§14.1.3 APPROACH 3: INCREASING THE NUMBER OF ALTO GUITARS USED

If one has two altos available, one may, for example, decide to use one of them for

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

and the other one for

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

and play every guitar along its sorted repertoire.

The effect would be, that
..the number of strings involved in retuning decreases from 6 to 5
..the number of retuning activities per playing cycle (i.e. one playing cycle of the first alto plus one playing cycle of the second alto) decreases from 12 to 10.

One can switch back and forth between the two sub-repertoires (by exchanging the guitar) at any time without affecting the frequency of retuning.

It stands to reason that only 'adjacent' tunings should be grouped together. If, for example, the natural tuning of key signature 1b would be added to the first group of tunings (i.e. a tuning which is not adjacent to the natural tuning of 3#), we would have an adverse effect (number of strings involved increasing to 8, number of retuning activities per playing cycle increasing to 16):

First group of tunings:
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 1b => AA-tuning : E D C B1B A1 G1 F1 E1

Second group of tunings:
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

It can be proved that, provided that only adjacent tunings are grouped together and n is less than or equal to 7, any partitioning of the original set of tunings in n subsets (for n guitars) reduces
..the number of strings touched by retuning to 7 - n
..the number of retuning activities by playing cycle to (7-n) *2 .

In other words, any additional guitar 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:
..This is the most expensive of all approaches
..Increasing the number of alto guitars increases the number of guitars (and hence total number of strings) which must be kept in tune.

Using 7 altos completely eliminates retuning and playing cycles. But again, in this case the disadvantages weigh heavily. One must take into consideration how many bass strings could be bought for the equivalent of 6 additional altos.

§14.2 HOW TO AVOID FREQUENT RETUNING OF 11 STRING ALTOS

For 11 string altos, the convenient tunings are:

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

There is no difference between the tunings of AA-key signatures 1# and blank.
String 6 is not touched by retuning, so 5 of the 6 variably tuned strings are in fact involved in scordatura.
The difference between two adjacent tunings in this order is one semitone.

§14.2.1 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, the groups 1# and blank joined, and the result sorted in the order 5#, 4#, 3#, 2#, 1# or blank, 1b.

Playing cycles:

1st variant:
One playing cycle consists of 5#, 4#, 3#, 2#, 1# or blank, and 1b.

2nd variant:
One playing cycle consists of 5#, 4#, 3#, 2#, 1# or blank, 1b, 1 # or blank, 2#, 3#, 4#.

Evaluation:
Proceeding from group to group in a random way: avnstr = about 2.3.
1st variant: avnstr = 10 / 6 = about 1.7.
2nd variant: avnstr = 1

§14.2.2 APPROACH 2: REDUCING THE SET OF TUNINGS USED

Again, provided that tunings are only discarded 'at the edges', 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. Using just one natural tuning completely eliminates retuning and playing cycles.

§14.2.3 APPROACH 3: INCREASING THE NUMBER OF ALTO GUITARS USED

Again, provided that only 'adjacent' tunings are grouped together, any additional guitar reduces the number of strings involved in retuning by 1 and the number of retuning activities per playing cycle by 2. Using 6 altos completely eliminates retuning and playing cycles.

END OF THEORY OF SCORDATURA

The theory has been reworked and supplemented by some new insights.

Essentially involved were §1, §2, §8, §9, §10.

Silvanig,
This is interesting!
Sten

Silvanig,
A major effort, congratulations! I have run off the complete paper so that it is easier to study.
Like Sten, I do not like retuning basses, but will do so if I find the piece is very attractive. In any case, frequent retuning tends to fatigue a metal string resulting in a dull tone.
I suspect that like many players of older music, I prefer to stay with the original key, but recognise that modern pitch is probably about half a tone sharp. For this reason, I have a number of instruments tuned in F# to simulate old G pitch. Although originally done to play renaissance and later renaissance music, this pitch is useful for some baroque pieces. For example, a piece in D, would typically require retuning of three alto basses for the transposing key of B, where all basses are used. A piece in D can be played on an F# tuned instrument using a transposing key of C, as F# is just a tone higher than a standard guitar E-tuning. I do realise this is still half a tone sharp, but life is full of compromises, and half a tone is not a big compromise ... .
Still, Sivang, you have made a major study of the issues of performing Weiss on an alto; and have produced an algorithm to expedite transcription. I do not think anyone else has done anything remotely comparable.
James.

Sten and James,
Thanks for your feedback!

James,
Your idea to use an F# tuned instrument especially offers an elegant way to achieve 'Baroque sound' in all the cases where a transposition of three semitones down results in a convenient key, but not of four semitones down.

Examples:

Original key =G, 3 semitones down -> E, F# tuned instrument -> Eb.

Likewise:
F -> D -> Db
Bb -> G -> Gb
C -> A -> Ab

A new chapter has been added:

§14 HOW TO AVOID FREQUENT RETUNING OF ALTO GUITARS