Lesson 8: The ukulele and the bass

The first seven lessons were written on a six-string guitar in standard tuning, and almost every sentence in them is still true on a ukulele or a bass: a pitch class is a pitch class, a major triad is a major triad. What changes is the map — the grid of strings and frets that the notes are laid on. Two of GA’s own constants, Tuning.Ukulele and Tuning.Bass, define that map for the two instruments most people pick up after the guitar, and they are enough to show the two facts that guitar-shaped code tends to get wrong.

• A ukulele is re-entrant: its fourth string is higher than its third. String number and pitch stop agreeing.
• A bass is tuned entirely in fourths: it has none of the guitar’s one irregular pair, so every shape moves between strings unchanged.

All GA links point to commit a826864.

dotnet run --project code/music-theory-ga/GaTheory -c Release -- l8

The two tunings

A ukulele’s four strings are G C E A, and a bass’s four are E A D G, the guitar’s four lowest strings an octave down. Strings are numbered from the highest-sounding one, so string 1 is the A on a ukulele and the G on a bass.

== Tunings: the ukulele and the bass, string 1 first
string       course       GA           check
ukulele 1    A4           A4           ok
ukulele 2    E4           E4           ok
ukulele 3    C4           C4           ok
ukulele 4    G4           G4           ok
bass 1       G2           G2           ok
bass 2       D2           D2           ok
bass 3       A1           A1           ok
bass 4       E1           E1           ok

The course writes each tuning as a list of pitches and reads string n off it; GA stores a Tuning and indexes it with a Str, a range value object from 1 to 26. Both agree on all eight strings.

Re-entrant: the fourth string is not the lowest

On a guitar, string 6 is the lowest and string 1 the highest, and the pitches rise all the way. A standard ukulele does not do that. Its fourth string is a G above middle C — above the third string and above the second.

== Re-entrant: on a ukulele the fourth string is not the lowest
string       course         GA             check
ukulele 1    A4 midi 69     A4 midi 69     ok
ukulele 2    E4 midi 64     E4 midi 64     ok
ukulele 3    C4 midi 60     C4 midi 60     ok
ukulele 4    G4 midi 67     G4 midi 67     ok
steps from string 4 to string 1: -7 4 5  (the negative step is what makes it re-entrant)
the low G ukulele instead:      5 4 5  (only rising: not re-entrant)

The steps between consecutive strings are the giveaway: −7, +4, +5. A tuning is re-entrant when the pitches, read in string order, neither only rise nor only fall (Re-entrant tuning, Wikipedia). The ukulele’s high fourth string is what gives it its sound: a strummed chord has no bass note, and the lowest note you hear is not the one your lowest finger is on.

Some players restring the fourth string an octave down — “low G” — and then the ukulele is an ordinary rising instrument. GA’s constant is the re-entrant one.

Why this matters in code

Anything that reasons about a fretboard by assuming that a higher string number means a lower pitch is right on a guitar, a bass, a mandolin and a violin, and wrong on a ukulele, a five-string banjo, a charango and a Tahitian ukulele. That assumption is easy to write by accident, because on a guitar the two orders coincide.

Where GA decides which end of a tuning is string 1

GA builds its array of strings in Tuning.BuildPitchArray. A tuning may be written either way round, so the constructor guesses: if the first pitch is lower than the last, reverse the list; otherwise keep it.

if (buffer[0] < buffer[^1])
{
    for (var i = 0; i < buffer.Length; i++)
    {
        result[i] = buffer[buffer.Length - 1 - i];
    }
}

It compares the two ends only. On any tuning that rises or falls all the way, the ends tell you the direction and the guess is right. On a re-entrant tuning they do not.

== How GA decides which end of a tuning is string 1
instrument             course           GA               check
guitar                 E4               E4               ok
ukulele                A4               A4               ok
bass                   G2               G2               ok
banjo, tenor jazz      A4               A4               ok
banjo, 5 strings       D4               G4               DIFF
banjo, C tuning        D4               G4               DIFF
GA, banjo strings 1 to 5: G4 D3 G3 B3 D4
The fifth string of a 5-string banjo is the short drone, the highest of the five, so
"string 1 is the highest" is not a definition: here GA returns the file's order, reversed.

