§8.6 SUMMARY

Lutes:

There are relatively few pieces for an 11 course lute. There are pieces for a 13 course lute which use course 12, but not course 13. Details:

Music for an 11 course lute:
Dresden MS: 1 suite
London MS: 1 suite, 1 sequence of pieces, 11 single pieces

Music for a 13 course lute, without course 13 being used:
Dresden MS: 3 suites
London MS: 5 suites, 5 single pieces

Music for a 13 course lute, with course 13 being used:
Dresden MS: 30 suites
London MS: 21 suites, 2 sequences of pieces, 18 single pieces

Key signatures:

Mostly the key signatures of the pieces in a suite or sequence of pieces are identical.
There are only 2 suites and 1 sequence of pieces with mixed key signatures. Details:

Music with one key signature in the Dresden MS:
3#: 7 suites
Blank: 7 suites
1b: 8 suites
2b: 6 suites
3b: 4 suites
4b: 1 suite

Music with one key signature in the London MS:
3#: 2 suites
2#: 3 suites, 1 sequence of pieces, 3 single pieces
1#: 2 suites, 3 single pieces
Blank: 2 suites, 8 single pieces
1b: 8 suites, 10 single pieces
2b: 4 suites, 1 sequence of pieces, 6 single pieces
3b: 4 suites, 2 single pieces
4b: 1 suite
5b: 1 single piece
6b: 1 single piece

Music with two key signatures in the Dresden MS:
Blank and 3b: 1 suite

Music with two key signatures in the London MS:
Blank and 3b: 1 suite
1# and 2b: 1 sequence of pieces

Key of a suite

In any suite, there is a predominant key. It is common practice, to call this key the key of the suite. Actually often one of the pieces of a suite is composed in the corresponding minor / major key, and this piece is typically a Sarabande.

Key-conform tuning:

Key-conform tuning is predominant. Details:

Music with key-conform tuning:
Dresden MS: 28 out of 34 suites
London MS: 21 out of 27 suites, 2 out of 3 sequences of pieces, 31 out of 34 single pieces

Standard tuning:

The key signatures, which have standard tuning as natural tuning, are 2b for 11 and 13 string lute and 3b for 11 string lute. In addition, Weiss uses standard tuning as a tuning for all music in the two manuscripts that has a key signature of 3b or 4b. This is a strong argument to define standard tuning for a baroque lute the way it has been done in this paper. Details:

Music with standard tuning:
Dresden MS: 11 out of 34 suites
London MS: 9 out of 27 suites, 1 out of 3 sequences of pieces, 8 out of 34 single pieces

Applied tunings:

In the two manuscripts there are only two tunings which are not a natural tuning of any key, and each of these tunings is only applied in one single piece.

These are the tunings applied in the two manuscripts to an 11 course baroque lute (in O-notation):

Tuning for key signature =3#: (g#G#) (f#F#) (eE) (dD) (c#C#)
Tuning for key signature =2#: (gG) (f#F#) (eE) (dD) (c#C#)
Tuning for key signature =1#: (gG) (f#F#) (eE) (dD) (cC)
Tuning for key signature =1b: (gG) (fF) (eE) (dD) (cC)
Tuning for key signature =2b or 3b: (gG) (fF) (ebEb) (dD) (cC)

These are the tunings applied in the two manuscripts to a 13 course baroque lute (in O-notation):

Tuning for key signature =3#: (g#G#) (f#F#) (eE) (dD) (c#C#) (BB1) (AA1)
Tuning for key signature =2#: (gG) (f#F#) (eE) (dD) (c#C#) (BB1) (AA1)
Tuning for key signature =1#: (gG) (f#F#) (eE) (dD) (cC) (BB1) (AA1)
Tuning for key signature = : (gG) (fF) (eE) (dD) (cC) (BB1) (AA1)
Tuning for key signature =1b: (gG) (fF) (eE) (dD) (cC) (BbB1b) (AA1)
Tuning for key signature =2b: (gG) (fF) (ebEb) (dD) (cC) (BbB1b) (AA1)

Tuning for London no 60: (gG) (fF) (ebEb) (dbDb) (cC) (BbB1b) (AA1)
Tuning for London no 34: (gG) (fF) (ebEb) (dbDb) (cbCb) (BbB1b) (AbA1b)

Maximum interval of retuning of a string:

Inspecting all tunings in the two manuscripts reveals an interesting property: there is only one course the maximum interval of retuning of which is two semitones (namely course 11), whereas the maximum interval of retuning for the other courses is one semitone.

Example:
Course 9 is tuned either to (eE) or (ebEb).
The interval between e and eb (and likewise between E and Eb) is one semitone.

§9 VARIABLY TUNED STRINGS OF AN ALTO GUITAR

The comparison of the standard tuning of the 11 course baroque lute in O-notation
f1 d1 (aa) (ff) (dd) (aA) (gG) (fF) (ebEb) (dD) (cC)
to the standard tuning of the 11 string alto in A-notation
g1 d1 a f c G F Eb D C B1b
shows a correspondence between courses 7 to 11 of the baroque lute and strings 6 to 10 of the alto.
As the courses 7 to 11 of the baroque lute are the variably tuned courses, it suggests itself to consider the strings 6 to 11 as the variably tuned alto strings.

Likewise, the comparison of the standard tuning of the 13 course baroque lute in O-notation
f1 d1(aa) (ff) (dd) (aA) (gG) (fF) (ebEb) (dD) (cC) (BbB1b) (AA1)
to the standard tuning of the 13 string alto in A-notation
g1 d1 a f c G F Eb D C B1b A1 G1
shows a correspondence between courses 7 to 13 of the baroque lute and strings 6 to 12 of the alto.
As the courses 7 to 13 of the baroque lute are the are the variably tuned courses, it suggests itself to consider the strings 6 to 13 as the variably tuned alto strings.

