67. The best minimal, instruments for chords, 4-voice harmony and mainly harmonic melodic songs: The Harmonic Tetrachord (4-double strings or 4-voices ) in various sizes.

(This post has not bee written completely yet)

Although in the next I describe 4-courses string instruments with the stanrd ukulele (cuatro) tuning, I have set to them now days alternative tunings that sound better

I have usuallythe next tuning on my 4-courses string instruments

(Here 5-interval of 5th, 4=interval of 4th , 6m=interval of minor 6th, 4M-interval of major 3rd
3m=interval of minor 3rd) 

1) Standard guitar tuning and ukulele (Venezolano cuatro) tuning 4-3M-4

2) Cavaco (Brazilian  cuatro)  tuning 4-3M-3m

3) 1st overtones tuning 5-4-3M-3m-4 (as in post 191 and 301) for 6 strings, or less

4) Troll or Black mountain or Calico open tuning 5-4-3M on 4 (courses) strings instruments (see post 301)

5) Inverted  ukulele tuning   5-6m-5 on 4 (courses) strings instruments as in post 308

6) 5th overtones open tuning 4-3M-3m-4-5 as in post 301, for 6 strings or less
For less than 6 strings we keep the higher part of the tuning so as to have the interval of 5th at the two highest notes E.g. for 4 strings it would be 3m-4-5

7) 6th overtones tuning 5-4-3M-4-5 as in post 301, for 6 strings or less

8) The alternating minor-major 3rds of harmonic guitra tuning 3m-3M-3m as in post 164 for 6 strings or less and in particularthe panduri tuning 3M-3m for 3-course string instruments 

We highlight here instruments made from 4-pairs of strings, and of special and well known tuning defined as intervals among these 4-pairs of strings, as they have special advantages compared to other instruments, when the size of the scale length is between  50 cm to 77 cm. We call such instruments Harmonic Tetrachords. The term Harmonic refers to their optimal design  in playing chords, and melodies that are more than 70% harmonic and less that 30% chromatic. In the Greek Language χορδη (chord) means string and tetra means quartet. When saying that it is appropriate for melodies more than 70% harmonic and less that 30% chromatic, we do not mean that it cannot play, all melodies in the 12-notes chromatic scale. We mean  that if the melodies are accompanied by chords and the chords are important in the sounding of the song, and thus the instrument player has to alternate melody playing with chords playing in the same song, then it is optimal for this. Hard as it is to realize it, it is not the guitar the minimal optimal instrument for this. As most beautiful melodies are 2/3 of the time with notes from the chords and 1/3 with notes outside the chord, the optimal of such -4-string instruments for the shapes of the chords and their inversions gives also optimal for soloing where 2/3 of the notes are within the shapes of chord! The guitar in its standard, as we have already mentioned in post 16, already contains in its 4 higher strings such an instrument, the harmonic tetrachord  but it  has 2 more strings that make things both of more possibilities but also more complications , forcing chord voicing that may be undesirable for a composer, and is thus not the minimal optimal such instrument! Anyone who knows how to play the guitar knows also how to play the harmonic tetrachord, but the converse does not hold! The harmonic tetrachords are 100% optimal for chords but less than 30% optimal for solos, as the tuning is not uniform. If one of them would be kept entirely for solos, then, if it is of scale length less than 50 cm, then it must be tuned like a mandolin or violin by pure 5ths (7 semitones), e.g. (G3:G3, D4:D4, A4:A4 ,E5:E5), because the size f the human palm can cover 7 frets of such small fretboards.   But if its scale length is between 50 cm and 77 cm it should be tuned in all pairs of strings by pure 4ths (5 semitones) because of the size of the human palm, as the tuning of the realizations we discuss in this post. It is all about ergonomics of the instrument!

It can be proved that for a 4-string double or single strings,  instrument, the optimal tuning so as to play , all minor. major, diminished , augmented, 7nth and 6th chords, within only 3 frets, the tuning must be from lower string to higher , 1) a perfect 4th (5 semitones) 2)  a major third (4 semitones 3) and again a perfect 4th (5 semitones). This trick to alternate a major 3rd with a perfect 4th, was used in the Medieval  Instruments like Lutes, and in some type of tuning of the  Viola da Gamba, and of course in the tuning of the modern guitar The sound of the chords are enhanced if the 4-strings are double, so we are talking for 8-string instrument. Such an instrument for chords it is better than a 6-string or 12-string guitar, as it does not force the voicing of chords with 6 strings, and undesirable sometimes repetitions of certain notes. The sound of the chords is the clear musically and harmonically intended without unnecessary repetitions, and the playing is easier! When playing the harmonic tetrachords you have the pleasure to realize that you achieve the required harmonic result with a minimal number of courses  (4-note chords with 4-double strings).

Such instruments are very popular e.g. The 4-string double  or single strings Ukulele with tuning  (G4:G3,  C5:C4 , E4:E4, A4:A4) which is already an open  chord the A minor dominant 7nth,  Am7 , or the Tenor 4-double strings guitar with the Chicago tuning (D4:D3, G4:G3, B3:B3, E4:E4) which is the open chord E minor dominant 7nth ,Em7. Such instruments can easily be created in various sizes by modifications from other existing instruments. For example one can convert to such an instrument , the octave mandolin, 4-double string Greek or Irish Bouzouki, Alto mandola, Cretan 4-double string Lute, Greek 4-double string Lute, Constantinople 4-double string Lute, Sazi 3-groups of strings, Tambur 3-groups of strings , Tenor 4-string guitar , Guitar 4-string bass etc. The tuning by pure 5ths, which is standard for the Octave mandolin, Alto mandola, Irish Bouzouki, Greek, Cretan and Constantinople Lute , mandocello etc, is optimal only for the size of the mandolin, or violin, and from the moment it goes to sizes of 50 cm and larger it is not convenient anymore and it should be substituted with tuning based on the pure 4th and alternations of them. The main reason is that the human hand at scale lengths from 50 cm to 77 cm its can cover 5 frets but no more (like 7 that tuning by 5th would require). 

For scale lengths of  34cm which is the violin , the mandolin, and the soprano ukulele, the optimal tuning is the standard by 5ths (7 semitones) eg. (g3:g3, d4:d4, a4:a4, e5:e5)  or (g2:g3, d3:d4, a3:a4, e4:e5) This is so ,because 7 frets can be covered easily by the human palm, 4-frets chords are easy (violin and mandolin chords) and the  tuning is uniform among the strings which is easy for the solos. such tetrachords are 100% optimal for melodies and 70% optimal for chords too. The standard soprano ukulele tuning which is not uniform is 80% optimal for chords and 80% optimal for melodies.

for scale lengths of  43cm-50cm which is the tenor ukullele and the puerto-rican cuatro, the optimal tuning is the standard of cuatro by 4ths (5 semitones)  eg. (d3:d3, g4:g4, c4:c4, f4:f4 )  or (d2:d3, g3:g4, c3:c4, f3:f4 ) . This is so , because 5 frets can be covered easily by the human palm in this size, 4-frets chords are still easy (cuatro  chords) and the  tuning is uniform among the strings which is easy for the solos. Such tetrachords are 100% optimal for melodies and 70% optimal for chords too. The standard tenor ukulele tuning which is not uniform is 80% optimal for chords and 80% optimal for melodies.

For scale lengths of 50cm and larger apply what we write below.

Of course even for the scale lengths of 34cm and 43 cm , it is convenient to keep a tuning which is not the uniform above (all 5ths or all 4ths) but one 4th them major 3rd then 4th again (that we describe mainly below) if someone wants to be able to play these smaller instruments once he knows how to play the guitar. 

It is also known that tuning by fifths (e.g. mandolin or Irish bouzouki) has much more difficult chords that very often use 4 frets, sometimes ...5 which is more difficult the larger the instrument is (for the small mandolin or the violin it is easy because the human hand does not stretch too much) .

While with the tuning of  the 4-strings by forth-major third- forth (e.g. as in the Greek 4-double strings Bouzouki C3F3A3E4, or as in baritone ukulele D3G3B3E4, or in tenor ukulele G3C4E4A4 or as in Colombian cuatro A3,D4,F♯4,B3 , or higher 4 -strings of the guitar  etc) then you have the optimal that all chords major minor and even dominant seventh with all their three inversions  can be played within 3 or 4 frets at most, which is by far easier. That is why this tuning is widespread among the centuries and continents all over the planet (after the discovery of the  harmonic music ).