The ukulele survives, by luck: its highest string happens to be the last one written. The five-string banjo does not. Its fifth string is a short drone tuned to the G above the first string, and GA’s own Instruments.yaml writes it first, G4 D3 G3 B3 D4. The first pitch is higher than the last, so the list is kept as written, and GA answers that string 1 is the drone and string 5 is the first string — the instrument’s numbering, reversed.

The deeper problem is in the documentation rather than in the arithmetic. Str’s comment reads “String 1 is the string with the highest pitch”, and Tuning.AsSpan promises “highest string first”. On a five-string banjo no such rule can hold: string 1 is D4 and string 5 is G4, higher. String numbering is a fact about the instrument — which string your first finger falls on — not a sort order. A tuning that carries its strings in playing order, and never re-sorts them, has nothing to guess.

The ukulele has the interval pattern of four guitar strings

Forget the pitches for a moment and look only at the gaps between strings.

== The ukulele has the interval pattern of four guitar strings
instrument               tuning                 steps between strings
guitar, strings 4 to 1   D3 G3 B3 E4            5 4 5
ukulele, low G           G3 C4 E4 A4            5 4 5
ukulele, baritone        D3 G3 B3 E4            5 4 5
guitar, strings 6 to 3   E2 A2 D3 G3            5 5 5
bass                     E1 A1 D2 G2            5 5 5
ukulele - guitar         5 5 5 5                semitones, string by string
baritone - guitar        0 0 0 0                semitones
bass - guitar 6 to 3     -12 -12 -12 -12        semitones

Three results, each worth a sentence.

• A ukulele’s gaps are 5 4 5 semitones, and so are the gaps of the guitar’s strings 4-3-2-1. A ukulele is the top four strings of a guitar, tuned up a perfect fourth.
• A baritone ukulele is those four strings exactly: D3 G3 B3 E4, difference zero. Every chord shape you know on the top four strings of a guitar is the same chord, with the same name, on a baritone ukulele.
• A bass’s gaps are 5 5 5, and so are the guitar’s strings 6-5-4-3. A bass is the guitar’s bottom four strings an octave down.

That is why the ukulele feels familiar: the one irregular gap of the guitar, the major third between strings 3 and 2, is also on the ukulele, between its strings 3 and 2. The shapes carry over with their kinks intact.

A guitar shape played on a ukulele

Take the top four strings of the open C chord — frets 2, 0, 1, 0 on strings 4, 3, 2, 1 — and put the same fingers on a ukulele.

== A guitar shape played on a ukulele: the same fingers, a fourth higher
instrument               sounds                 pitch classes
guitar, strings 4 to 1   E3 G3 C4 E4            0 4 7  C major
ukulele, low G           A3 C4 F4 A4            0 5 9  F major
ukulele, re-entrant      A4 C4 F4 A4            0 5 9  F major
difference               5 5 5 5                semitones: a perfect fourth
The re-entrant ukulele plays the same four pitch classes, with its bass note an octave up.

The same fingering, a perfect fourth higher: a C major shape becomes F major. The third line is the interesting one. On a re-entrant ukulele the fourth string is an octave up, so the shape sounds A4 C4 F4 A4 instead of A3 C4 F4 A4: the same four pitch classes, the same chord name, a different voicing, with the root no longer at the bottom. Lesson 3’s point about inversions applies here at the level of the instrument — a re-entrant ukulele has no root-position voicings on all four strings.

A3 C4 F4 A4

E3 G3 C4 E4

== Pitch class C on frets 0 to 12
instrument   course                             GA                                 check
ukulele      4/5 3/0 3/12 2/8 1/3               4/5 3/0 3/12 2/8 1/3               ok
bass         4/8 3/3 2/10 1/5                   4/8 3/3 2/10 1/5                   ok

Five Cs on a ukulele’s first twelve frets against six on a guitar’s, and the course and GA agree on all of them: Fretboard takes any Tuning and any fret count, so new Fretboard(Tuning.Ukulele, 12) works with no other change.

