CONTENTS:

§1 PREFACE

§2 NOTATION

§3 TUNING OF AN ALTO GUITAR

§4 TUNING OF A BAROQUE LUTE

§5 SCORDATURA

§6 NATURAL TUNINGS OF AN 11 COURSE BAROQUE LUTE
§6.1 ALGORITHM
§6.2 COMPLETE LIST OF NATURAL TUNINGS OF AN 11 COURSE BAROQUE LUTE
§6.3 UNIFORMITY OF THE O-NOTATION FOR VARIABLY TUNED COURSES

§7 NATURAL TUNINGS OF A 13 COURSE BAROQUE LUTE
§7.1 ALGORITHM
§7.2 COMPLETE LIST OF NATURAL TUNINGS OF A 13 COURSE BAROQUE LUTE
§7.3 UNIFORMITY OF THE O-NOTATION FOR VARIABLY TUNED COURSES

§8 TUNINGS IN THE DRESDEN AND LONDON WEISS MANUSCRIPTS
§8.1 CLASSIFICATION OF THE SCORES
§8.2 KEY-CONFORM AND INDIVIDUAL TUNING
§8.3 LEVEL OF ASSIGNMENT OF A TUNING
§8.4 WEISS MUSIC FOR AN 11 COURSE BAROQUE LUTE
§8.5 WEISS MUSIC FOR A 13 COURSE BAROQUE LUTE
§8.6 SUMMARY

§9 VARIABLY TUNED STRINGS OF AN ALTO GUITAR

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

§11 NATURAL TUNINGS OF AN 11 STRING ALTO GUITAR

§12 DERIVATION OF THE TUNING OF AN ALTO ARRANGEMENT
§21.1 BASIC TRANSPOSING MECHANISMS
§12.2 GUIDELINE TO DERIVING ALTO TUNINGS
§12.3 ALGORITHMS
§12.3.1 DERIVATION OF THE TUNING OF AN ALTO ARRANGEMENT FROM A KEY-CONFORM LUTE TUNING
§12.3.2 DERIVATION OF THE TUNING OF AN ALTO ARRANGEMENT FROM A NATURAL, BUT NOT KEY-CONFORM LUTE TUNING
§12.3.3 DERIVATION OF THE SCORDATURA OF AN ALTO ARRANGEMENT FROM A LUTE TUNING WHICH IS NOT A NATURAL TUNING
§12.4 TRANSPOSING DOWNWARDS WITHOUT EXCEEDING BASS RANGES

§13 CONVENIENT TUNINGS FOR ALTO GUITARS
§13.1 CONVENIENT KEY SIGNATURES FOR ALTO GUITARS
§13.2 ALTO TUNINGS FOR KEY-CONFORM LUTE MUSIC
§13.3 ALTO TUNINGS FOR LUTE MUSIC WITH NATURAL, BUT NOT KEY-CONFORM TUNING
§13.4 DIVERSITY OF TUNINGS

§14 HOW TO AVOID FREQUENT RETUNING OF ALTO GUITARS
§14.1 HOW TO AVOID FREQUENT RETUNING OF 13 STRING ALTO GUITARS
§14.1.1 APPROACH 1: PLAYING ALONG A SORTED REPERTOIRE
§14.1.2 APPROACH 2: REDUCING THE SET OF TUNINGS USED
§14.1.3 APPROACH 3: INCREASING THE NUMBER OF ALTO GUITARS USED
§14.2 HOW TO AVOID FREQUENT RETUNING OF 11 STRING ALTO GUITARS
§14.2.1 APPROACH 1: PLAYING ALONG A SORTED REPERTOIRE
§14.2.2 APPROACH 2: REDUCING THE SET OF TUNINGS USED
§14.2.3 APPROACH 3: INCREASING THE NUMBER OF ALTO GUITARS USED

§1 PREFACE

This paper is a result of my experiences with arranging lute music by the great master of the baroque lute, Silvius Leopold Weiss, on the alto guitar, and my efforts to clarify my thoughts on scordatura and connected topics such as notation and tuning.