Digression :

There is also a correspondence between courses 2, 3 and 4 of the lute and strings 2, 3 and 4 of the alto guitar.

An important consequence of this is, that, when arranging a piece for lute on an alto guitar, in a way that A-key = O-key (which means transposing the scores 3 semitones down), fretting positions concerning these courses can be preserved, provided that the musical context (i.e. the notes simultaneously played on the other strings) allows it.

This can be made transparent by the following consideration:
Let for example p be an arbitrary fretting position on course 3 (p = 0 representing the open course, p = 1 representing the first fret, p=2 representing the second fret, and so on).
Then the corresponding note is
a + p semitones (in O-notation).
Transposing this note 3 semitones down yields
(a + p semitones) - 3 semitones =
(a - 3 semitones) + p semitones =
f# + p semitones.
As f# is the note of the open empty 3rd alto string (in AA-notation), the fretting position of the note f# + p semitones is again p, i.e. the fretting position is preserved.

Any other transposing strategy is worse with respect to preserving fretting positions for courses which are not variably tuned.
Transposing one semitone down, the fretting positions for the first course are preserved.
Transposing five semitones down, the fretting positions for the fifth course are preserved.

§10 NATURAL TUNINGS OF A 13 STRING ALTO GUITAR

As there are two relevant notations for the alto guitar, natural tunings can be expressed in two ways, with respect to AA-notation and with respect to A-notation.

§10.1 ALGORITHMS

The natural tuning of the variably tuned strings for an AA-key on a 13 string alto guitar is determined as follows:

Step 1:
Determine the start note in AA-notation as follows:
= E#, if the key signature consists of 6#
= Eb, if the key signature consists of 2b, 3b, 4b, 5b or 6b
= E, else

Step 2:
The tuning of the variably tuned strings is identical with the key-specific descending
note sequence for the key, start note and length 8.

The natural tuning of the variably tuned strings for key an A-key on a 13 string alto guitar is determined as follows:

Step 1:
Determine the correspondent AA-key as A-key minus 3 semitones.

Step 2:
Determine the natural tuning for the AA-key

Step 3:
Transform the tuning from AA-notation to A-notation (by adding 3 semitones to every note).

Comment:

This design of the algorithms focuses on ascertaining the uniformity of the AA-notation for variably tuned strings over all natural tunings. This makes sense, as this is the notation which is relevant for sight-reading (confer §6.3).
A consequence is, that this uniformity does not hold for the A-notation.
For example string 8 is tuned to C#, C or Cb in AA-notation, but E, Eb, D# or D in A-notation. The problems are caused by the cases, where the counterpart of a #-A-key is a b-AA-Key, concretely, by
A-key F# with counterpart AA-key Eb
A-key B with counterpart AA-key Ab
A-key E with counterpart AA-key Db
A-key A with counterpart AA-key Gb.

As A-key A has a second counterpart, which is F#, there is no need to use Gb as a transposing key for A-key A.

AA-keys Eb, Ab and Db are inconvenient transposing keys, so they will hardly ever be used.

Even if one would be willing to use them and apply transposing strategy A-key = O-key for Weiss music, there would not exist a practical example:
From where we stand, there exists no Weiss piece with O-key = F#, B, E, d#, g# or c# in any known manuscript.

§10.2 COMPLETE LIST OF NATURAL TUNINGS OF A 13 STRING ALTO GUITAR

Definition of abbreviations:
AA-tuning: Tuning in AA-notation
A-tuning: Tuning in A-notation
AA-keysig: Signature of AA-key
A-keysig: Signature of A-key

I) TUNING OF VARIABLY TUNED STRINGS:

AA-keysig: 6# => AA-tuning = E# D# C# B1 A1# G1# F1# E1#
=> A-keysig: 3#, A-tuning = G# F# E D C# B1 A1 G1#
AA-keysig: 5# => AA-tuning = E D# C# B1 A1# G1# F1# E1
=> A-keysig: 2#, A-tuning = G F# E D C# B1 A1 G1
AA-keysig: 4# => AA-tuning = E D# C# B1 A1 G1# F1# E1
=> A-keysig: 1#, A-tuning = G F# E D C B1 A1 G1
AA-keysig: 3# => AA-tuning = E D C# B1 A1 G1# F1# E1
=> A-keysig: , A-tuning = G F E D C B1 A1 G1
AA-keysig: 2# => AA-tuning = E D C# B1 A1 G1 F1# E1
=> A-keysig: 1b, A-tuning = G F E D C B1b A1 G1
AA-keysig: 1# => AA-tuning = E D C B1 A1 G1 F1# E1
=> A-keysig: 2b, A-tuning = G F Eb D C B1b A1 G1
AA-keysig: => AA-tuning = E D C B1 A1 G1 F1 E1
=> A-keysig: 3b, A-tuning = G F Eb D C B1b A1b G1
AA-keysig: 1b => AA-tuning = E D C B1b A1 G1 F1 E1
=> A-keysig: 4b, A-tuning = G F Eb Db C B1b A1b G1
AA-keysig: 2b => AA-tuning = Eb D C B1b A1 G1 F1 E1b
=> A-keysig: 5b, A-tuning = Gb F Eb Db C B1b A1b G1b
AA-keysig: 3b => AA-tuning = Eb D C B1b A1b G1 F1 E1b
=> A-keysig: 6b, A-tuning = Gb F Eb Db Cb B1b A1b G1b
or A-keysig: 6#, A-tuning = F# E# D# C# B1 A1# G1# F1#
AA-keysig: 4b => AA-tuning = Eb Db C B1b A1b G1 F1 E1b
=> A-keysig: 5#, A-tuning = F# E D# C# B1 A1# G1# F1#
AA-keysig: 5b => AA-tuning = Eb Db C B1b A1b G1b F1 E1b
=> A-keysig: 4#, A-tuning = F# E D# C# B1 A1 G1# F1#
AA-keysig: 6b => AA-tuning = Eb Db Cb B1b A1b G1b F1 E1b
=> A-keysig: 3#, A-tuning = F# E D C# B1 A1 G1# F1#

So standard tuning is connected with AA-keysig = 1#.