A comment that contradicts its code

Fretboard.GetNote(int stringIndex, int fret) documents its first parameter as “Zero-based string index (0 = lowest string)”, then calls Tuning[new(stringIndex + 1)] — and string 1 is the highest. Index 0 is the highest string, not the lowest. The code is consistent with Str; only the comment is wrong, which is the more dangerous way round.

A bass is tuned in fourths

== A bass is tuned in fourths: one shape, every pair of strings
pair             semitones    interval
bass 4-3         5            a fourth
bass 3-2         5            a fourth
bass 2-1         5            a fourth
guitar 6-5       5            a fourth
guitar 5-4       5            a fourth
guitar 4-3       5            a fourth
guitar 3-2       4            a major third
guitar 2-1       5            a fourth
Every pair on the bass is a fourth, so a shape keeps its sound on any pair of strings.
On the guitar the 3-2 pair is a major third: that one gap is what makes the shapes irregular.

A guitar has one irregular pair. Strings 3 and 2 are a major third apart, four semitones, where every other pair is a perfect fourth. That single gap is the reason the CAGED shapes of lesson 3 are five different shapes rather than one shape moved around, and the reason a shape moved across the 3-2 boundary has to be adjusted by a fret.

A bass has no such pair. Every interval is a fourth, so an interval shape — a root and a fifth, an octave, a scale fragment — keeps its sound wherever you move it, sideways or across strings. The same is true of the seven-string guitar’s B string, of AllFourthsguitar in GA’s catalogue, and of a bass with five or six strings.

The ukulele and the bass in GA’s catalogue

Tuning.Ukulele and Tuning.Bass are C# constants. GA also has a 1589-line YAML catalogue, Instruments.yaml, with 122 instruments and 280 tunings, and the two do not say the same thing. Appendix A reads the whole file; here are the rows for these two instruments.

== The ukulele and the bass in GA's Instruments.yaml
entry                                tuning as written              read by the course
BassGuitar.Standard                  E1 A1 D2 G2                    4 pitches
BassGuitar.FiveStrings               B0 E1 A1 D2 G2                 5 pitches
BassGuitar.SixStrings                B0 E1 A1 D2 G2 C3              6 pitches
Ukulele.SopranoConcertAndTenorC      C6 G4 C4 E4 A4                 5 pitches
Ukulele.SopranoConcertAndTenorD      D6 A4 D4 F#4 B4                5 pitches
Ukulele.Baritone                     D4 G3 B4 E4                    4 pitches
Ukulele.Tahitian                     G4 G4 C5 C5 E5 E5 A4 A4        8 pitches
Ukulele.BanjoOrBanjoleteC            C6 G4 C4 E4 A4                 5 pitches
Ukulele.BanjoOrBanjoleteD            D6 A4 D4 F#4 B4                5 pitches
BassGuitarExtended.DropD             D1 A1 D2 G2                    4 pitches
BassGuitarExtended.BEAD              B0 E1 A1 D2                    4 pitches
BassGuitarExtended.Tenor             A1 D2 G2 C3                    4 pitches
BassGuitarExtended.Piccolo           E2 A2 D3 G3                    4 pitches
BassGuitarExtended.FretlessStandard  E1 A1 D2 G2                    4 pitches
BaritoneUkulele.Standard             D4 G3 B3 E4                    4 pitches
BaritoneUkulele.LowGReentrant        D4 G4 B3 E4                    4 pitches
SopraninoUkulele.Standard            D5 G4 B4 E5                    4 pitches

GA's own constants: Tuning.Ukulele = G4 C4 E4 A4 (4 strings), Tuning.Bass = E1 A1 D2 G2 (4 strings)
A ukulele has four strings: in the five C6 and D6 entries, the tuning's name, C or D, is written as a pitch.
A baritone ukulele is tuned like the guitar's top four strings, D3 G3 B3 E4; the two entries
disagree with each other and with that: D4 G3 B4 E4 and D4 G3 B3 E4.

The bass rows are all correct — standard, five-string, six-string, drop D, BEAD, tenor, piccolo — and BassGuitarExtended.Piccolo, E2 A2 D3 G3, is exactly the guitar’s bottom four strings, which is what a piccolo bass is.

The ukulele rows are not.