That is why I call this tuning the best for large or small 4-strings instruments that want to play 70% chords and 30% melodies.

I insist also that for minimal perfect chord voicing (as major and minor chords including 7nths and 6ths are only 4 notes) the 4-strings instruments with the above tuning forth-major third-forth are the best minimal instruments because the do not force unequal and unbalanced chord-voicing as e.g. the 6-string guitar (6 notes for a 3 or 4 notes chords and .....the player usually cannot chose the voicing of the chord by 6 notes because the standard guitar tuning is forcing it its 6-note voicing.).

If one wants to play inversely 70% melodies and 30% chords , and the instrument is  large then the optimal tuning of the 4-strings  is

 forth-forth-forth (e.g.  FCGE, or CGEA, or EARG etc) like the 4-string bass  ) as its symmetric for the hand and you do not have to think at what string you are to change string while playing the melody.

The disadvantage of course is that indeed you can play minor and major chords on three only strings with only 2 frets, but only two of the three inversions not all the three inversions as in the previous mixed tuning.

Exceptions to the above optimal are the very small size instruments like mandolin and violin , that are small enough so that the human hand can cover easily up to 7-frets, and where melodies mainly  are played  (more than 80% melodies , less than 20% chords) therefore the optimal tuning is by fifths which is the next best interval after the octave. But not so for large instruments like ....Irish Bouzouki or...Violoncello (in spite the fact that it is so the usual).

Irish Bouzouki was copied by the 3-double string Greek Bouzouki (by adding one more string)  with tuning fifth-forth D3-A3-D4 , which is ancient tuning that goes back 7 thousand years to Sumerians. At that time though the instrument accompanying the singing voice was mainly with single notes or by two simultaneous sounding notes. The 3 and 4 notes chords had not been discovered yet. And as the best intervals are the octave and fifth the tuning e.g. D3-A3-E4 is the best for one or two notes accompanying.  While for 3 only strings and playing more than 80% melodies less than 20% chords it is good , it is not good if one want to play more than 80% chords and less than 20% melodies. That is why during the 20th century it was changed by very skillful Bouzouki players (like Manolis Chiotis) who wanted to play a lot of chords and initially the were playing the guitar , to the C3F3A3E4.


THIS CONCEPT OF 4-COURSES (DOUBLE STRINGS) INSTRUMENTS TUNED BY
FORTH-MAJOR THIRD-FORTH, CAN BE CONSIDERED AS PARTS OF  THE BAROQUE GUITARS AND LUTES, WHERE ONLY THE MIDDLE 4-COURSES ARE USED FOR REASONS OF MINIMALISM  AND BEST CHORD VOICING

THE SHAPES OF THE CHORDS IN THE HARMONIC TETRACHORD INSTRUMENTS ARE DESCRIBED BY THE DAE SYSTEM (see post 3)

ARPEGGIOS AND DEA SYSTEM OF 4-STRING INSTRUMENTS (SEE POST 67)


For the 4-string (double or simple strings) instruments of post 67, that are most of the ethnic music instruments , the chord shapes theory simplifies to the DEA instead of the CAGE of the 6-string guitar. Similarly the arpeggios of the chords, although are not identical with the shapes of the chords in a 6-string guitar, for the above 4-string instruments , they are identical with the chord shape! Thus knowing the chords means knowing their arpeggios of them too, which gives immediately a way of easy improvisation along a chord progression! 

In this post we shall show also how to convert some of  the above instruments to an harmonic Tetrachord of various sizes.

MEZZO SOPRANO VOCAL RANGE

1) I have a 4-double plastic string tenor ukulele, for small sizes (43 cm scale length) tuning (G4:G3,  C5:C4 , E4:E4, A4:A4) . It has the advantage of finding each note one wants when improvising easier. And practically as it has only 4-pairs of strings (as all other Harmonic tetrachords of course) it is almost impossible to make a mistake , when playing as almost all 4-notes can be a type of not very unusual  extended chord! We must let the fingers dance in the improvisation, which is advised to utilize all 4-pairs of strings, in a harping (finger picking) , strumming or soloing. And even a if an undesirable  dissonance appears, it is softened by the octave distance of the strings in the pairs.  (see e.g. https://www.youtube.com/watch?v=a6sM9zWc9Tg)

The next tuning makes the tenor 8-string ukulele to sound more bass and sweet:
(G4:G2, C5:C3, E3:E4, A3:A4)! Notice that we do not alter the high pitch strings in each pair but we alter the second string to be one octave lower! For the high pitch strings we use the usual strings of the ukulele. For the lower pitch strings we use nylon strings of the guitar. For the A3 of the A3:A4 pair a D3 or G3 guitar string. For the E3 of the pair E3:E4, a A2 guitar string or D3 . For the C3 of the pair C5:C3, we use a E2 or A2 string of the guitar. And for the G2 string of the pair G4:G2, again a E2 string of the guitar. 

Finally other optimal ways of tuning are the harmonic tuning that is alternating intervals of major and minor thirds, that make a 3-notes chord (see also Harmonic tuning of the guitar in post 90 ) Here it would be 


A3A4-C3C4-E4E4-G4G4  


Such tuning (see also post 90 about the harmonic tuning of the guitar ) has the next advantages:

1) the highest density per a number of frets of major or minor triad chords in normal position .
2) They are easier for chords in normal position rather than for inversions(Compared to the standard tuning o 4-double strings ukulele which is easier for chords in  inversions rather than in normal position).

3) The solos are in closer frets to the fingers and we do not need to shift the hand a lot to find the appropriate note.


2) I have converted a TENOR 4-PAIRS OF STRINGS GUITAR AND THE GUITAR-LELE  (58 cm scale length) to tenor 4-double string  Harmonic Tetra-chord or contralto guitar (tuning (D4:D3, G4:G3, B3:B3, E4:E4) , which is my favorite size. Notice that it could be tuned also as  (D3:D3, G3:G3, B3:B3, E4:E4). For the strings I used he standard strings of a 12-string guitar. But it is a bit moe hard to play, as it is of 58 cm only compared to the 66 cm of the classical guitar.so one could use slightly different strings. E.g. for the G3 of the G4:G3 pair instead of the standard 0.61mm string  one could use a 0.56 mm string of the Bouzouki , which in the standard tuning of the 4-string bouzouki would be the F3 string. And for the D3, string of the pair D4:D3,  instead of the standard 0.81mm D3-string of the guitar ,  it could be used the 0.61mm Guitar G3-string, or the 0.71mm Bouzouki C3-string, or the 0.76mm 12-string guitar metal covered E3-string which is put as high secondary to the 6th lowest E2-string. 
This tuning of a tenor guitar, gives a sweet and high frequencies sound, as the lower two strings are "octaved" at an octave higher.It is suitable for small instrument bodies.

 4-DOUBLE STRINGS GUITAR (GUITAR-LELE)
(The range is about the vocal range of a female contralto singer)
An alternative way to tune and play it, is by utilizing nylon strings and tune the 1st string as A. In more detail I use the tuning (G2:G4, C3:C4, E3:E4, A3:A4),  which is appropriate with nylon strings for a tenor guitar which has scale length usually 58 cm. So for the A4 of the 1st pair, I use an E4-guitar nylon string. For the A3 of the 1st string I use an nylon G3-string of the guitar. For the E4 of the 2nd pair, I use a nylon B3-string of the guitar. For the E3 of the 2nd pair, I use a nylon D3-string of the guitar. For the C4 of the 3rd pair, I use again a nylon G3-string of the guitar. For the C3 of the 3rd pair, I use a nylon A2 string of the guitar. For the G4 of the 4th pair, I use a nylon A4-string of the tenor ukulele . And for the G2 of the 4th pair I use a nylon E2-string of the guitar. If we want to have thsi tuning with metal strings (and scale length 58 cm) then one solution is to utilize Saz metal strings (For the instrument of saz see below) 

The above tuning is best done on a baritone ukulele which usually has scale length of 48 cm, thus one can utilize directly guitar nylon strings , without stretching the strings at all above their usual tension. 
 If we want to tune the baritone ukulele in its normal tuning it is
(D4:D3, G4:G3, B3:B3, E4:E4). If we tune in the normal tuning to make it sound more vivid, with normal tension of strings of the guitar or tenor ukulele (the baritone ukulele has usually 2/3 of the normal tension only) we may use for the pair E4:E4, a pair of B3:B3 of the guitar (with the smaller size of the baritone ukulele sound exactly as E4E4), for the pair of  B3:B3, we use a pair of G3:G3 of the guitar strings, for the G3 of the pair G4:G3, we use a normal G3 string of the baritone ukulele, for the G4, of the pair G4:G3, we use a A4 string of a tenor ukulele, which because of the difference of the sizes in the baritone it can be tuned easily to sound G4. For the D3 of the pair D4:D3, we use a normal D3 string of the baritone ukulele, for the D4, of the pair D4:D3, we use a E4 string of a tenor ukulele, which because of the difference of the sizes in the baritone it can be tuned easily to sound D4.