II) RETUNING BASED ON STANDARD TUNING:

Tuning for AA-keysig =6#: Sharpen strings 6, 7, 8, 10, 11, 13
Tuning for AA-keysig =5#: Sharpen strings 7, 8, 10, 11
Tuning for AA-keysig =4#: Sharpen strings 7, 8, 11
Tuning for AA-keysig =3#: Sharpen strings 8, 11
Tuning for AA-keysig =2#: Sharpen string 8,
Tuning for AA-keysig =1#:
Tuning for AA-keysig = : Flatten string 12
Tuning for AA-keysig =1b: Flatten strings 9, 12
Tuning for AA-keysig =2b: Flatten strings 6, 9, 12, 13
Tuning for AA-keysig =3b: Flatten strings 6, 9, 10, 12, 13
Tuning for AA-keysig =4b: Flatten strings 6, 7, 9, 10, 12, 13
Tuning for AA-keysig =5b: Flatten strings 6, 7, 9, 10, 11, 12, 13
Tuning for AA-keysig =6b: Flatten strings 6, 7, 8, 9, 10, 11, 12, 13

§11 NATURAL TUNINGS OF AN 11 STRING ALTO GUITAR

The natural tuning of an 11 string alto guitar for an arbitrary key is determined by omitting everything with respect to strings 12 and 13 from the corresponding tuning of a 13 string alto for the same key.

§12 DERIVATION OF THE TUNING OF AN ALTO ARRANGEMENT

§12.1 BASIC TRANSPOSING MECHANISMS

In this chapter some insights will be derived from the following two transposing schemes for keys:

C -> Db -> D -> Eb -> E -> F -> F# or Gb -> G -> Ab -> A -> Bb -> B -> ..

c -> c# -> d -> d# or eb -> e -> f -> f# -> g -> g# -> a -> bb -> b -> ..

The first scheme concerns major, the second minor keys.
Any arrow -> in the schemes represents a transposition of a semitone.

For example the detail D -> Eb has to be interpreted as:
'Transposing a piece in D major one semitone upwards leads to Eb major'
or
'Transposing a piece in Eb major one semitone downwards leads to D major'.

Transposing an arbitrary number n of semitones upwards / downwards means following n successive arrows forwards / backwards starting from the source key.

For example, transposing 3 semitones downwards from source key f minor, we have to follow 3 arrows backwards:
d -> d# or eb -> e -> f
So the resulting target key is d minor.

These schemes have to be interpreted in a circular way, i.e. for instance transposing a semitone upwards from B leads to C.

Definition:

The forward distance from an arbitrary source key k1 to an arbitrary target key k2, abbreviated by fdist(k1,k2), is the minimum number of arrows leading from k1 to k2 in the corresponding transposing scheme, if arrows are followed forwards. So fdist(k1, k2) is the minimum number of semitones leading from k1 to k2 when transposing upwards.

Example:

C is reached from A by following 3 arrows forwards. Of course it is also reached by following 15 arrows forwards. But according to the definition fdist(A,C) = 3.

Definition:

The backward distance from an arbitrary source key k1 to an arbitrary target key k2, abbreviated by bdist(k1,k2), is the minimum number of arrows leading from k1 to k2 in the corresponding transposing scheme, if arrows are followed backwards. So bdist(k1,k2) is the minimum number of semitones leading from k1 to k2 when transposing downwards.

Example:

bdist(A,C) = 9

Proposition1:

If a piece is transposed from an arbitrary source key k1 n semitones upwards to a target key k2, then fdist(k1,k2) = n mod 12.

Here mod is the modulo function (n mod 12 is the rest remaining, when n is divided by 12; hence 12 mod 12 = 0, 13 mod 12 = 1, 2 mod 12 = 2 and so on).

Background: Complete octaves don't contribute to fdist.

Example:

Transposing from C major 14 semitones upwards results in target key D major. According to the definition of fdist we have fdist(C major, D major) = 2. 2 is equal to 14 mod 12.

Proposition2:

If a piece is transposed from an arbitrary source key k1 n semitones downwards to a target key k2, then bdist(k1,k2) = n mod 12.

Example:

Transposing from C major 2 semitones downwards results in target key Bb major. According to the definition of bdist we have bdist(C major, Bb major) = 2. 2 is equal to 2 mod 12.

Proposition3:

For any pair of keys k1, k2, where k1 is not equal to k2,
fdist(k1,k2) + bdist(k1,k2) = 12

Background:

One complete circle consists of 12 arrows, i.e. 12 semitone steps constitute one octave.

Example:

fdist(A,C) = 3.
bdist(A,C) = 9.
fdist(k1,k2) + bdist(k1,k2) = 12

Proposition4:

Transposition does not change key distances.

In other words:
If there are two arbitrary keys k11 and k21 with distance d and we transpose from both keys the same number of semitones upwards or downwards, then the resulting keys again have a distance of d. The distance can be measured alternatively in terms of fdist or bdist.

Example 1:

k11 = Eb, k12= Bb, according to transposing scheme fdist(Eb,Bb) = 7.
Transposing 3 semitones down from Eb yields C.
Transposing 3 semitones down from Bb yields G.
fdist(C,G) = 7, too.

Proof:

Let
- s be the number of semitones
- k12 the key resulting from the transposition of k11
- k22 the key resulting from the transposition of k21.