• Five pitches for a four-string instrument, four times over. C6 G4 C4 E4 A4 is the correct tuning G4 C4 E4 A4 with C6 in front of it. That C6 is the tuning’s name — ukulele players call G C E A “C tuning” — written where a pitch goes, and it parses as the C two octaves above middle C. The same mistake produced D6 A4 D4 F#4 B4, and both are repeated for the banjolele. It is not confined to the ukulele: the pedal steel guitar’s two tunings begin C6 and E9, which are also tuning names, C6 and E9 being the two standard pedal steel setups.
• A baritone ukulele in the wrong octave, twice, differently. A baritone is tuned D3 G3 B3 E4, the guitar’s top four strings. Ukulele.Baritone says D4 G3 B4 E4 and BaritoneUkulele.Standard says D4 G3 B3 E4: two entries for one instrument, disagreeing with each other and both putting the D an octave too high.
• A name that contradicts itself. BaritoneUkulele.LowGReentrant, “Re-entrant (dGBE)”, is D4 G4 B3 E4. Low G and re-entrant are opposites: a low G is what you restring a re-entrant ukulele to in order to stop it being re-entrant.

None of this reaches a program that asks GA for a ukulele tuning, because nothing reads the file — which is Appendix A’s subject.

Exercises

1. A ukulele’s open C chord is 0003: nothing on strings 4, 3 and 2, third fret on string 1. Which notes sound, and what chord is it?
2. Play the same 0003 on a baritone ukulele. Which chord is it now, and why is it not the same one?
3. How many of the 280 tunings in GA’s catalogue are written neither only rising nor only falling? Is every one of them a re-entrant instrument?

Solutions

== Exercise solutions
question                     answer
1. ukulele 0003              G4 C4 E4 C5                    pitch classes 0 4 7: C major
2. 0003 on a baritone        D3 G3 B3 G4                    pitch classes 2 7 E: G major
3. re-entrant entries        107 of 273                     tunings that rise and fall
   first ten: Baglama.Greek, Bajo.Quinto, Bajo.Sexto, Bandola.Central, Bandola.Guayanesa, Bandora.Standard, Bandurria.Peruvian, Banjo.TenorIrish, Banjo.Bluegrass5Strings, Banjo.CTuning5Strings

1. G4 C4 E4 C5 — pitch classes 0, 4 and 7, a C major triad. The re-entrant G4 is the highest note but sits on the fourth string, so the chord sounds with G on top and C doubled.
2. D3 G3 B3 G4 — pitch classes 2, 7 and 11, a G major triad. The baritone is tuned a perfect fourth below the standard ukulele, so every shape names a chord a fourth lower: C becomes G.
3. 107 of the 273 readable tunings. But no: most of them are courses, pairs of strings tuned in octaves or unisons, written as A2 A1 D3 D2 …, where the fall is inside a pair rather than across the instrument. The test used here — “the steps are not all of one sign” — finds every re-entrant tuning and also every doubled-course instrument, and the file gives no way to tell them apart, because it stores pitches and nothing about which of them share a course. Telling the twelve-string guitar from the ukulele would need a field the file does not have.

Key takeaways

• Strings are numbered from the highest-sounding string on a guitar, but that is a convention about typical instruments, not a definition: on a five-string banjo, string 5 is the highest.
• A tuning is re-entrant when its pitches, in string order, neither only rise nor only fall. Guessing the direction from the two ends works only for the instruments that make the guess unnecessary.
• The gaps matter more than the pitches. 5 4 5 is a guitar’s top four strings, a ukulele and a baritone ukulele; 5 5 5 is a bass and a guitar’s bottom four.
• A baritone ukulele is a guitar’s top four strings, exactly. A standard ukulele is the same, up a fourth. A bass is the bottom four, down an octave.

Sources

• Re-entrant tuning, Ukulele, Banjo, Bass guitar and Pedal steel guitar, Wikipedia.
• GuitarAlchemist/ga at a826864: Common/GA.Domain.Core/Instruments (Tuning.cs, Primitives/Str.cs, Primitives/Fretboard.cs), Common/GA.Business.Config/Instruments.yaml.
• The course program: Lesson8.cs, Instruments.cs, expected/l8.txt.

Next: Appendix A reads all 122 instruments of GA’s catalogue, and what GA’s own loader returns for them.