather, forward moves are listed, then backward moves. In each category, they are listed by how often they occur in the scale. In case of a tie, the largest step size is listed first. The first move in the list is the primary forward move, and the first negative number is the primary backward move. All other moves are secondary. Because there are only 3 string-hops in an octave, there are at most 3 backwards moves. There are at most 2 secondary backwards moves, and usually only 1. A scale that doesn't have any secondary moves (i.e. has only two step sizes) is usually one of the rare MOS scales. | The moves column is perhaps the most practical information in the table. It says how many frets to move up or down as you ascend the scale. Positive numbers refer to forward moves that move up the fretboard on a single string. Negative numbers refer to backwards moves that move up a string, then down the fretboard. The moves are not listed in order of size. Rather, forward moves are listed, then backward moves. In each category, they are listed by how often they occur in the scale. In case of a tie, the largest step size is listed first. The first move in the list is the primary forward move, and the first negative number is the primary backward move. All other moves are secondary. Because there are only 3 string-hops in an octave, there are at most 3 backwards moves. There are at most 2 secondary backwards moves, and usually only 1. A scale that doesn't have any secondary moves (i.e. has only two step sizes) is usually one of the rare MOS scales. | ||
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== Pentatonic Scales == | == Pentatonic Scales == | ||
Every pentatonic scale has 5 modes, but only those modes with a non- | Every pentatonic scale has 5 modes, but only those modes with a non-dual 5th are listed. | ||
=== Major and minor scales === | === Major and minor scales === | ||
The za scales are nearly [[5-edo|equipentatonic]], dividing the P4 into two nearly equal steps of 8 and 9 edosteps (^M2 and vm3). Since one is a 2nd and the other a 3rd, this can conflict with one's categorical perception of intervals. Best to think of them both as "penta-2nds". | The za scales are nearly [[5-edo|equipentatonic]], dividing the P4 into two nearly equal steps of 8 and 9 edosteps (^M2 and vm3). Since one is a 2nd and the other a 3rd, this can conflict with one's categorical perception of intervals. Best to think of them both as "penta-2nds". | ||
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=== Altered minor scales === | === Altered minor scales === | ||
The conventional 12-edo melodic minor and harmonic minor scales are [[MODMOS]] scales, modified MOS scales. This is a sort of "macro- | The conventional 12-edo melodic minor and harmonic minor scales are [[MODMOS]] scales, modified MOS scales. This is a sort of "macro-dual-ness", where one scale step can vary by an entire semitone. The two harmonic downminor scales have an enormous L/s ratio! | ||
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<br> | <br> | ||
</tt> | </tt> | ||
In both cases, the D is | In both cases, the D is dual. But the two dorian scales and the two locrian scales are not from these lattices, and are not actually modes of the other scales. | ||
To be consistent, the two dorian scales should have a | To be consistent, the two dorian scales should have a dual tonic. To avoid this, and to provide all six triads, there are ''two'' dual notes. Note that the 6th of the <u>up</u>dorian scale can be <u>downed</u>. Note that this dual-ness affects the step sizes, and 74<u>67</u>-<u>74</u>6 can become 74<u>67</u>-<u>65</u>6. Thus the moves vary. | ||
To be consistent, the two locrian scales should have an upflat or downflat 5th. To get a plain flat 5th, and thus a more consonant 5:6:7 or 7/(7:6:5) tonic triad, the 5th is | To be consistent, the two locrian scales should have an upflat or downflat 5th. To get a plain flat 5th, and thus a more consonant 5:6:7 or 7/(7:6:5) tonic triad, the 5th is dual as well as the 3rd. Again, this dual-ness affects the step sizes and the moves. | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
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Tritonic scales are augmented triads. The moves are -0 and --1, meaning same fret up 1 string, and up 1 fret up 1 string. Tetratonic scales are dim6/dim7 tetrads. Both augmented and dim6/dim7 chords are discussed on the [[Kite Guitar Chord Shapes (downmajor tuning)|chords page]]. | Tritonic scales are augmented triads. The moves are -0 and --1, meaning same fret up 1 string, and up 1 fret up 1 string. Tetratonic scales are dim6/dim7 tetrads. Both augmented and dim6/dim7 chords are discussed on the [[Kite Guitar Chord Shapes (downmajor tuning)|chords page]]. | ||
=== Pentatonic (2L 2s | === Pentatonic (2L 2s 1xs) === | ||