The same if it is used a 51cm -52 cm scale length Puerto-Rican Cuatro where only the higher  4 pairs of strings are used at all. 


The next 3 pictures are from a baritone ukulele 

 


The next pictures are from a Puerto-Rican Cuatro as 4-double strings instrument, with the previous tuning that makes it a guitar-lele.

We may utilize a E4 guitar string for the A4 of the A3:A4. We may utilize a D3  guitar string for the A3 of the A3:A4. We may utilize a  A2 or better for lower tension a D3 guitar string for the E3 of the E3:E4. We may utilize a B3 guitar string for the E4 of the E3:E4. We may utilize a G3  guitar string for the C4 of the C3:C4. We may utilize a  A2 guitar string for the C3 of the C3:C4. We may utilize a D3  guitar string for the G3 of the G3:G2. We may utilize a E2  guitar string for the G2 of the G3:G2. With the above choice of strings, and given that the scale length of this shorter guitar (baritone ukulele and the quatro with 48-52 cm scale length) is about at the A4 note of the e4 string of  a normal guitar, the strings are either exactly at the intended tension given the scale length or one note above or below the normal tension! 

Some advantages of this kind of tuning where in all the pairs  the two strings are one octave apart are the next  
1) it covers larger range of octaves than the tuning of an ordinary (tenor) ukulele! 
2) It also gives the opportunity with sufficient skill in the right hand fingernails, to play only the higher strings of each pair or only the lower strings of each pair moving therefore in two different octaves the same melody or harp-ism. 
3) Furthermore when singing, the instrument supports the voice in two octaves rather than one octave which makes singing more well supported by the instrument thus easier and more comfortable and accurate! 

While a tenor ukulele is about the range of a human mezzo soprano, and a charango-ukulele, about the range of a soprano human voice, the above tuning for a baritone ukulele or a Quatro it is about the range  Contralto human voice. 


2.1) THE 8-STRING GUITAR-LUTE

(OR NORMAL AND BARITONE 8-STRING GUITAR (8-string Baroque guitar) FROM A 12-STRING GUITAR).
I have converted a 12-string guitar to a 8-string guitar by simply , eliminating the 6th and 5th pairs of strings. The reason to use an 8-string guitar instead ofa 12-string guitar, is that a 12-string guitar requires much force in the left hand, and in addition , an 8-string guitar has more perfect voicing of the chords. The elimination of the 2 lower pairs of the 12-string guitar , makes it easier also to play as far a muscular effort of the left hand is concerned.
The inherited tuning from the 12-string guitar is of course, the (D4:D3, G4:G3, B3:B3, E4:E4). The sound is very fine and I believe it sounds better than a standard 4-double string Greek bouzouki with tuning (C4:C3, F4:F3, A3:A3, D4:D4) . I call this instrument a normal 8-string or 4-double strings guitar from a 12-string guitar. It is tuned as a baritone ukulele, but of course due to the larger size of the wooden body, the sound is richer and deeper.  

I have nevertheless a second 12-string guitar which again I have converted to a 8-string guitar, but with different tuning. The tuning I use is the (D3:D2, G2:G3, B2:B3, E3:E4). The reason that I set this tuning and not the (D4:D3, G4:G3, B3:B3, E4:E4) as previously , is that the former has a range at the lower frequencies the same and  actually slightly lower than a usual 6-string guitar. I call thsi instrument a baritone 8-string or 4-double strings guitar from a 12-string guitar.   This range of this guitar is very close to the range of the cello and mandocello !

(A slight variation of this tuning is the (D3:D2, G3:G4, B2:B3, E3:E4). The reason for such a variation is to make it a bit higher in the range and make the chords sound clearer. We may call it the light baritone -8string guitar from  a 12-string guitar. )
The advantages-disadvanges  of this tuning are

1) because the 3 higher in pitch pairs of strings have high notes , like G3, B3, E4, the chords are heard clear

2) Because of the D2 string it has sufficient bass range , in fact larger than the 6-string guitar , and it goes one tone lower than it, So it sounds close to the mandocello.

3) Compared to the tuning (D4:D3, G4:G3, B3:B3, E4:E4), the tuning (D3:D2, G3:G4, B2:B3, E3:E4), makes the instrument sound closer in range to the 6-string guitar, as it has notes in the 2nd octave too! 

4) Compared to the tuning baritone (D3:D2, G3:G2, B2:B3, E3:E4), the light baritone tuning (D3:D2, G3:G4, B2:B3, E3:E4), is softer in the pressures of the hand.

5) Because the two lower in pitch pairs D3:D2, G3:G4, have opposite octaving, that is the G3:G4,  has re-entrance and shift the 2nd string one octave higher, while the first pair D3:D2,  shifts the 2nd string one octave lower, when playing melodies that use both these two strings, the sound of the pitch sounds as if it jumps one octave . This can be improved if we eliminate the G4 in the pair G3:G4 and we turn it to G3:G3, in other words with the tuning (D3:D2, G3:G3, B2:B3, E3:E4). Also because the G4-string in the pair G4:G3 breaks often, we may replace this pair with the G3:G3.   Finally because the total sound is a bit more bass than usual we may try the next tuning (D4:D2, G3:G3, B2:B3, E3:E4)! Notice that the pair D4:D2 has 2-octaves distance! In this way at the cost a bit of  the pitch uniformity, we succeed to have bass notes like D2 which is lower than a normal 6-string guitar, while at the same time high pitch strings like D4, B3, E4, which also exist in a normal 12-string tuned guitar.

So for the E4 of the 1st pair, I use an E4-guitar n string of the 12-string guitar. For the E3 of the 1st string I use an  D3-string of the guitar. For the B3 of the 2nd pair, I use a  B3-string of the guitar. For the B2 of the 2nd pair, I use a  A2-string of the guitar. For the G3 of the 3rd pair, I use again a  G3-string of the guitar. For the G2 of the 3rd pair, I use a A2-string of the guitar. For the D3 of the 4th pair I use a  D3-string of the guitar (or a E3 string of the 12-string guitar)  . For the D2 of the 6th pair, I use a E2 string of the guitar. 

This tuning of a converted 12-string  guitar, gives a deep bass  frequencies sound, as all the  strings are "octaved" at an octave lower.It is suitable for large body instrument bodies. The disadvantage is that the chords are voiced with low frequencies and thus are not so vivid and easy to grasp by the ear.  This is partially corrected by the next alternative tuning based on A3 on the 1str string rather than E4. But the advantage of the current tuning based on E4 is that the 1st string, still has an E4, and the usual mapping of the chords on the 6-string guitar, (after truncation) still are the same on this 8-string tuning. 

This tuning (D3:D2, G2:G3, B2:B3, E3:E4) can be done also in the 12-string guitar with nylon string also. In which case again for the E4 of the 1st pair, I use an E4-guitar n string of the nylon 6-string guitar. For the E3 of the 1st string I use an  D3-string of the guitar. For the B3 of the 2nd pair, I use a  B3-string of the guitar. For the B2 of the 2nd pair, I use a  A2-string of the guitar. For the G3 of the 3rd pair, I use again a  G3-string of the guitar. For the G2 of the 3rd pair, I use a A2-string of the guitar. For the D3 of the 4th pair I use a  D3-string of the guitar (or a E3 string of the 12-string guitar)  . For the D2 of the 6th pair, I use a E2 string of the guitar. 

