For guitarists a good basis for understanding ukulele tuning is the fact that the uke's four strings are directly related to the top four strings of the guitar as if fretted at the 5th fret. That makes it G C E A from lowest pitch guitar string to highest pitch in order.
The C on the 5th fret of the guitar's G (3rd) string is the same register as the "Middle" C of a piano. Middle C can usually be found in the middle of the keyboard nearest the makers name or logo mark. (For us geeks that's C4 or the C that is 261.6Hz)
It is also good to be aware of four things here:
Simply put a re-entrant tuning is when the strings or courses of strings don’t follow in order from highest to lowest pitch or lowest to highest pitch without interruption. Banjos, ukulele tuning and charrangos are classic examples of this type of break in tuning. The tuning does not normally ascend or descend all in order of pitch.
Whatever order we put the letters in as long as the pitches don't go from low to high or visa versa without some interruption we have re-entrant tuning and not linear tuning. Look very closely at the photo. You may be able to see the 4th string (the one on the left) is smaller. That is because it is in the standard re-entrant tuning with the 4th or G string of a gauge that can be pitched/tuned one octave higher. So this is a "C6 high 4" tuning short for "GCEA with the 4th string G one octave higher in a re-entrant" tuning. I think "C6 high 4" is less of a mouthful!
If the above is true then it follows that whatever order we put the letters in as long as the pitches do go from low to high or visa versa without any interruption we have a Linear Ukulele Tuning and not Re-entrant Tuning.
As far as tuning interval structure when we talk about the C6 tuning the notes that the strings are tuned to (in order from closest to your chin to closet to your feet) are G C E A. That is actually a C6 chord. The A string is considered string One and the G string is string Four regardless of the the octave that the G string inhabits. So in both re-entrant and linear tuning the letter order is the same.
The chord harmony analysis follows standard western theory, that is:
(C6 = C the Root, E the 3rd , G the 5th and A the 6th)
It is just arranged by string order as:
(G the 5th, C the Root, E the 3rd, A the 6th)
The D6 ukulele tunings would comprise the letters (notes) of the D major 6 Chord i.e. DF#AB (or Root 3rd 5th 6th) in the string order ADF#B (or 5th Root 3rd 6th).
If you don't understand the chord theory have a look at this video. There are other videos on this site that may help. Just go to the Music Theory Section and the Guitar Theory section. Don't forget you can look through the whole catalogue of videos on youtube or Vimeo.
There are really six sizes that I've heard of but we will stick to the four pretty standard sizes and tunings for the ukulele. From the smallest in size and highest in pitch to the largest size and lowest pitch they are as follows:
The Soprano
D6 Tuning: A D F# B (either re-entrant or linear “low 4th A”)
C6 tuning: G C E A (either re-entrant or linear “low 4th G”)
The Concert Soprano
D6 Tuning: A D F# B (either re-entrant or linear “low 4th A”)
C6 tuning: G C E A (either re-entrant or linear “low 4th G”)
The Tenor
C6 tuning: G C E A (linear “low 4th G”)
The Baritone
G6 tuning: D G B E (linear “low 4th D”)
A really great way to get to grips with this instrument if you already play guitar is to put a capo on the fifth fret of your guitar and ignore the bottom two strings (the low E and A strings). At the 5th fret the capo gives you the C6 low 4 tuning! At the 7th fret you will get the D6. Of course the Baritone ukulele tuning is the G6 that is just like your guitar without any low E or A strings and no capo!
Happy playing!
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