To keep the theory compact, some definitions (of terms), abbreviations and representations will be presented. If not explicitly otherwise stated, any such definition, abbreviation or representation is my own.

The analysis of Weiss music, the results of which are presented in this paper, is based on the tablatures and scores provided by Jean-Daniel Forget. Many thanks to him for the tremendous work he dedicated to Weiss, for the benefit of all of us.

§2 NOTATION

In this paper, notes are named by a variant of the Helmholtz notation.
This variant differs from the original Helmholtz notation by replacing subscripts and superscripts by numbers.

Examples:
c'' is replaced by c2
C, is replaced by C1.

Using this variant avoids the occurrence of a certain problem in the editor software.
The same variant is used by Per-Olof Johnson in his description of the alto guitar.

In connection with lutes and guitars there exist two types of musical notation, namely tablature and stave. In this paper we focus on stave.
In the baroque period lute music was usually notated by tablature. The description of the rules for the transformation of lute music from tablature to stave is not in the scope of this paper, so we consider the result of this transformation to be the base of an alto arrangement.

In stave notation, a note is represented by a note symbol, possibly modified by a sharp or flat sign. The corresponding note name is determined by the clef. The clef which is normally used in scores for the alto (as well as for the classical guitar and the baroque lute) is the suboctave treble clef, which is written as a treble clef with an eight below it. In comparison to the treble clef note symbols are placed one octave higher. So the note symbol between the third and fourth line represents an a and not an a1.
In scores for the classical guitar, it is common practice to omit the eight below the treble clef. So what appears to be a treble clef is in reality a suboctave treble clef.
However, Jean-Daniel Forget never omits the eight in his Weiss scores.

The term note name used above does not include a precise synchronization with tone frequencies. In connection with classical guitars and alto guitars it is normal, to synchronize according to the modern concert pitch (with 440 hz for a1). In the Baroque Era, there was no universally recognized pitch standard, but musicologists tend to consider about 415 hz for a1 to be realistic for Weiss music.

As the alto guitar is used as a transposing instrument, scores for the alto guitar are normally notated three semitones below sound.

In this paper, the following abbreviations will be used:

AA-notation:
Notation used in the scores of alto arrangements.
The key in the scores of an alto arrangement is called nominal key, transposing key, arranging key or AA-key.
The formulation 'tuning in AA-notation' which will be used in the following, means, that the notes specified have to be interpreted relatively to AA-notation, i.e. belong to AA-notation.

A-notation:
Notation three semitones obove AA-notation, representing the sound of an alto guitar.
The key of a piece of music in A-notation is called A-key.
The formulation 'tuning in A-notation' means, that the notes specified belong to A-notation.

O-notation:
Notation used in the scores for the lute (original instrument).
The key in the scores of a piece of music for the lute is called O-key.
The formulation 'tuning in O-notation' means, that the notes specified belong to O-notation.

§3 TUNING OF AN ALTO GUITAR

Representation:
A TUNING OF AN 11 STRING ALTO GUITAR is represented by a sequence of note names n1 n2 n3 ... n11, where n1 = tuning of string 1, n2 = tuning of string 2 and so on.

Definition:
The STANDARD TUNING OF AN 11 STRING ALTO GUITAR in A-notation is
g1 d1 a f c G F Eb D C B1b

This is precisely the 'tuning' of an 11 string alto guitar given by Per-Olof Johnson in the book 'Altgitarren' (on page II of the translation of the first 12 pages into English by Sten Edebäck).

Conclusion:
The STANDARD TUNING OF AN 11 STRING ALTO GUITAR in AA-notation is
e1 b f# d A E D C B1 A1 G1


Representation:
A TUNING OF A 13 STRING ALTO GUITAR is represented by a sequence of note names n1 n2 n3 ... n13, where n1 = tuning of string 1, n2 = tuning of string 2 and so on.