An alternative way to tune and play it, maybe to tune it is the 8-string tenor guitrar with nylon strings but here with metal strings, or as the tenor mandola is 3) below.

 In more detail we may use the tuning (G3:G2, C3:C4, E3:E4, A3:A3), 

with metal strings of the 12-string guitar. So for the A3 of the 1st pair, we use an G3-guitar metal string.  For the E4 of the 2nd pair, we may use a metal  E4-string of the guitar. For the E3 of the 2nd pair, we may use a metal D3-string or better a G3-string of the guitar. For the C4 of the 3rd pair, we may use a metal B3-string or better a E4-string of the guitar. For the C3 of the 3rd pair, we may use use a metal A3 string or better a D3-string of the guitar. For the G3 of the 4th pair, we may use  a metal G3. And for the G2 of the 4th pair we may use a metal E2-string or better a A2 string of the guitar.
The advantage of this tuning is that the chords are vivid and easier to grasp by the ear compared to the previous tuning based on E4, and also still the bass frequencies and range of a guitar exists in this tuning. The disadvantage is that the 1st pair of strings has to have a re-entrace on A3 rather than A4. Maybe with special E4 strings the pair, A3:A3 could become A3:A4, but with stanrad E4-strings of a guitar this is most often not possible as the E4 string brakes after some re-tunings. 

The tuning (D3:D2, G2:G3, B2:B3, E3:E4) as we said should be considered a Baritone 8-string guitar. The chords are at a lower range and are not so vivid. Therefore for an accompanying guitar may not be recommended this tuning. On the other hand a slightly different way to octave the strings, may result to a higher range, therefore not a baritone 8-string guitar, which is appropriate for the chords and accompanying, and this is the tuning (D4:D3, G4:G3, B2:B3, E3:E4). Notice that the two higher strings B3, E4 are octaved at a lower octave and the two lower strings D3, G3 at a higher octave. The sounding creates the feeling of a slight re-entrance, but the over all sound of the chords is vivid, and the bass are preserved, compared to the tuning
(D4:D3, G4:G3, B4:B3, E4:E4), which has not bass notes and the range is limited. I have converted a 12-string Taylor guitar to an 8-string guitar (because it is played with less muscular effort than the 12-string guitar and had more perfect chord voicing) with the tuning (D4:D3, G4:G3, B2:B3, E3:E4).
I have used a d3-string for the 2nd lower string E3 of the 1st pair E3:E4. I have used also a d3-string for the 2nd lower string B3 of the 1st pair B2:B3. It is better a d3-string for the 2nd lower string B3 of the 1st pair B2:B3 , than a a3-string as the diameter of the string is less, thus less difference from the diameter of the b3-string, and also the tension lower which makes it softer to play. The strings for the  pairs D4:D3, G4:G3, are exactly as in a 12-string guitar. 

After some time, I changed the tuning to the (D4:D2, G4:G2, B2:B3, E3:E4). The sound is very smooth and beautiful! Notice that the lower pitch string in all pairs are a continuity without re-entrance while only the higher pitch strings make re-entrance. The lower pitch string range in the octaves 2, 3 , while the higher pitch strings in the pairs in the octave 4. In this way this 4-double strings guitar has wider range than the 6-string guitar, (in the open strings) , while its is softer in muscular effort and chord shapes to play it compared to a 12-string guitar! It is a bit more muscular effort though than a 6-string guitar, but is has also richer sound! From all the tuning of the 4 double strings guitar  I prefer this one, the (D4:D2, G4:G2, B2:B3, E3:E4)! This is  my favorite tuning for the 4-double string LUTE-guitar or Cello-guitar

Some advantages of this kind of tuning where in all the pairs , the two strings are one octave apart are the next  
1) it covers larger range of octaves than the tuning of an ordinary (tenor) ukulele! 
2) It also gives the opportunity with sufficient skill in the right hand fingernails, to play only the higher strings of each pair or only the lower strings of each pair moving therefore in two different octaves the same melody or harp-ism. 
3) Furthermore when singing, the instrument supports the voice in two octaves rather than one octave which makes singing more well supported by the instrument thus easier and more comfortable and accurate! 

Still an alternative tuning for such a body of a 12-string guitar is like the Greek 4-double string Bouzouki, in other words (C4:C3, F4:F3, A3:A3, D4:D4) . or baritone version (C4:C3, F4:F3, A2:A3, D3:D4). This tuning still does not go as low in pitch as  a cello and mandocello! But the corresponing shift of the tuning (D3:D2, G3:G4, B2:B3, E3:E4), one tone lower , in other words the (C3:C2, F3:F4, A2:A3, D3:D4), goes exactly as deep in pitch as a cello and mandocello! 


Notice also that the lower octaving at the higher strings and higher octaving at the lower strings is also how the saz instruments (see below) is octaved. 

We may compare the previous tuning with the Greek lute tuning. Notice that the range of the 8-string guitar tuning (D4:D3, G4:G3, B2:B3, E3:E4) is just one tone higher than the Greek island range of the Lute.
The Greek Islands lute tuning which has re-entrance at the lower pair and it  is 

(C3:C4, G2:G3, D3:D4, A3:A3)

and the Cretan Lute which again has re-entrance at the lower pair and it is 


(G3:G3, D3:D3, A3:A3, E3:E3)

I have also a 12-string guitar but with nylon strings! It is softer to play, and the slight out of tune cases of a 12-string guitar , are not highlighted as with the metal string 12-string guitar. I used a e4-string for the 2nd higher g4 of the g3-string, again an e4-string for the 2nd higher d4 of the d3-string, an  g3-string for the 2nd higher a3 of the a2-string, and an d3-string for the 2nd higher e3 of the e2-string. The nylon strings have larger amplitudes of the metal strings, so the necessary 12-string guitar body, must have the strings of pair of strings not too close. I also raised the right bridge to be sure that the nylon strings do not hit the fretboard at strong strumming. I used at first to test that is feasible to have a 12-string guitar with nylon strings, a J&D 12-string guitar, because it had larger distances among the strings in each pair. 

This nylon strings Greek bouzouki below  is good for solos or combination of chords with bridge-solos, because the one notes sounds concentrated in time , and louder compared e.g. to a 8-nylon string guitar derived from a 12 nylon string guitar.

But when one only needs to play the chords with strumming the a 8-nylon string guitar derived from a 12 nylon string guitar is better as the sounds of the notes last longer.

2.1.1) The GUITAR-CELLO (or Guitar-Oud) :

I have converted an ordinary 6-string guitar to a 4-string guitar-cello with tuning

(C2, F2, A2, D3) which is in the lower range of a cello. It may be also considered a type of Guitar-oud. 


THE PANDURI-TAMOURAS-BOUZOUKI GUITAR TUNING:

This is a very efficient tuning of a 6-string guitar,which combines an open D major (or minor ) panduri on the lower 3 strings inother words D2-F2#-A2 (or D2-F2-A2)  , a 3-string tampoura on the 4th,3rd-2nd  strings in other words A2-D3-A3, and a 3-string bouzouki on the 1st 2nd and 3rd string in other words D3-A3-D3 .Inover all the 6-string tuning is 

D2-F2#-A2-D3-A3-D4  (or D2-F2-A2-D3-A3-D4 ) which is also an open D major tuning (or open D minor tuning). 

The chords are played by triads on the 3 lower strings only (as in Zither where the strings for accompanying chords are separte from the strings for solo) , and are easy as the need only 1 or 2 frets. The 3 or 4 higher strings are used for the isocratic melody playing wich is accompanyind by an intervalof 5,as it is usual in 3-string Bouzouki and tambouras



If the guitar is child's guitar of scale length 55-58.5 cm then the previous tuning can be raised to the (with standard guitar strings) 

G2-B2-D3-G3-D4-G4  (or G2-Bb2-D3-G3-D4-G4 ) which is also an open G major tuning (or open G minor tuning) which is very sweet in listening. 