We've already seen how the upmajor and downminor pentatonic scales are nearly equi-pentatonic. | We've already seen how the upmajor and downminor pentatonic scales are nearly equi-pentatonic. | ||
=== Hexatonic (whole tone) ( | === Hexatonic (whole tone) (1XL 3L 2s) === | ||
There are no perfect 5ths, only tritones. Thus there are no off-5ths, and no motivation for | There are no perfect 5ths, only tritones. Thus there are no off-5ths, and no motivation for dual-ness. There is only one scale which distributes the 3 large steps equally. There are six modes of this scale. Each mode is a pair of augmented triads. The three 4thward modes have a triad 3 frets above the tonic triad, and the three 5thward modes have it below. All six modes sound similar and are not named individually. | ||
The whole-tone scale is very comfortable to play physically with its alternating +3 and -3 moves. It only spans 4-5 frets and each string has exactly two notes. | The whole-tone scale is very comfortable to play physically with its alternating +3 and -3 moves. It only spans 4-5 frets and each string has exactly two notes. | ||
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|787-676 | |787-676 | ||
|} | |} | ||
=== Heptatonic ( | |||
=== Heptatonic (1XL 4L 2s) === | |||
These are reminiscent of [[7edo|7-edo]]. The 4th is divided into three nearly equal steps of two vM2's and a ~2 (6 6 5), thus it's also reminiscent of the third-4th [[pergen]] and the [[Porcupine|Triyo]] temperament. Unfortunately, obvious near-equal scales like P1 ~2 ~3 P4 P5 ~6 ~7 P8 = 5757-575 are very awkward to play. | These are reminiscent of [[7edo|7-edo]]. The 4th is divided into three nearly equal steps of two vM2's and a ~2 (6 6 5), thus it's also reminiscent of the third-4th [[pergen]] and the [[Porcupine|Triyo]] temperament. Unfortunately, obvious near-equal scales like P1 ~2 ~3 P4 P5 ~6 ~7 P8 = 5757-575 are very awkward to play. | ||
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These scales can be derived from the seven ya modes by widening the two smallest steps by 1 or 2 edosteps, from an upminor 2nd to a mid or downmajor 2nd. The tonic triad is never altered by the widening, thus equi-lydian and equi-mixolydian would be the same as equi-major, and equi-phrygian the same as equi-minor. | These scales can be derived from the seven ya modes by widening the two smallest steps by 1 or 2 edosteps, from an upminor 2nd to a mid or downmajor 2nd. The tonic triad is never altered by the widening, thus equi-lydian and equi-mixolydian would be the same as equi-major, and equi-phrygian the same as equi-minor. | ||
The standard downmajor and upminor scales have two chains of 5ths. Only one | The standard downmajor and upminor scales have two chains of 5ths. Only one dual note linking them is needed to avoid a wolf fifth. But these scales have three chains, and would need two dual notes. They are shown here with only one dual note, thus they have a wolf 5th. To avoid the wolf, either the 3rd must be dual, or else there must be a step of 4 edosteps which inflates the L/s ratio and destroys the near-equal feel. The two equi-mid scales each have two wolf 5ths. | ||
These scales are harmonic or subharmonic series fragments. Equi-major is (8:9:10:11:12)/8 plus (9:10:11:12)/6. Equi-mid is (9:10:11:12)/9 + (8:9:10:11:12)/6. Equi-minor is 12/(12:11:10:9:8) + 18/(12:11:10:9). | These scales are harmonic or subharmonic series fragments. Equi-major is (8:9:10:11:12)/8 plus (9:10:11:12)/6. Equi-mid is (9:10:11:12)/9 + (8:9:10:11:12)/6. Equi-minor is 12/(12:11:10:9:8) + 18/(12:11:10:9). | ||
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=== Octotonic (3L 3m 2s) === | === Octotonic (3L 3m 2s) === | ||
The [[Bohlen-Pierce]] or B-P 13-edt scale is a non-octave scale that is contained in 41-edo. (Technically, the 41edo scale is 13-edt stretched by half a cent.) B-P has only one step size, 5 edosteps = ~2. These steps are very near to one-eighth of an octave, so it can be thought of as a near-8edo scale. Unfortunately B-P is very awkward to play on the Kite guitar. It also has the wolfy ^5 and | The [[Bohlen-Pierce]] or B-P 13-edt scale is a non-octave scale that is contained in 41-edo. (Technically, the 41edo scale is 13-edt stretched by half a cent.) B-P has only one step size, 5 edosteps = ~2. These steps are very near to one-eighth of an octave, so it can be thought of as a near-8edo scale. Unfortunately B-P is very awkward to play on the Kite guitar. It also has the wolfy ^5, v8 and M10 intervals. As with the other scales, we can avoid the wolves by shifting some of the notes by a single edostep. We must shift three notes, and shifting one more also makes the scale non-awkward. Instead of 13 equal steps 5 5 5 5 5 5 5 5 5 5 5 5 5, we have 5 6 4 5 6 4 5 6 4 5 6 4 5. The moves are +3 +2 -4. This scale now includes a perfect 8ve, which implies an octave-repeating version of it that is 5 6 4 5 6 4 5 6. This is one form of the octotonic 3L 3m 2s scale. | ||