There are four cases:
Case 1: Transposing upwards, distance measured in fdist
Case 2: Transposing upwards, distance measured in bdist
Case 3: Transposing downwards, distance measured in bdist
Case 4: Transposing downwards, distance measured in fdist

Proof of case 1:

As fdist(k11,k21) = d, we reach k21 from k11 by following forwards d arrows in the scheme:

k11 + d arrows => k21

As k22 is the key resulting from the transposition of k21, we reach k22, if we follow further s arrows. So in total we reach k22 by following forwards d + s arrows, starting from k11:

k11 + d arrows + s arrows => k22.

It is clear that we also reach k22 from k11 by first following forwards s arrows and then d arrows.

k11 +s arrows + d arrows => k22.

According to the presupposition we reach k12, if we follow s arrows forwards starting from k11:

k11 +s arrows => k12.

So d arrows lead from k12 to k22:

k12 +d arrows => k22.

I.e. fdist(k12,k22) = d. This had to be proved.

Proof of case 2:

If bdist(k11, k21) = d, then fdist(k11, k21) = 12 - bdist(k11, k21) , see proposition 3.
Let us denote 12 - bdist(k11, k21) by d'.

Now we have transformed case 2 to case 1.

From the proof of case 1 follows, that fdist(k12,k22) = d'.
Now bdist(k12,k22) = 12 – d' = 12 – (12 - bdist(k11, k21)) = bdist(k11,k21). This had to be proved.

Proof of case 3:

This proof can derived from the proof of case 1 by replacing every occurrence of fbist by bdist and every occurrence of the word forwards by backwards.

Proof of case 4:

Can be derived from the proof of case 3, analogously to case 2.

§12.2 A GUIDELINE TO DERIVING ALTO TUNINGS

Transposing a piece of lute music, any source note is transformed to a target note which lies a fixed number s of semitones above or below the source note. This, of course, especially applies to the source notes, which are played on variably tuned courses.

Mostly, variably tuned courses are played as open courses. In order not to increase the risk of creating unfrettable constellations on the alto guitar, an alto tuning is favourable, if it fulfils this

Requirement:

The counterpart of any note of a piece which is played by striking an open variably tuned lute course can be played on the alto by striking an open variably tuned string.

In almost all Weiss pieces O-tuning is a natural tuning of some O-tuning-key. In this case the requirement can be fulfilled by tuning the alto according to the natural tuning of the AA-tuning-key, which lies s semitones above/below O-tuning-key, provided that all these counterparts lie within the range of the variably tuned alto strings.

Especially if O-tuning is key-conform (i.e. O-tuning-key = O-key), adding/subtracting s semitones to/from both O-tuning-key and O-key results in the same key, i.e. AA- tuning-key = AA-key. In other words, key-conformity is preserved.

But this is not necessarily the only way to fulfil the requirement. There are lute pieces with natural tuning, for which an alto tuning can be devised, which fulfils the requirement formulated above, but is not a natural tuning. The arrangement of the well known passagaille by Weiss (from suite no XIII, London MS) for an 11 string alto, contained in Per-Olof Johnson's Altgitarren book, is an example of such an arrangement:

O-key =D.
O-tuning is key-conform:
(gG) (f#F#) (eE) (dD) (c#C#) (BB1) (AA1)
The notes of the piece which are played on the variably tuned courses, are: G F# D C# B1 A1.
I.e. the pieces doesn't contain any E.

Transposing three semitones down to AA-key = B, key-conform AA-tuning on a 13 string alto would be:
E D# C# B1 A1# G1# F1# E1
The notes of the piece which are played on the variably tuned courses, are (in AA-notation):
E D# B1 A1# G1# F1# (these are the counterparts of G F# D C# B1 A1, O-notation).
I.e. two of the variably tuned alto strings are not used.

Instead the piece is arranged on a 11 string alto tuned like this:
E D# B1 A1# G1# F1#.
As all counterparts of the notes played on the open variably tuned lute courses can be played as open variably tuned alto strings, the requirement is fulfilled. But this tuning isn't a natural tuning.

It is true that this way the piece can be arranged on an 11 string alto in B without disrupting bass lines.
On the other hand doing so has quite a few disadvantages:
- It is unlikely that this exotic tuning can be used for a lot of other pieces. So the total number of tunings of one's repertoire is inflated.
- This tuning is inappropriate for playing the whole suite, because there are pieces in the suite which do have an E (in O-notation)
- Sight reading is disturbed, as standard representation of strings 8 to 11 is not valid.

So on balance there are strong arguments to derive alto tunings according to this general guideline:

If a lute piece has natural tuning of some O-tuning-key , and the piece is transposed s semitones upwards/downwards, then the tuning of the alto arrangement should be the natural tuning of the key which lies exactly s semitones above/below O-tuning-key. A consequence of this approach is, that a key-conform lute tuning is transformed to a key-conform alto tuning.

§12.3 ALGORITHMS

All algorithms presented in the following are based on these data:
..O-key
..O-tuning
..O-tuning-key (only existent, if O-tuning is a natural tuning)
..AA-key
..Direction of transposition: upwards/downwards
..Number of semitones of the transposition (abbreviated by transp-dist)

§12.3.1 DERIVATION OF THE TUNING OF AN ALTO ARRANGEMENT FROM A KEY-CONFORM LUTE TUNING

According to the general guideline, a key-conform lute tuning should be transformed to a key-conform alto tuning. So AA-tuning-key = AA-key.


§12.3.2 DERIVATION OF THE TUNING OF AN ALTO ARRANGEMENT FROM A NATURAL, BUT NOT KEY-CONFORM LUTE TUNING

One may choose among the following two algorithms.