THE 4-COURSES AND 3-COURSES BOUZOUKI GUITAR TUNING 

This is a tuning of the guitar where the lower 4 strings are as in a 4-courses Greek Bouzouki in other words C2-F2-A2-D3 and the higher 3 as in a 3-courses bouzouki in other words D3-A3-D4 . In total   C2-F2-A2-D3-D3-A3-D4. It is an open Dm7 tuning

Similarly for  If the guitar is child's guitar of scale length 55-58.5 cm then the previous tuning can be raised to the (with standard guitar strings) 

F2-Bb2-D3-G3-D4-G4 (or F2-B2-D3-G3-D4-G4 It is an open Gm7 tuning) 

THE  BOUZOUKI-TAMPOYRAS-BOUZOUKI TUNING OF THE 6-STRING GUITAR

It is a tuning applied in to chidrens 53 cm scale length guitar where the lower 3-strings is a 3-courses bouzouki D2-A2-D3 bouzouki, the hiher 3 are a 3-courses A3-E4-A4 bouzouki and the 2nd 3rd and 4th strings are tampouras tuned in D3-A3-E4 . In total

D2-A2-D3-A3-E4-A4

Of course there are lower versions of it on an ordinary 66 cm scale lenghth guitar

THE MABDOCELLO-BOUZOUKI TUNING OF THE OUD

This tuning of the oud utlizes only the higher 5-courses and it is 

C2C2-G2G2-D3D3-A3A3-D3D4

The lower 4-courses C2C2-G2G2-D3D3-A3A3 is mandocello while the  higher 3-courses is a 3-courses bouzouki.


THE PANDURI-BOUZOUKI TUNING OF THE OUD

This tuning of the oud utlizes only the higher 5-courses and it is 

B1B1-G2G2-D3D3-A3A3-D3D4


The lower 3-courses B1B1-G2G2-D3D3 is an open G major pandiuri tuning (inverted form of the chord) while the  higher 3-courses is a 3-courses bouzouki.

SOPRANO VOCAL RANGE

2.2) THE MANDOLELE WITH METAL STRINGS I have converted  also a mandolin , and tune it to (D5:D4, G4:G4, B4:B4, E5:E5),  so as to have the previous tenor guitar one octave higher! Normally the mandolin is tuned like (G3:G3, D4:D4, A4:A4, E5:E5). One may use for the B4:B4 strings the A4:A4 strings of the mandolin, for the 3rd pair G4:G4, again the 2nd pair A4:A4 of the mandolin and for the 4th pair D5:D4, the D4 of 3rd pair of the original strings of mandolin, where for the D5 is used the old E5.  Still another alternative tuning would be the (D3:D4, G3:G4, B3:B4, E4:E5), where for the E4 and B3 we may use the old D4 of the mandolin, for the G3 and D3 the old G3 of the mandoline. The same use of strings would cover also the tuning (D4:D4, G4:G4, B3:B4, E4:E5). The previous choices of strings can cover also the tuning (D4:D4, G4:G4, B4:B4, E5:E5). This instr

2.3) THE MANDO-LELE OR GUITARLINO WITH NYLON STRINGS I have also a charango-ukulele or charango-mandolin (GUITARLINO, TAROPATCH ) of 37.5 cm scale length (same as he concert ukulele) , and with nylon strings, with the same tuning   (D5:D4, G4:G4, B4:B4, E5:E5). in other words with the tuning of  double string soprano ukulele 

Notice that the normal tuning of a charango is (G4 G4 - C5 C5 - E5 E4 - A4 A4 -E5 E5)
Therefore the range of the suggested tuning (D5:D4, G4:G4, B4:B4, E5:E5) has as lowest the D4, which is lower than the lowest of the normal charango tuning the E4. 

For the strings I used strings from the charango set of strings. I used the charango high 1st pair E5:E5, for the high 1st pair of the charango-mandolin. I used the pair C5:C5 of the charango strings for the pair B4:B4 of the charango-mandolin. I used the pair A4:A4 of the charango or the pair G4:G4 for the pair G4:G4 of the new charango-mandolin. And I used the re-entrance pair E5:E4 for the re-entrance pair D5:D4 of the charango mandolin. In order to have the range of the classical tenor ukulele  or of the mandolin one may resort to the next octaved tuning:  (D5:D4, G3:G4, B4:B4, E5:E5). In which case only the G3 string of the 3rd pair G3:G4 is different from the original tuning (D5:D4, G4:G4, B4:B4, E5:E5). We may use a G3 string of a ukulele.

The fretboard of this charango-mandolin is of larger width than the width of the fretboard of a mandolin, which makes it easier to play chords. The sound of the nylon strings somehow in the chords is better too and hides better the slight out of tuning that may occur.
In order to have even lower the range one may resort to the next octaved tunings (D3:D4, G3:G4, B4:B4, E5:E5). or even lower to the (D3:D4, G3:G4, B3:B4, E4:E5). For the E4 of the E4:E5 one may utilize an guitar D3 string. For the B3 of the B3:B4 one may utilize a guitar A2-string. For the G3 of the G3:G4 one may utilize a guitar E2-string. For the D3 of the D3:D4 one may utilize a guitar E2-string.

The body can be also of a concert-size ukulele with 8-strings or TAROPATCH, but instead of tuning this taropatch as a ukulele we tune as above at the range of mandolin and charango, in other words as a tuning of the sopranino ukulele but with double strings.

CONTRALTO VOCAL RANGE

3) THE MANDOLA-LELE 
(OR TENOR MANDOLA OR OCTAVE MANDOLIN (CHICAGO TUNING)) The same size (58 cm scale length in other words exactly as long as the scale length of a tenor guitar ) is  also an Octave mandolin which can be tuned in a different way (G3:G2,  C3:C4 , E3:E4, A3:A4).  As alternative tuning is the  (D3:D4, G4:G3, B3:B3, E4:E4) (alternative tuning is (C3:C4, F4:F3, A3:A3, D4:D4) which an open chord, D minor dominant 7nth, Dm7.  If there would exist thinner strings at this length it could also be tuned (D3:D4, G4:G3, B3:B4, E4:E5). or (C3:C4, F4:F3, A3:A4, D4:D5). But from all these alternatives the tuning (G3:G2,  C3:C4 , E3:E4, A3:A4) seems better, because is has all pairs of strings with one octave difference which obviously makes he sound more rich compared say to a tenor ukulele. At the same time one can use existing strings e.g. from a 12-string guitar as it ranges approximately as a  guitar if all 8 strings are taken in to account. The standard tuning of an octave mandolin (in Europe also called tenor mandola) is the (G2:G2 D3:D3 A3:A3 ,E4:E4). Notice that at the lower pair of strings G3:G2, the higher note string is put first and then he lower. The reason is that I play it mainly with the finger nails. The three finger nails struck usually from down to up, so the convenient setting of the strings in the first 3 strings is down the higher up the lower.But the 4th lower pair is struck usually with the thump nail, and it is usually from up to down, that is inversely. 

For the A3 of the pair A3:A4, and for the E3 of the pair E3:E4 I used an 0.56 mm metal covered string of Bouzouki, which in its standard tuning would be the F3 string. For the C3 of the pair C3:C4,  I used an 0.71 mm metal covered string of Bouzouki, which in its standard tuning would be the C3 string. For the G3 of the pair G3:G2,  I used an 0.76 mm metal covered string of Guitar, which in its standard tuning would be the G3 string. For the G2 of the pair G3:G2,  I used an 1.07 mm metal covered string of Guitar, which in its standard tuning would be the A2 string. 

Normally the tenor mandola would take strings of diameter 0.30mm, 0.40mm, 0.65mm, 0.90 mm I used some of the strings of tenor mandola (or octave mandolin) especially the E4 of diameters 0.3 mm.

I have tried to modify the above (G3:G2,  C3:C4 , E3:E4, A3:A4) tuning to the more deep and smooth (G2:G4,  C5:C3 , E3:E4, A3:A4), still I have not find a an appropriate metal string for 58cm scale length that wold give the C5. Nevertheless the tuning (G2:G4,  C4:C3 , E3:E4, A3:A4)  is entirely feasible. 

Obviously a standard ukulele tuning would be possible too, which is the 

(G3:G4,  C4:C4 , E4:E4, A4:A4) 


For such a tuning, we may use for the A4, the  G4 strings of a 12-string guitar  or the D4 strings (highest) of Saz instrument (see below end of post), for E4 , the  E4 strings of a 12-string guitar, for C4 , the B4 strings of a 12-string guitar, for G4, a E4 or G4 string of a 12-string guitar, and for the G3, a G3 string or an A3 string of a 12-string guitar. 