Definition:
The STANDARD TUNING OF A 13 STRING ALTO GUITAR in A-notation is
g1 d1 a f c G F Eb D C B1b A1 G1

This is precisely the 'tuning' of a 13 string alto guitar given by Per-Olof Johnson in the book 'Altgitarren' (on page II of the translation of the first 12 pages into English by Sten Edebäck).

Conclusion:
The STANDARD TUNING OF A 13 STRING ALTO GUITAR in AA-notation is
e1 b f# d A E D C B1 A1 G1 F1# E1

§4 TUNING OF A BAROQUE LUTE

The strings of a baroque lute are grouped by courses.
Courses 1 to 2 are single-strung.
Each of the courses 3 to 6 consist of two strings which are tuned in unison.
Each of the other courses consists of two strings which are tuned in octave.

Representation:
A tuning of an 11 course baroque lute is represented by an expression t1 t2 (t3) (t4) ... (t11), where
t1 = tuning of the single string of course1
t2 = tuning of the single string of course2
t3 = tuning of the two strings of course 3, in ascending order of strings
t4 = tuning of the two strings of course 4, in ascending order of strings
...
t11 = tuning of the two strings of course 11, in ascending order of strings

Definition:
The standard tuning of an 11 course baroque lute in O-notation is
f1 d1 (aa) (ff) (dd) (AA) (gG) (fF) (ebEb) (dD) (cC)

If a course contains two strings, both are simultaneously fretted by the same finger, and the corresponding pair of notes is represented in the scores by a single note. If the two strings are tuned in octave, the lower of the two notes is represented in the scores. For instance, the course (gG) is represented in the scores by a G.

Representation:
A tuning of a 13 course baroque lute is represented by an expression t1 t2 (t3) (t4) ... (t13), where
t1 = tuning of the single string of course1
t2 = tuning of the single string of course2
t3 = tuning of the two strings of course 3, in ascending order of strings
t4 = tuning of the two strings of course 4, in ascending order of strings
...
t13 = tuning of the two strings of course 13, in ascending order of strings

Definition:
The standard tuning of a 13 course baroque lute in O-notation is
f1 d1 (aa) (ff) (dd) (AA) (gG) (fF) (ebEb) (dD) (cC) (BbB1b) (AA1)

There seems to be a consensus with regard to the standard tuning of courses 1 to 6, but not with regard to the standard tuning of the other courses.
The definitions chosen here facilitate showing a certain parallelism of baroque lute and alto guitar. In addition, the 13 course variant of this tuning obviously plays an outstanding role among the tunings used by Weiss for the 13 course lute. Details of these aspects will be presented later.

§5 SCORDATURA

According to the Concise Oxford Dictionary of Music (1996) a scordatura is an abnormal tuning of a string instrument in order to obtain special chordal effects and changes of tonal quality.

So, focussing on technical aspects, according to our terminology, we have the following definition:
A scordatura is a tuning of a string instrument, which differs from the standard tuning of this instrument.

Usually an instrument has strings, which keep their standard tuning, whatever scordatura is applied, whereas the other strings may be involved in retuning. The latter are called the variably tuned strings.
Courses of a baroque lute with variably tuned strings are called variably tuned courses.

Obviously Weiss keeps the tuning of courses 1 to 6 constant. So the variably tuned courses are the courses 7 to 11 and 7 to 13 for an 11 course baroque lute and a 13 course baroque lute respectively.

The further development of the theory will deal with different tunings for an instrument. Doing so has the advantage, that standard tuning does not have to be treated as an exception.

In order to avoid superfluous repetitions, the representation of the tunings will be restricted to the variably tuned strings.
Besides specifying the note names for the variably tuned strings, any such tuning will be characterized in terms of retuning based on standard tuning.

The following definition will be used for the description of the tunings of the variably tuned strings both of baroque lutes and altos.