§12.3.2.1 ALGORITHM 1

Determine AA-tuning-key by
.. transposing transp-dist semitones downwards from O-tuning-key, if direction of transposition is downwards.
.. transposing transp-dist semitones upwards from O-tuning-key, if direction of transposition is upwards

§12.3.2.2 ALGORITHM 2

This algorithm derives straightforwardly from §12.1, proposition 4.

Step 1:
Calculate alternatively d =fdist(O-key,O-tuning-key) or d = bdist(O-key,O-tuning-key).

Step 2:
Determine AA-tuning-key by transposing from AA-key d semitones upwards or downwards, respectively.

§12.3.2.3 EXAMPLE 1

Suite no 34 in the Dresden MS:
O-key = Eb,
O-tuning-key = Bb.
Let us assume that AA-key = C, direction of transposition = downwards, transp-dist = 3.

Algorithm 1:
As direction of transposition is downwards, we have to transpose 3 semitones downwards form Bb. So AA-tuning-key = G.

Algorithm 2:
Step 1: bdist(O-key,O-tuning-key) = 5 semitones
Step 2: AA-tuning-key is determined by transposing 5 semitones downwards from C, i.e. AA-tuning-key = G.

§12.3.2.4 EXAMPLE 2

Suite no 28 in the Dresden MS:
O-key =f,
O-tuning-key = g.
Let us assume that AA-key = d, direction of transposition = downwards, transp-dist = 3.

Algorithm 1:
As direction of transposition is downwards, we have to transpose 3 semitones downwards from g. So AA-tuning-key = e.

Algorithm 2:
Step 1: fdist(O-key,O-tuning-key) = 2 semitones
Step 2: AA-tuning-key is determined by transposing 2 semitones upwards from d, i.e. AA-tuning-key = e.

§12.3.3 DERIVATION OF THE TUNING OF AN ALTO ARRANGEMENT FROM A LUTE TUNING, WHICH IS NOT A NATURAL TUNING

The idea is to consider such a tuning as a modification of key-conform tuning.

Step 1:
Compare O-tuning with key-conform tuning. Identify all variably tuned courses of key-conform tuning, which differ from O-tuning. The result consists of
triples (c,(n11 n12),d),
where c is a number of a course, (n11 n12) the key-conform tuning of this course in O-notation and d the amount of semitones between the key-conform tuning of c and the real tuning of the same course.

Step 2:
For every tripel (c,(n11 n12),d) from the result of step 1 do:
Step 2.1: Determine n2 = the note which lies transp-dist semitones above/below n12.
Step 2.2: If n2 is the tuning of a variably tuned string from the key-conform tuning of AA-key, then retune this string by d semitones

Example:
London MS, no 60. O-key signature = 5b. O-key = bb
O-tuning = (gG) (fF) (ebEb) (dbDb) (cC) (BbB1b) (AA1)
Let us assume that the piece has to be transposed 4 semitones down, i.e. direction of transposition = downwards, transp-dist = 4, so AA-key = f#, and that transposition is made for a 13 string alto.

Step 1:
The key-conform lute tuning would be :
(gbGb) (fF) (ebEb) (dDb) (cC) (BbB1b) (AbA1b).
So we can consider O-tuning as key-conform tuning with additional modification of courses 7 (one semitone upwards) and 13 (also one semitone upwards).

So there are two relevant triples:
(7,(gbGb),1) and (13,(AbA1b),1)

Step 2:
For AA-key = f#, the key-conform alto tuning is
E D C# B1 A1 G1# F1# E1.

Dealing with triple (7,(gbGb),1):
Step 2.1: n2 = n12 minus 4 semitones = Gb minus 4 semitones = D
Step 2.2: D is contained in E D C# B1 A1 G1# F1# E1, so string 7 has to be retuned to D#.

Dealing with triple (13,(AbA1b),1):
Step 2.1: n2 = n12 minus 4 semitones = A1b minus 4 semitones = E1.
Step 2.2: E1 is contained in E D C# B1 A1 G1# F1# E1, so string 13 has to be retuned to E1#.

So AA-tuning = E D# C# B1 A1 G1# F1# E1#.

§12.4 TRANSPOSING DOWNWARDS WITHOUT EXCEEDING BASS RANGES

Transposing downwards, one runs the risk that some of the resulting notes lie below the bass range of the used alto guitar. A method to solve this problem, is to octavate the problematic notes upwards. But this may have the consequence, that bass lines are disrupted. So it is favourable to limit the number of semitones of the transposition so that the bass range of the alto is not exceeded.

In this paragraph this topic will be discussed for lute pieces with natural tuning. It is presupposed that the corresponding alto tunings are derived according to the guideline from §12.2.

Whether an alto bass range is sufficient, depends on
... the range of used courses of the lute piece (11 courses, 12 courses or 13 courses)
... the number of semitones of the transposition
... the key signature of the tuning for the lute piece
... the kind of alto used (11 string or 13 string)

§12.4.1 MUSIC FOR A 13 COURSE LUTE, WITH COURSE 13 BEING USED

Transposing 3 semitones down

Example:

Key signature of O-tuning = 2#.
As course 13 is being used, the deepest note of the piece is A1, in O-notation.
Transposing 3 semitones down, the counterpart of A1 is F1# and the resulting key signature of AA-tuning is 5#.
With respect to this tuning, F1# is the note of the open 12th alto string (if existent).
So an 11 string alto is not sufficient for this transposition, provided that one does not accept octavating basses as a means to solve problems of bass range.