Alternatively so as to play many folk Greek or Byzantine songs that are written in G minor (either natural or harmonic or double harmonic) a tuning at Bb major is required which is the 


(F3:F4,  Bb3:Bb4 , D4:D4, G4:G4) .



We could call his instrument with this tuning, Bouzoukolino, or Bouzoukulele as it is a 5th higher than an ordinary Bouzouki, although as range it starts from the lower of the Bouzouki (Greek) but goes higher than this. Or it could be called Mandolalele  as it is in the body of an octave mandolin but with tuning as a ukulele. 

We may compare the range of  this tuning with the range of the Portuguese guitar Lisbon tuning , and realize that it is larger with less pairs of strings, and still it goes (except one tone) as high as that of the Portuguese guitar! (D3D4, A3A4, B3B4, E4E4, A4A4, B4B4).

Some advantages of this kind of tuning where in all the pairs  the two strings are one octave apart, are the next  
1) it covers larger range of octaves than the tuning of an ordinary (tenor) ukulele! 
2) It also gives the opportunity with sufficient skill in the right hand fingernails, to play only the higher strings of each pair or only the lower strings of each pair moving therefore in two different octaves the same melody or harp-ism. 
3) Furthermore when singing, the instrument supports the voice in two octaves rather than one octave which makes singing more well supported by the instrument thus easier and more comfortable and accurate! 

  Notice that for  the Greek 4-string Bouzouki, the standard tuning is (C4:C3, F4:F3, A3:A3, D4:D4) )  .To get the full advantage of this tuning you  have to play the instrument with the finger nails rather than a plastic pick! After all the 4 finger nails are like 4 picks and the hand and fingers has the skill to combine playing all these 4 picks with various way! Now that instruments a are electro-acoustic, the goal of loud sound is not important so the main advantage of the plastic pick to give loud sound is not significant.  

4) BARITONE 4-PAIRS OF STRINGS BOYZOUKI. I have converted also a Greek 4-double string bouzouki (66 cm scale length, it can very well be done with an Irish Bouzouki too) to an alto such Harmonic Tetra-chord with tuning  (C3:C2, F3:F2, A2:A3, D3:D4) or to have the chords sounding better the tuning (C3:C3, F3:F3, A2:A3, D3:D4).   (alternatively (D3:D2, G3:G2, B2:B3, E3:E4)) which has a better deeper and more appropriate sound for the body of such a Bouzouki instrument sounding like a mandocello, (one octave lower than the standard 4-string Bouzouki  (C4:C3, F4:F3, A3:A3, D4:D4) ).

Notice that we could have also a half-baritone bouzouki with tuning (C3:C4, F3:F4, A2:A3, D3:D4) or a re-entrance tuning (C3:C4, F2:F3, A2:A3, D3:D4). When playing with the finger nails, it is more convenient to have the lower string up and the higher string down in the 3 higher pairs of strings and inverselly for the 4th lower pair of strings that the tump-nail is used (thus  (C4:C3, F3:F4, A2:A3, D3:D4) rather than (C4:C3, F4:F3, A3:A2, D4:D3)). This is so because the natural move of the finger nail is from lower to up, and in order to be sure that the high string is played, it must be put in the lower position in the pair.

For  the D3 of the pair D3:D4 I used an 0.56 mm metal covered string of Bouzouki, which in its standard tuning would be the F3 string. For  the A3 of the pair A3:A4 I used an 0.71 mm metal covered string of Bouzouki, which in its standard tuning would be the C3 string. 

A more bright sound for the chords and solos of the Baritone Bouzouki is achieved if we apply re-entrance tuning as the ukulele does! In other words we may use the next tuning (C4:C3, F2:F3, A2:A3, D3:D4). This tuning is like the baritone-3-pairs of strings Bouzouki below in the 3-pairs of strings (F3:F2, A2:A3, D3:D4), but at the lower pair instead of the (C3:C2), that would be the standard, and it would make the chords not sounding well we substitute the pair (C3:C2) with the pair  (C4:C3). In other words the pair is still one octave apart, but also one octave higher than the expected! I consider the tuning (C4:C3, F2:F3, A2:A3, D3:D4) the best for a baritone 4-pairs of strings Bozouki, as 1) it has the lower sound and the low F2 , which is the lowest F of the guitar, but 2) at the same time with re-entrance at the lowest pair (C4:C3) which is an octave higher and remains in the octaves 3, 4, the chords still sound clear and bright. (as we said two or more notes ofthe chord at the octave 2, makes the chord sound blur and not bright) and finally 3) the 4-pairs of strings and 3-pairs of strings baritone Bouzoukis coincide in tuning in the high 3 pairs.

For the F2 of the pair F2:F3,  in this tuning I used an 1.07 mm metal covered string of Guitar, which in the guitar standard tuning would be the A2 string. The other strings in this tuning of the baritone Bouzouki, are described before, or are the standard of the Bouzouki strings and tuning. 


THE CELLO-BOUZOUKO Recently I have changed the tuning to the more sweet with one octave difference to the the more sweet, with one octave difference to the two higher pairs and 2-octaves difference to the lower pairs. In other words to the tuning

(C2:C4, F2:F4, A2:A3, D3:D4) 

 Again similar things can be said as previously. This instrument sound much like a bass-guitar or Mandocello, or Greek Lute. In all pairs of strings the strings differ by an octave! To get the full advantage of this tuning you  have to play the instrument with the finger nails rather than a plastic pick! After all the 4 finger nails are like 4 picks and the hand and fingers has the skill to combine playing all these 4 picks with various way!   An advantage of the tuning by octaves in all pairs of strings, is that of one string from a pair goes slightly out of tuning, the audio effect is not that bad as when both strings of the pair have identical frequencies. This tuning allows for larger range of the instrument, that covers the standard 4-double string  Greek Bouzouki  but also that of the guitar in lower notes.  Another advantage is that playing it with the finger nails one can play a single only string from each pair instead of both, thus allowing for more degrees of freedom and sound effects in melodies.  
We summarize

Some advantages of this kind of tuning where in all the pairs  the two strings are one octave apart are the next  
1) it covers larger range of octaves than the tuning of an ordinary (tenor) ukulele! 
2) It also gives the opportunity with sufficient skill in the right hand fingernails, to play only the higher strings of each pair or only the lower strings of each pair moving therefore in two different octaves the same melody or harp-ism. 

3) Furthermore when singing, the instrument supports the voice in two octaves rather than one octave which makes singing more well supported by the instrument thus easier and more comfortable and accurate! 

We may compare the previous tuning with the Greek lute tuning.
The Greek Islands lute tuning which has re-entrance t the lower pair and it  is 

(C3:C4, G2:G3, D3:D4, A3:A3)

and the Cretan Lute which again has re-entrance at the lower pair and it is 


(G3:G3, D3:D3, A3:A3, E3:E3)







3.1) I have also converted an ordinary Greek 4-double strings Bouzouki so as to admit nylon strings and sound like the old centuries bouzouki when there where not metal strings. It is also easier and better  played with the finger nails rather than a plastic pick. 

 Notice that for  the Greek 4-string Bouzouki, the standard tuning is (C4:C3, F4:F3, A3:A3, D4:D4) . This can be considered the 4 middle double strings of a 6 courses  Baroque Lute!  But for the Bouzouki with nylon strings we may shift the tuning one note higher as in the 12-string guitar to become
(D4:D3, G4:G3, B3:B3, E4:E4) as it sound brighter but still more sweet like a lute compared to the standard metal-strings bouzouki. This can also be considered the 4 middle double strings of a 6-courses Baroque Guitar! In this way we may play this bouzouki at the 4th high neighborhood with a capo at 12 or no capo (see post 13) , in which case it sounds as the 4-double strings  charango or taropatch ukulele or guitarlino, described previously!
If we shift to this tuning then usually the G4 string of the pair G4:G3 will often break. So we substitute it with G3 string making thus the final tuning to (D4:D3, G3:G3, B3:B3, E4:E4).
This nylon strings Greek bouzouki is good for solos or combination of chords with bridge-solos, because the one notes sounds concentrated in time , and louder compared e.g. to a 8-nylon string guitar derived from a 12 nylon string guitar.
But when one only needs to play the chords with strumming the a 8-nylon string guitar derived from a 12 nylon string guitar is better as the sounds of the notes last longer.

