Magic Guitar

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Guitars Designed for Magic Temperament

Magic Temperament is a good, mainstream choice for an unequal guitar fretting. It's almost as simple as Meantone but with much improved 9-limit harmony. It can be close enough to just intonation to sound clean within the inherent imprecision of a physical guitar. It has well tuned perfect fifths that will be familiar from 12-edo. It can be tuned to 41-edo while minimizing the number of frets spaced by a single step. As it isn't a Meantone, it will have comma problems when playing conventional music. Magic supports full 11-limit harmony with acceptable tuning within the appropriate gamut for a guitar.

Fretting Options

A simple Magic fretting can follow either a 19 note (Pengcheng) or 22 note (Haizhou) MOS. A Pengcheng fretting is good ergonomically because the frets will be spaced at intervals of (approximately) two or three steps from 41-edo. It is poor for playing diatonic music because it allows for no comma shifts. A Haizhou fretting has a few frets spaced a comma apart and still allows the 19 notes of the Pengcheng scale to be played on frets spaced a semitoe, 2 steps from 41-edo, apart.

Most frets on either a Haizhou or Pengcheng are spaced a semitoe apart. In both cases, there are three discontinuities to an octave. An isomorphic fretting with equal semitoes right down the neck is possible. In the general case, each string sits on a different foot of Tripod Notation. With equal temperament, a string can morph from one foot to another to become a skip fretting. Given 41-edo, this leads to a fret spacing of 2 steps from 41-edo or 20½-edo or the Kite Guitar.

Old Spear's standard fretting pattern places the discontinuity of a Haizhou fretting such that both 10/9 and 9/8 are available from the nut but allows for Pengcheng fretting on the rest of the neck. The closer to the nut, the wider the frets are spaced, so the first comma-spaced fret is acceptable. Comma shifts can still be reached at other positions by bending the strings.

Standard Tuning

Old Spear's standard tuning of the open strings follows the standard guitar tuning, also known as EADGBE or EBGDAE, adapted to magic temperament. The highest four strings (corresponding to the strings on a ukulele) are tuned to an Em⁷ chord, giving a poor voicing of Em⁷ on the open strings while allowing for E minor or G major chords across three strings and transpositions on other frets. The low E is naturally two octaves from the high E. This leaves the tuning of the A string wild. It must have a wolf with either E or D. You can choose the tuning for the key you want to play. Either way, some chords won't work with their conventional shapes and others will work but give you no choice about how to resolve comma shifts. Of course, the advantage of a standard tuning variant is that you put your fingers roughly where you expect given experience of a standard guitar. With your electronic tuner, make B and E and maybe A about 20 cents flat relative to D and G and maybe A. If tuning to 41-edo, the fifths should also be widened slightly relative to 12-edo, and your ears can probably guide you with this.

In Tricycle Notation with wheel numbers as subscripts, the open strings from the bass are E₂ A D₂ G₃ B₁ E₂. Tuning A as A₃ gives perfect fourths in the bass and places the wolf between A₃ and D₂. Where B is played on the A string, Old Spear's comma-doubled frets give two different tunings a comma apart. An E major chord takes the higher B using the narrowly spaced frets, an E on the D string using the other whole tone fret, and G♯ obviously on the first fret of the G string. A G major chord uses the same pitch. An A minor chord works with the open E and A strings, E on the wider-spaced whole tone fret on the D string, A on the same fret on the G string, and C on the obvious second fret on the B string. An A major chord has three pitches all on the same whole tone fret. A D minor chord can't use the open A string and so is restricted to 4 strings.

Tricycle Notation guides you on how to resolve the commas. E major is E₂ G♯₃ B₁. The first three frets conserve the wheel number. A₃ to B₁ changes wheel, so use the 9/8 fret, the one with narrow spacing. D₂ to E₂ doesn't change wheel, so use the 10/9 fret. G♯₃ is one semitoe from G♯ so use the first fret. (G₃ is correct for E minor, so use the open string there.) B₁ and E₂ are correct, so use the open strings.

A minor is A₃ C₁ E₂. E and A work on the open strings. D₂ to E₂ stays on the same wheel and so uses the 10/9 fret. G₃ to A₃ stays on the same wheel and so uses the 10/9 fret. B₁ to C₁ stays on the same wheel and so uses the second fret. B₁ to C♯₁ also stays on the same fret, so the 10/9 fret is consistently used for C major. Chords that only use the first three frets of Old Spear's layout have a special property: they are consistent with the Kite Guitar and can be played on 10 different positions of a Pengcheng guitar or 13 positions of a Haizhou guitar, provided the comma frets don't get too close to comfort. These 13 chords might not be arranged as convenient substitutes for 12 chords of chromatic music.

G major is G₃ B₁ D₂ and already uses three open strings. E₂ to G₃ is a minor third and correctly moves up by 1 wheel. A₃ to B₁ changes wheel so use the 9/8 fret, the one with the narrow spacing.

The alternative tuning of A gives E₂ A₁ D₂ G₃ B₁ E₂. The wolf is now between E₂ and A₁. Chords that use B₁ now keep the 10/9 fret on the A string. D chords can be played in tune. A major can be used with the open A string but it means shoving three fingers between the comma-spaced frets. E chords only use the first three frets, so these chord shapes work on 10 different Pengcheng and 13 different Haizhou roots.

E major chords can also be played with an added harmonic seventh fretted to turn B₁ into the semitoe below D₂. The same fret gives a C harmonic seventh chord. In both cases, the approximate 7/6 fret turns a 6 into a 7. Sadly, the standard shape for G7 is inconsistent with a harmonic seventh.

The two tunings of the A string give every pitch of 41-edo between them somewhere on the fretboard.

There are variations of standard tuning that retune the two bass strings to fit the home key and tend not to fret them, giving temporary drones. Logic like this is fine with a magic fretting. If you're going to retune the A string to suit the key, why not go the whole hog and tune them how you like so that you only have 4 strings to think about fretting?

Major Thirds Tuning

The simplest tuning for magic temperament is to tune the open strings to the 5:4 major third that generates the temperament. This minimizes the discrepancies between the strings. Old Spear considers this to be interesting theoretically but tricky in practice. It uses the same chord shapes as on the first three frets of the Kite Guitar. However, most Kite Guitar chords use more than the first three frets, so Old Spear's magic frettings work better with different tunings.

An alternative is to tune three strings to 5:4 and set pairs of three strings an octave apart. This is simpler yet and corresponds more closely to the lines of Tripod Notation.

Open Tunings

Open tunings will generally work with a magic guitar. You know that at least one chord will work. The frets are chosen to give approximations to simple JI intervals which means that other notes of the key should be there.

As a variation, you could fret a guitar to just intonation ratios that the magic guitar approximates and then tune the open strings to just intonation. For example, frets at 25/24, 16/15, 10/9, 9/8, 7/6, 6/5, 5/4, 4/3, etc, or equivalents with 225:224 tempered out. This will work for a few chords relative to an open tuning but break some transpositions (in the sense of being a little bit out of tune).

Fourths Tuning

Sorry, fourths tuning doesn't really work with Magic Temperament. Chaining Fourths like that assumes Meantone. Old Spear's Standard Tunings work because they break the chain low down and keep a major third between G and B. You can also include a wolf in a fourths tuning but doing that breaks the isomorphic pattern you wanted and only complicates matters.