The following tables show for all natural O-tunings
.. the key signature of O-tuning ( abbreviated by <O-tuning-keysig> )
..the deepest note of the piece in O-notation ( abbreviated by <Deep. note lute> )
.. the key signature of AA-tuning ( abbreviated by < AA-tuning-keysig> )
..the deepest note of the piece in AA-notation ( abbreviated by <Deep. note alto> )
..the number of the corresponding alto string ( abbreviated by <Alto string> )

O-tuning-keysig.....6#.....5#....4#......3#............2#....1#.......bl
Deep. note lute......A1#..A1#..A1......A1............A1.....A1.....A1
AA-tuning-keysig...3b.....4b....5b.....6b/6#........5#.....4#......3#
Deep. note alto.....G1....G1...G1b..G1b/F1#....F1#...F1#....F1#
Alto string..............11....11.....11......11/12........12.....12......12

O-tuning-keysig.....1b.....2b.....3b....4b.....5b....6b
Deep. note lute......A1.....A1... A1b..A1b..A1b..A1b
AA-tuning-keysig....2#.....1#.....bl.....1b....2b....3b
Deep. note alto......F1#...F1#...F1...F1.....F1....F1
Alto string..............12......12.....12....12....12....12

Result: A 13 string alto is sufficient in any case. In fact a 12 string alto would do. An 11 string alto is insufficient in most cases.

Transposing 2 semitones down

O-tuning-keysig....6#.....5#......4#.....3#.....2#....1#.....bl
Deep. note lute.....A1#..A1#....A1.....A1.....A1....A1....A1
AA-tuning-keysig...4#.....3#......2#.....1#.....bl.....1b....2b
Deep. note alto.....G1#...G1#...G1....G1... G1...G1...G1
Alto string..............11......11.....11......11.....11....11....11

O-tuning-keysig....1b....2b......3b.......4b...........5b.....6b
Deep. note lute......A1....A1.....A1b....A1b.........A1b...A1b
AA-tuning-keysig...3b....4b......5b.....6b/6#.........5#....4#
Deep. note alto.....G1....G1....G1b...G1b/F1#...F1#....F1#
Alto string..............11.....11.....11......11/12........12......12

Result: In most cases, an 11 string alto is sufficient.

Transposing 1 semitone down

O-tuning-keysig.....6#....5#....4#......3#......2#.......1#.............bl
Deep. note lute......A1#..A1#..A1.....A1......A1.......A1............A1
AA-tuning-keysig...1b.....2b....3b......4b......5b.....6b/6#.........5#
Deep. note alto......A1....A1....A1b...A1b....A1b...A1b/G1#...G1#
Alto string..............10.....10....10......10......10......10/11........11

O-tuning-keysig....1b......2b.....3b......4b.....5b......6b
Deep. note lute......A1.....A1.... A1b...A1b...A1b...A1b
AA-tuning-keysig...4#.......3#.....2#.....1#.....bl......1b
Deep. note alto.....G1#...G1#....G1....G1....G1.....G1
Alto string..............11.......11......11.....11.....11.....11

Result: In any case, an 11 string alto is sufficient.

§12.4.2 MUSIC FOR A 13 COURSE LUTE, WITHOUT COURSE 13 BEING USED

Transposing 3 semitones down

O-tuning-keysig.....6#.....5#.....4#.......3#............2#....1#......bl
Deep. note lute......B1....B1......B1......B1...........B1....B1....B1
AA-tuning-keysig...3b.....4b......5b.....6b/6#........5#.....4#.....3#
Deep. note alto.....A1b...A1b...A1b..A1b/G1#...G1#..G1#...G1#
Alto string..............10.....10......10......10/11........11.....11.....11

O-tuning-keysig....1b....2b.....3b.....4b.....5b....6b
Deep. note lute.....1b...B1b...B1b..B1b..B1b..B1b
AA-tuning-keysig...2#....1#......bl.....1b....2b.....3b
Deep. note alto......G1...G1....G1....G1....G1....G1
Alto string..............11.....11.....11.....11.....11....11

Result: In any case, an 11 string alto is sufficient.

§12.4.3 MUSIC FOR AN 11 COURSE LUTE

Transposing 4 semitones down

O-tuning-keysig....6#.....5#.....4#....3#.....2#....1#......bl
Deep. note lute.....C#....C#....C#...C#....C#.....C......C
AA-tuning-keysig...2#.....1#.....bl....1b.....2b.....3b.....4b
Deep. note alto.....A1.....A1.....A1....A1....A1...A1b...A1b
Alto string.............10.....10......10....10....10.....10.....10

O-tuning-keysig....1b.........2b..........3b......4b.......5b......6b
Deep. note lute.....C..........C...........C........C.........C......Cb
AA-tuning-keysig...5b......6b/6#.......5#......4#........3#......2#
Deep. note alto.....A1b...A1b/G1#...G1#...G1#....G1#....G1
Alto string.............10.......10/11........11......11........11......11

Result: In any case, an 11 string alto is sufficient.


§13 CONVENIENT TUNINGS FOR ALTO GUITARS

§13.1 CONVENIENT KEY-SIGNATURES FOR ALTO GUITARS

There are AA-key signatures which are convenient for alto guitars. The following is based on the assumption that there is a consensus on these being 5#, 4#, 3#, 2#, 1#, blank and 1b.

§13.2 ALTO TUNINGS FOR KEY-CONFORM LUTE MUSIC

Preserving key-conformity of tuning when transposing, the alto tunings for key-conform lute pieces are the natural tunings of the AA-key signatures
5#, 4#, 3#, 2#, 1#, blank and 1b.
These tunings will be referred to as convenient tunings for alto guitars (or short: convenient alto tunings).

§13.3 ALTO TUNINGS FOR LUTE MUSIC WITH NATURAL, BUT NOT KEY-CONFORM TUNING

The analysis of the two manuscripts has shown, that there is music, the tuning of which is a natural tuning of some key signature (O-tuning-keysig), but this key signature differs from the key signature in the scores (O-keysig).
Following the rule, that a natural lute tuning is transformed to a natural alto tuning, transposing such music to a convenient key signature (AA-keysig) requires a tuning, which is a natural tuning of some other key signature (AA-tuning-keysig). But it is not guaranteed, that this tuning is a convenient tuning.