For Irish Guitar-Bouzouki with naylon strings see also the examples

https://www.youtube.com/watch?v=NMpCoNPZy-g



and

http://www.fyldeguitars.com/custom_guitars/guitar-bouzouki.html




 THE CELLO-BOUZOUKO-GUITAR WITH NYLON STRINGS.
Recently I have changed the tuning to a more deep with 2-octaves difference in the lowest pair and one octave to the other pairs. In other words in to the tuning

(D2:D4, G2:G3, B2:B3, E3:E4)


Finally other optimal tuning are the harmonic tuning that is alternating intervals of major and minor thirds, that make a 3-notes chord (see also Harmonic tuning of the guitar in post 90 ) Here it would be 


E3E4-G3G4-B3B3-D4D4  or F3F4-A3A4-C4C4-E4E4 or 

F3F4-A3A4-C3C4-E3E4 or G3G4-B3B4-D4D4-F4F4

G3G4-Bb3Bb4-D3D4-F3F4
 etc  

Such tuning (see also post 90 about the harmonic tuning of the guitar ) has the next advantages:

1) the highest density per a number of frets of major or minor triad chords in normal position .
2) They are easier for chords in normal position rather than for inversions(Compared to the standard tuning o 4-double strings bouzouki which is easier for chords in  inversions rather than in normal position).
3) The solos are in closer frets to the fingers and we do not need to shift the hand a lot to find the appropriate note.

BARITONE 3-PAIRS OF STRINGS BOYZOUKI OR CELLO-BOUZOUKO. 

This is a 3-pairs of strings bouzouki which I have tuned it to

 ( F2:F3 , A2:A3,  D3:D4 ) ! 
This tuning is optimal for chords as it follows the DAE system (see post 3) and has all 3 inversions of major and minor chords possible to play in only 3 frets. 


I used for the F3 , A2:A3,  D3:D4 the strings of the 4-double string bouzouki, while for F2, the A2-string of a guitar.

For the D3 of the pair D3:D4,  I used an 0.56 mm metal covered string of Bouzouki, which in its standard tuning would be the F3 string. For the A2 of the pair A2:A3,  I used an 0.71 mm metal covered string of Bouzouki, which in its standard tuning would be the C3 string. For the F2 of the pair F2:F3,  I used an 1.07 mm metal covered string of Guitar, which in its standard tuning would be the A2 string. For the F3 of the pair F2:F3,  I used an 0.71 mm metal covered string of Bouzouki, which in its standard tuning would be also the F3 string. 

The advantage is that the two high pairs of strings are not repeating but are an octave part, which makes solos more sweet on them, while the 3rd lower string F2:F3 makes chords still sounding well, and again more sweet. This 3-pairs of strings Bouzouki, ranges almost as a guitar when all 6 strings are taken in to account!

Also another important advantage of this tuning of the 3-pairs of strings Bouzouki, is that all the three inversions of a major or minor chord exist as triads (3-notes chords) as the shapes of the guitar chords D, A, E, inherited on the 3-pairs of strings only. (see DAE system post 3) So for any solo that one plays, he can quickly try a major or minor chord in one of the above 3-inversions or shapes that contains the particular note of the melody (even as root of the chord shape). The simplicity of an instrument when it is adequate harmonically and melodically ,sometimes can contribute as  major factor for easy and fruitful improvisation.

We summarize

Some advantages of this kind of tuning where in all the pairs  the two strings are one octave apart are the next  
1) it covers larger range of octaves than the tuning of an ordinary (tenor) ukulele! 
2) It also gives the opportunity with sufficient skill in the right hand fingernails, to play only the higher strings of each pair or only the lower strings of each pair moving therefore in two different octaves the same melody or harp-ism. 

3) Furthermore when singing, the instrument supports the voice in two octaves rather than one octave which makes singing more well supported by the instrument thus easier and more comfortable and accurate! 

PANDURI TUNING FOR BOUZOUKI (IN THE RANGE OF TAMBOURAS AND BOULGARI)

This tuning is the open D minor or open D major 

D2D4-F3F3-A3A3  or D2D4-F#3F#3-A3A3


It has 3 inversions triads for each major o minor chord but the normal form is the simplest. 

I used  4-course bouzouki A3 strings for the pair A3 and 4-course bouzouki F3 strings for the pair F3F3. I used a guitar E2 string for D2 and a bouzouki string D4 for the D4 i the pair D2D4.


OPTIMAL TUNING FOR POWER CHORDS (see post 35) 
The standard tuning D3D4-A4A4-D4D4 is optimal of course for power chords.But if we want a lower pitch Bouzouki for accompanying with power chords the next 3 tunings seem better

D2D3-A3A3-D3D4   (which is a pattern of octaves taken from Sazi and tambouras and somehow is in the range of them too ) 

or

D2D4-A3A3-D3D3 which is sweet but it has rentrance on A3A3 or

D2D4-A2A3-D3D3 which is sweet and  it  does not have rentrance on A2A3 

(For the latter I used a E2 guitar string for D2, a D4 bouzouki string for D4 in D2D4 a C2 string (from 4-courses bouzouki) for the A2 of A2A3, a A3 string of bouzouki for the A3 in A2A3 and two C2 strings (of 4-courses bouzouki) for the D3D3 or of course two D3 strings of a 3-courses bouzouki)

The tuning D2D4-A2A3-D3D3  deserves the name Cello-Bouzouko

BUT if we want emphasis on solos rather than chords then for the 3-double strings Buzuki the next alternative tuning is optimal 
( E2:E3 , A2:A3,  D3:D4 )
This tuning is optimal not for chords as it follows the AE system (see post 26) and has only  2 inversions of major and minor chords possible to play in only 3 frets (compared to the DAE system in post 3), but it is optimal for solos as it is uniform tuning among the strings!

The reasons of being solo optimal  are the next 
1) The uniformity of tuning of the 1st to 2nd and 2nd to 3rd string as distance by an interval of 4th (5 semitones) it helps much in solos, where the fingers are moving the same among the strings (As the bass is also tuned or Puerto Rican cuatro)
2) There are 4 simple chord shapes in the EA system (see post 26 ). The EA system is perfect when there are only 3-strings! Thsi gives easier chords than the tuning D3-A3-D4, although less inversions than the tuning F3-A3-D4
3) The range is a bit larger than the F3-A3-D4 tuning.

For these reasons, for solos among many scales (and not only for D-major) the tuning by 4ths that is E3-A3-D4, is better than the standard D3-A3-D4. 

Finally other optimal tuning are the harmonic tuning that is alternating intervals of major and minor thirds, that make a 3-notes chord (see also Harmonic tuning of the guitar in post 90 ) Here it would be 

G3G4-B3B3-D4D4 or G3G4-B3B4-D3D4 or G3G4-Bb3Bb3-D4D4 or 
G3G4-Bb3Bb4-D3D4 or A3A4-C4C4-E4E4 or A3A4-C#4C#4-E4E4  or  
 F3F4-A3A4-C3C4 or F3F4-A4A4-C4C4 

etc

Some Russian bass or alto Balalaikas have such tuning inherited from Russian (gypsy) guitar open tuning

Such tuning (see also post 90 about the harmonic tuning of the guitar ) has the next advantages:

1) the highest density per a number of frets of major or minor triad chords in normal position .
2) They are easier for chords in normal position rather than for inversions(Compared to the standard tuning o 3-double strings bouzouki which is easier for chords in  inversions rather than in normal position).
3) The solos are in closer frets to the fingers and we do not need to shift the hand a lot to find the appropriate note.

5) And I have converted a saz (https://en.wikipedia.org/wiki/Ba%C4%9Flama 79 cm scale length)  to an Harmonic Threechord (with tuning   G3:G4, B3:B3, E3:E4:E4  or F3:F4, A3:A3, D3:D4:D4.

Notice that the highest string E4 is octaved one octave lower by E3, while the lowest string G3  is octaved by one octave higher.  This instrument  has a very fine and sweet sound and can be played with the most swell and easiest way compared to the other conversions, because of 

1) the narrow width of the neck 

2)  very low height of the (movable) frets 

3) very low tension of the strings 

4) very small diameter of the strings of such instruments. 

I consider this instrument a divine sound and easiness  when playing it!