Definition:
The key-specific descending note sequence of length n for a key and a start note (where the start note is a note of the musical scale of the key) is a sequence of notes, which is determined as follows:
. The first note is the start note
. The second note is the next lower note to the start note, in the scale of the key
. The third note is the next lower note to the second note, in the scale of the key
...
. The nth note is the next lower note to the (n-1)th note, in the scale of the key
For minor keys the natural scale is the relevant scale.

Example:
The key-specific descending note sequence of length 8 for major key A and start note E is
E D C# B1 A1 G1# F1# E1
As will be shown later, the tuning of the variably tuned strings for the performance of a piece of music by Weiss is mostly determined by the key of the piece, insofar as the variably tuned courses reflect a contiguous part of the musical scale of the key (plus octaves).
This is the motivation, to define for every combination of instrument type (11 course baroque lute, 13 course baroque lute) and key a special tuning, which is called the natural tuning of this key for the given instrument type.

§6 NATURAL TUNINGS OF AN 11 COURSE BAROQUE LUTE

§6.1 ALGORITHM

The natural tuning of the variably tuned strings for a key on an 11 course baroque lute is determined as follows:

Step 1:
Determine the start note in O-notation as follows:
= G#, if the key signature consists of 3#, 4#, 5#, 6#
= Gb, if the key signature consists of 5b or 6b
= G, else

Step 2:
Determine the key-specific descending note sequence n1 n2 n3 n4 n5 for the key and the start note

Step 3:
Determine the natural tuning of the variably tuned strings as
(o1 n1) (o2 n2) (o3 n3) (o4 n4) (o5 n5)
where o1 = n1 + one octave, o2 = n2 + one octave and so on.

Example:
Given: Key = E (key signature = 4#)
=> Start note = G#
Key-specific descending note-sequence of length 5 = G# F# E D# C#
Tuning = (g#G#) (f#F#) (eE) (d#D#) (c#C#)

As the musical scales for the major and minor keys with the same key signature contain the same notes, the natural tunings for a major key and the minor key with the same key signature are identical.

§6.2 COMPLETE LIST OF NATURAL TUNINGS OF AN 11 COURSE BAROQUE LUTE

I) TUNING OF VARIABLY TUNED STRINGS (in O-Notation):

Tuning for key signature =6#: (g#G#) (f#F#) (e#E#) (d#D#) (c#C#)
Tuning for key signature =5#: (g#G#) (f#F#) (eE) (d#D#) (c#C#)
Tuning for key signature =4#: (g#G#) (f#F#) (eE) (d#D#) (c#C#)
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 = : (gG) (fF) (eE) (dD) (cC)
Tuning for key signature =1b: (gG) (fF) (eE) (dD) (cC)
Tuning for key signature =2b: (gG) (fF) (ebEb) (dD) (cC)
Tuning for key signature =3b: (gG) (fF) (ebEb) (dD) (cC)
Tuning for key signature =4b: (gG) (fF) (ebEb) (dbDb) (cC)
Tuning for key signature =5b: (gbGb) (fF) (ebEb) (dbDb) (cC)
Tuning for key signature =6b: (gbGb) (fF) (ebEb) (dbDb) (cbCb)
So the tunings for key signatures 2b and 3b are identical with standard tuning.

II) RETUNING BASED ON STANDARD TUNING:

Retuning for key signature =6#: Sharpen courses 7, 8, 9(twice), 10, 11
Retuning for key signature =5#: Sharpen courses 7, 8, 9, 10, 11
Retuning for key signature =4#: Sharpen courses 7, 8, 9, 10, 11
Retuning for key signature =3#: Sharpen courses 7, 8, 9, 11
Retuning for key signature =2#: Sharpen courses 8, 9, 11
Retuning for key signature =1#: Sharpen courses 8, 9
Retuning for key signature = : Sharpen course 9
Retuning for key signature =1b: Sharpen course 9
Retuning for key signature =2b:
Retuning for key signature =3b:
Retuning for key signature =4b: Flatten course 10
Retuning for key signature =5b: Flatten courses 7, 10
Retuning for key signature =6b: Flatten courses 7, 10, 11