In order not to inflate the set of alto tunings it would be fortunate, if the convenient alto tunings were sufficient as alto tunings in these cases, too.
So for each of the relevant combinations (O-keysig, O-tuning-keysig) it has to be verified, that there is at least one convenient AA-keysig such that the resulting tuning is a convenient tuning, in other words: that AA-tuning-keysig is a convenient key signature, too.

This applies especially to the suites with mixed key signatures. In such a suite there are pieces with key-conform tuning and others without.

These combinations occur:
Combination 1: O-keysig = 3b, O-tuning-keysig = 2b
Combination 2: O-keysig = 4b, O-tuning-keysig = 2b
Combination 3: O-keysig = 3b, O-tuning-keysig = blank
Combination 4: O-keysig = 2b, O-tuning-keysig = 1#

In alto arrangements, mostly the AA-key is not higher than the O-key and not lower than O-key minus 4 semitones.
So we will check the key signatures resulting from transposing 0, 1, 2, 3 and 4 semitones down.

Results:

Combination 1: Transposing
.. 0 semitones down: AA-keysig= 3b, AA-tuning-keysig = 2b, validation = (-,-)
.. 1 semitone down: AA-keysig= 2#, AA-tuning-keysig = 3#, validation = (+,+)
.. 2 semitones down: AA-keysig= 5b, AA-tuning-keysig = 4b, validation = (-,-)
.. 3 semitones down: AA-keysig= blank, AA-tuning-keysig = 1#, validation = (+,+)
.. 4 semitones down: AA-keysig= 5#, AA-tuning-keysig = 6#, validation = (+,-)

Explanation:
Validation = (-,-) means, that neither AA-keysig nor AA-tuning-keysig are convenient key signatures.
Validation = (+,+) means, that AA-keysig is a convenient key signature and the tuning is a convenient tuning.

So there are two 'good' arrangements for combination 1.

Combination 2: Transposing
.. 0 semitones down: AA-keysig= 4b, AA-tuning-keysig = 2b, validation = (-,-)
.. 1 semitone down: AA-keysig= 1#, AA-tuning-keysig = 3#, validation = (+,+)
.. 2 semitones down: AA-keysig= 6b, AA-tuning-keysig = 4b, validation = (-,-)
.. 3 semitones down: AA-keysig= 1b, AA-tuning-keysig = 1#, validation = (+,+)
.. 4 semitones down: AA-keysig= 4#, AA-tuning-keysig = 6#, validation = (+,-)
Again there are 2 'good' arrangements.

Combination 3: Transposing
.. 0 semitones down: AA-keysig= 3b, AA-tuning-keysig = blank, validation = (-,+)
.. 1 semitone down: AA-keysig= 2#, AA-tuning-keysig = 5#, validation = (+,+)
.. 2 semitones down: AA-keysig= 5b, AA-tuning-keysig = 2b, validation = (-,-)
.. 3 semitones down: AA-keysig= blank, AA-tuning-keysig = 3#, validation = (+,+)
.. 4 semitones down: AA-keysig= 5#, AA-tuning-keysig = 4b, validation = (+,-)
There are 2 'good' arrangements, too.

Combination 4: Transposing
.. 0 semitones down: AA-keysig= 2b, AA-tuning-keysig = 1#, validation = (-,+)
.. 1 semitone down: AA-keysig= 3#, AA-tuning-keysig = 6#, validation = (+,-)
.. 2 semitones down: AA-keysig= 4b, AA-tuning-keysig = 1b, validation = (-,+)
.. 3 semitones down: AA-keysig= 1#, AA-tuning-keysig = 4#, validation = (+,+)
.. 4 semitones down: AA-keysig= 6b, AA-tuning-keysig = 3b, validation = (-,-)
There is only one 'good' arrangement.

So the result of the verification is positive, i.e. that for each of the four cases there exists at least one convenient AA-key signature such that the resulting tuning is contained in the convenient tunings.
Especially, it becomes apparent, that in any constellation transposing 3 semitones down leads to a 'good' arrangement.

§13.4 DIVERSITY OF TUNINGS

The analysis of the two manuscripts has shown, that there are only two pieces the tuning of which is not a natural tuning of some key.
So we can assume, that alto arrangements of Weiss music essentially use the convenient tunings.

From an anecdote we know that Weiss spent much time with tuning his instruments (see http://www.classical.net/music/comp.lst/articles/weiss/bio.php ).

From the analysis above we also know that he used the natural tunings of six key signatures in the two manuscripts. He might have reduced the time spent for tuning by reducing the number of scordaturas. The fact, that he didn't do so, indicates that he esteemed the diversity of tunings.

The number of key signatures for convenient natural alto tunings is seven. So there is little difference between the diversity of the convenient natural alto tunings and the diversity of the natural lute tunings used by Weiss (as for number of different key signatures).

Further above natural tunings of alto guitars have been expressed in terms of both AA-tuning and A-tuning.
It makes sense, to consider a natural alto tuning to be the counterpart of a natural lute tuning, if O-tuning-keysig = key-signature of A-tuning (abbreviated by A-tuning-keysig).

So we have these correspondences between the convenient alto tunings and the natural tunings used by Weiss:

AA-tuning-keysig = 5# corresponds to O-tuning-keysig (= A-tuning-keysig) = 2#
AA-tuning-keysig = 4# corresponds to O-tuning-keysig (= A-tuning-keysig) = 1#
AA-tuning-keysig = 3# corresponds to O-tuning-keysig (= A-tuning-keysig) = blank
AA-tuning-keysig = 2# corresponds to O-tuning-keysig (= A-tuning-keysig) =1b
AA-tuning-keysig = 1# corresponds to O-tuning-keysig (= A-tuning-keysig) = 2b

For the following two convenient tunings there exists no counterpart among the natural tunings used by Weiss:
AA-tuning-keysig = blank
AA-tuning-keysig = 1b

The corresponding O-tuning-keysig is 3b and 4b, respectively, but Weiss didn't use these tunings in the two manuscripts.