Of course I have set the movable frets only to the 12-tone equal temperament  Bach scale and have get rid of the other additional movable frets. 

We notice that the rule ofthe octave strings in the pairs is that of teh half-baritone. In other words in the high triple string E3:E4:E4, the octave string E3 is lower, while in the lowests pair G3:G4, the octave string is the G4 which is higher. The main notes are the (G3, B3, E4).

Again an important advantage of this tuning of the 3-groups of strings Saz is that all the three inversions of a major or minor chord exist as triads (3-notes chords) as the shapes of the guitar chords D, A, E, inherited on the 3-pairs of strings only. (see DAE system post 3). So for any solo that one plays, he can quickly try a major or minor chord in one of the above 3-inversions or shapes that contains the particular note of the melody (even as root of the chord shape). The simplicity of an instrument when it is adequate harmonically and melodically ,sometimes can contribute as  major factor for easy and fruitful improvisation.

Finally other optimal tuning are the harmonic tuning that is alternating intervals of major and minor thirds, that make a 3-notes chord (see also Harmonic tuning of the guitar in post 90 ) Here it would be 

G3G4-B3B3-D4D4D4 or G3G4-Bb3Bb3-D4D4D4 or A3A4-C4C4-E4E4E4 or 

A3A4-C#4C#4-E4E4E4 etc  Such tuning has the highest density per a number of frets of major or minor triad chords.


Some Russian bass or alto Balalaikas have such tuning inherited from Russian (gypsy) guitar open tuning

PANDURI TUNING FOR SAZI (IN THE RANGE OF TAMBOURAS AND BOULGARI)

This tuning is the open G major chord

G2G3-B3B3-D3D4D4  

Because the D3D4D4 course of strings has two D4 it sounds like D4 finally rather than like D3! So the final sound is like G2-B3-D4.

It has 3 inversions triads for each major o minor chord but the normal form is the simplest. 


Such tuning (see also post 90 about the harmonic tuning of the guitar ) has the next advantages:

1) the highest density per a number of frets of major or minor triad chords in normal position .
2) They are easier for chords in normal position rather than for inversions(Compared to the standard tuning o 3-double strings bouzouki which is easier for chords in  inversions rather than in normal position).

3) The solos are in closer frets to the fingers and we do not need to shift the hand a lot to find the appropriate note.

G3G4-B3B3-D4D4D4 or G3G4-Bb3Bb3-D4D4D4 or A3A4-C4C4-E4E4E4 or A3A4-C#4C#4-E4E4E4 etc  Such tuning has the highest density per a number of frets of major or minor triad chords.

We give the  calculations of the right frets positions fora  saz  which has variable frets,of scale length 79 cm. We calculate the distance of each fret from the upper or lower end of the string (scale) , so that approximation errors are not added when realizing these positions. In these calculations the angle of the string relative to the surface ofthe fretboard is not included. If included the calculations might have very very small modifications. (For the formula for the  calculations see post 1) 

FRETBOARD DIMENSIONS	79	SCALE LENGTH IN CM
Number of fret	Lemgth from lower string end till fret	Multipler	Length from upper string end till fret
1	74.5661,	0.056125687	4.4339
2	70.3810	0.109101282	8.6190
3	66.4308	0.159103585	12.5692
4	62.7023	0.206299474	16.2977
5	59.1831	0.250846462	19.8169
6	55.8614	0.292893219	23.1386
7	52.7262	0.332580073	26.2738
8	49.7669	0.370039475	29.2331
9	46.9737	0.405396442	32.0263
10	44.3373	0.438768976	34.6627
11	41.8488	0.470268453	37.1512
12	39.5000	0.5	39.5000
13	37.2830	0.528062844	41.7170
14	35.1905	0.554550641	43.8095
15	33.2154	0.579551792	45.7846
16	31.3512	0.603149737	47.6488
17	29.5916	0.625423231	49.4084
18	27.9307	0.646446609	51.0693
19	26.3631	0.666290036	52.6369
20	24.8834	0.685019738	54.1166
21	23.4868	0.702698221	55.5132
22	22.1686	0.719384488	56.8314
23	20.9244	0.735134226	58.0756
24	19.7500	0.75	59.2500

6) But such a saz can also be converted easily by adding one more string , to an  harmonic Tetrachord with tuning (D4:D3, G4:G3, B3:B3, E4:E4)

See also about the 5-courses (but essentially 4-course, and much like the Ukulele ) instrument Charango and Kalampeano of Andes. The charango tuning ((C4:C4, F4:F4, A3:A4, D4:D4, A4:A4) or more often  (G4:G4, C5:C5, E5:E4, A4:A4, E5:E5) is like combining a 4-double string and a 3-double string Greek bouzouki in reverse order, with common the lower 2-double strings of the 3-double string bouzouki (only at smaller size and one octave higher) or like repeating the 2nd double string of a ukulele as highest double string.

The history of charango

https://www.youtube.com/watch?v=_VlXjj8sGOc

https://www.youtube.com/watch?v=sXSVRKK1gvY

https://en.wikipedia.org/wiki/Charango

Music of Andes (Charango)
https://www.youtube.com/watch?v=0xxrjalSkvw

And a......double guitar back-to-back ....
https://www.youtube.com/watch?v=ED2cFK9OEwA

Here is also a string tension calculator!

http://www.mcdonaldstrings.com/stringxxiii.html


ALTHOUGH IT IS NOT A 4-DOUBLE STRINGS INSTRUMENT I MAY MENTION HERE THAT I HAVE TUNES A CLASSICAL 6-STRING GUITAR WITH THE HARMONIC TUNING THAT IS   

An  optimal but unknown tuning for the 6-string guitar when chord-playing is the main target and not so much solo playing is and even better by alternating minor and major 3rds. In semitones for the 6 strings   4-3-4-3-4 or 3-4-3-4-3
E.g. Bb2- D3-F3-A3-C4-E4 or F2-A2-C3-E3-G3-B3 or A2-C3-E4-G4-B4-D4
THIS MAY BE CALLED THE HARMONIC TUNING OF THE GUITAR AS IT IS BASED ON THE HARMONIC 2-CTAVES 7-NOTES SCALE (see also post 79)
The latter is the most natural open tuning. There the same shape for major and minor chords and only 3 of them and in only one or frets compared to the 6 in the standard tuning guitar. If we want also dominant and major 7nth chords we use again only 2 frets. The same with the aug chords Only the dim7 chords require 3 frets. Because  of the symmetry of the tuning among the strings, the relations of relative chords and also chords in the wheel of 4ths is immediate to grasp also geometrically. Of course when we say shape of chords as it is standard in jazz, we do not play all 6-strings but only 3 or 4 strings.

The easiness with which one can improvise melodies within a diatonic scale (all notes within 3  frets and in a very symmetric zig-zag pattern) together with 3-notes chords of the scale (gain all chord patterns within 3-frets) is unsuprassed.
At the same time , the easiness with which one can me diatonic scale modulations, chromatic (1 semitone apart) or by changing a minor to a major chord and vice versa and continuing in a relevant diatonic scale is unsurpassed again! 

This harmonic tuning by alternating minor-major 3rds, allows, for all  4-notes chords of e.g. the D major scale in   the 3rd octave (c3,d3,e3,f3,g3,a3,b3), Cmaj7->Em7->G7->Bm6->Dm7->Fmaj7->Am7 in 1st normal position across the fretboard, something not possible with the standard tuning of the guitar. In the standard guitar it is possible only by 2nd or 3rd inversion, or by shifting to the 4th octave or 2nd octave. Therefore there are important very natural voicing of the 4-notes chords of the 3rd  octave that we miss with the standard tuning and it is in a single octave!

THERE ARE 4 VERY SYMMETRIC WAYS THAT THE CHORDS IN A WHEEL BY 4THS CAN BE REPRESENTED AND PLAYED IN THE FRETBOARD WITH THIS  TUNING. 

I WILL DEVOTE A SPECIAL POST FOR THIS REMARKABLE GUITAR TUNING WITH HIGHLY SYMMETRIC AND EASY TO PLAY HARMONY.

If one thinks that the above ideas as eccentric, then he only needs to watch and listen to the inventions of Linsey Pollak on easy to make instruments.

http://www.linseypollak.com/instruments/

https://www.youtube.com/watch?v=w4eziFH7VYc