The 3rd step size (4 edosteps) occurs twice in | The 3rd step size (4 edosteps) occurs twice in octotonic scales, making them less near-equal and less MOS-like than the other near-equal scales so far. Every scale contains a section of 2L 1m, making that section identical to an equi-heptatonic tetrachord. For example, the first scale in the table below has P5 ~6 ^m7 P8, as does the equi-minor scale. Every scale also contains a section of 2L 1s, making a down-4th, and thus an up-5th upon octave inversion. Every scale also contains 1L 2m, which makes another down-4th. Fortunately, octotonic chords are naturally constructed from stacking "octa-thirds", i.e. using every other note of the scale. Chords avoid both perfect and off-perfect 5ths in favor of the dim 5th. The down-4th is likewise avoided. | ||
Because of the prominence of the "octa-5th" (i.e. tritone) in octatonic chords, this interval plays a role analogous to the perfect 5th in other scales. Every octotonic scale contains eight tritones. The most consonant tritone is the dim 5th = 7/5. Of course all eight tritones can't be dim 5ths without | Because of the prominence of the "octa-5th" (i.e. tritone) in octatonic chords, this interval plays a role analogous to the perfect 5th in other scales. Every octotonic scale contains eight tritones. The most consonant tritone is the dim 5th = 7/5. Of course all eight tritones can't be dim 5ths without dual-ness, but half of them can be. In particular, the tonic chord can be a dim7 chord that contains two dim 5ths. The only two such chords that are playable are the ^dim7 and vdim7 chords. If we require that the remaining four notes of the scale make another such chord, there are only two near-equal octotonic scales. Each has two main modes, depending on which of the dim7 chords is considered to be the tonic chord. | ||
The scales are named after the root of the non-tonic dim7 chord. This chord is always upped or downed (^dim7 vs. vdim7) to match the root. If the tonic chord is upped or downed the opposite way, the two dim7 chords, and hence the entire scale, can easily be deduced from the name: the <u>up</u>flat-2 octotonic scale has an <u>up</u>dim7 chord on the ^bII and a <u>down</u>dim7 chord on the I. The octave inverse of ^b2 is vM7, thus the other main mode of the upflat-2 scale is the down-7 scale. If the tonic chord is upped or downed the same way, we must add that direction to the name: the up-3 up scale has an updim7 chord on ^III and an updim7 chord on I. | The scales are named after the root of the non-tonic dim7 chord. This chord is always upped or downed (^dim7 vs. vdim7) to match the root. If the tonic chord is upped or downed the opposite way, the two dim7 chords, and hence the entire scale, can easily be deduced from the name: the <u>up</u>flat-2 octotonic scale has an <u>up</u>dim7 chord on the ^bII and a <u>down</u>dim7 chord on the I. The octave inverse of ^b2 is vM7, thus the other main mode of the upflat-2 scale is the down-7 scale. If the tonic chord is upped or downed the same way, we must add that direction to the name: the up-3 up scale has an updim7 chord on ^III and an updim7 chord on I. | ||
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| +2, -5 | | +2, -5 | ||
|} | |} | ||
The obvious uneven scale is the [[Duodene|harmonic duodene]], with 3 | The obvious uneven scale is the [[Duodene|harmonic duodene]], with 3 dual notes to avoid wolf 5ths. Note that the dual major 2nd and dual minor 7th affect the step sizes. Both vM2-^m3 and vM6-^m7 are 5 edosteps. The step count and moves are affected as well. | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
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For an even scale with small steps that's not awkward, see the next section. | For an even scale with small steps that's not awkward, see the next section. | ||
=== Decatonic - the semitonal scale (2L 7s | === Decatonic - the semitonal scale (2L 7s 1xs) === | ||
Is there an easily playable chromatic-sounding scale with nearly equal steps? One such is the decatonic scale. However, the term for these scales is not chromatic but '''semitonal''', because the steps are roughly the size of a 12edo semitone. '''Chromatic''' refers to movement by a single fret, see the section on 19-tone scales. | Is there an easily playable chromatic-sounding scale with nearly equal steps? One such is the decatonic scale. However, the term for these scales is not chromatic but '''semitonal''', because the steps are roughly the size of a 12edo semitone. '''Chromatic''' refers to movement by a single fret, see the section on 19-tone scales. | ||