For the natural tuning with O-tuning-keysig = 3# used by Weiss there exists no counterpart among the convenient alto tunings.
The corresponding AA-tuning-keysig is 6#, but this is not a convenient alto tuning.

§14 HOW TO AVOID FREQUENT RETUNING OF ALTO GUITARS

Frequent retuning leads to fatiguing a string, and this has negative impact on sound.
In this chapter three approaches will be discussed to avoid frequent retuning.

§14.1 HOW TO AVOID FREQUENT RETUNING OF 13 STRING ALTOS

For 13 string altos, the convenient tunings are:

AA-key signature 5# => AA-tuning: E D# C# B1 A1# G1# F1# E1
AA-key signature 4# => AA-tuning: E D# C# B1 A1 G1# F1# E1
AA-key signature 3# => AA-tuning: E D C# B1 A1 G1# F1# E1
AA-key signature 2# => AA-tuning: E D C# B1 A1 G1 F1# E1
AA-key signature 1# => AA-tuning: E D C B1 A1 G1 F1# E1
AA-key signature blank => AA-tuning: E D C B1 A1 G1 F1 E1
AA-key signature 1b => AA-tuning: E D C B1B A1 G1 F1 E1

Strings 6 and 13 are not touched by retuning, so 6 of the 8 variably tuned strings are in fact involved in scordatura.
The difference between the tunings of two adjacent AA-key-signatures in this order is one semitone, i.e. one string has to be retuned by one semitone.

§14.1.1 APPROACH 1: PLAYING ALONG A SORTED REPERTOIRE

Approach 1 requires that all suites, sequences of pieces and single pieces of one’s repertoire be grouped by the natural tuning of the respective alto arrangement and these groups sorted in the order 5#, 4#, 3#, 2#, 1#, blank and 1b.
Playing takes place 'along' this sorted repertoire. There are essentially two variants of playing 'along':

1st variant:

One playing cycle consists of 5#, 4#, 3#, 2#, 1#, blank and 1b.
(I.e. at first one plays all pieces with tuning = natural tuning of AA-key signature 5#, then all pieces with tuning = natural tuning of AA-key signature 4# and so on. When one has played all pieces with tuning = natural tuning of AA-key-signature 1b, one starts again with 5#).

2nd variant:

One playing cycle consists of 5#, 4#, 3#, 2#, 1#, blank, 1b, blank, 1#, 2#, 3#, 4#.

Evaluation:

The degree of avoiding frequent retuning can be measured by the average number of strings involved in retuning when proceeding from a natural tuning to the next natural tuning (in the following abbreviated by avnstr).

Proceeding from group to group in a random way results in an avnstr of about 2.7. So using playing cycles, if of any value, must result in an avnstr which is significantly less than 2.7.

1st variant:

Proceeding from 5# to 4# requires retuning of one string, proceeding from 4# to 3# requires retuning of another string, and so on. Finally, proceeding from 1b to 5# requires retuning of 6 strings. So we have avnstr = 12 / 7 = about 1.7.

2nd variant:

Here the optimum is achieved (avnstr = 1).
But one must be aware, that there is also a con in comparison to the 1st variant. Using the 1st variant, each key signature reappears after exactly 6 other key signatures. Using the 2nd variant, the number of intermediate key signatures varies. For example 3# reappears after 7 or 3 intermediate key signatures, 4# after 1 or 9 intermediate key signatures.

§14.1.2 APPROACH 2: REDUCING THE SET OF TUNINGS USED

One might decide to reduce the set of natural tunings used in one's alto arrangements for example to

AA-key signature 3# => AA-tuning : E D C# B1 A1 G1# F1# E1
AA-key signature 2# => AA-tuning : E D C# B1 A1 G1 F1# E1
AA-key signature 1# => AA-tuning : E D C B1 A1 G1 F1# E1
AA-key signature blank => AA-tuning : E D C B1 A1 G1 F1 E1

The effect would be, that
..the average number of pieces per tuning increases
..the number of strings involved in retuning decreases from 6 to 3
..the number of retuning activities per playing cycle decreases from 12 to 6.

It is appropriate to discard tunings 'at the edges' of the list of tunings. If, for example, the natural tunings of key signature 2# and 1# would be eliminated (i.e. tunings in the middle of the list), we would have no effect with respect to decreasing the number of strings involved in scordatura or the number of retuning activities per playing cycle:

AA-key signature 5# => AA-tuning : E D# C# B1 A1# G1# F1# E1
AA-key signature 4# => AA-tuning : E D# C# B1 A1 G1# F1# E1
AA-key signature 3# => AA-tuning : E D C# B1 A1 G1# F1# E1
AA-key signature blank => AA-tuning : E D C B1 A1 G1 F1 E1
AA-key signature 1b => AA-tuning : E D C B1B A1 G1 F1 E1

Provided that tunings are only discarded 'at the edges' and n is less than or equal to 7, any reduction of the basical set of tunings to n tunings reduces
..the number of strings involved in retuning to n-1
..the number of retuning activities per playing cycle to (n-1) * 2.

In other words, any reduction by one tuning reduces the number of strings involved in retuning by 1 and the number of retuning activities per playing cycle by 2.

The cons of this approach are:
..Reduction of the richness of sonorities available
..Reduction of the chance to find optimal or at least good arrangements (in terms of playability, number of compromises, affinity to the original)

Using just one natural tuning completely eliminates retuning and playing cycles. But it is clear, that in this case the disadvantages weigh heavily.