Peter,
I fear errors dog us all, particularly when trying to do many other things!
Keep up the good work!
James.

Peter,
I forgot to say in my last post that it is possible to do a theoretical calculation of reduced line density using a Young's Modulus for the nylon strings. I note that manufacturers do not, to the best of my knowledge, publish such data. Lengthy searching produced values of YM, i.e., E, for nylon strings with likely values, 4.5, 5, 5.5, all times 10^9 N/m^2. The values for the metal wound strings have to be found experimentally. I guess careful "kitchen technology", e.g., suspending weights and measuring extension, would be adequate.
The results I got for nylon strings, mostly, D'Addario, were in the range: 5 to 10%, clustering around 7%. However, a discussion some time ago had a couple of investigators getting 10 to 15%. I think 15% is probably too high. My impression of manufacturer's stated tension levels is that they are not installed values, but ones based on the classical formula. Measuring installed tension is simple in principle, but requires very good technique to get plausible results.
James.

I should have made it clear in the last paragraph, that I was talking about the reduction in tension due to string extension.
James

A most interesting thread.

Becoming independent of string manufacturers's data

Peter, thank you for sharing the details of your string measurements and calculations. Great conceptual and practical work.

Your approach to derive tension data from your own measurements allows to become independent of data from the string manufacturers or suppliers, which may be non-uniform / questionable / inconsistent / incomplete / missing.

Further below I will add an example for such an inconsistency.

Something like this was in my mind since I started grappling with manufacturers's data, in order to get an impression of the tensions for different stringing scenarios for altos.

Based on these measurements, taken on neutral ground, one might finally be enabled to define for each alto string, what very low, low, medium, high, very high tension means and establish a data base, where strings from the market are ranked accordingly.


Differences of our results

The results of our calculations differ more or less. This is partly due to the fact that we use different tunings as a basis. After harmonizing the basis, there remain some significant differences on string level. I started compiling these differences, in order to get some further insights and maybe ideas for improvements. I'll post the results, when I have finished this analysis.


Inconsistencies in manufacturers's string tension data: An example

On the Hannabach internet site MHT (Medium High Tension) is defined as follows:
E/1st: 7.5 daN (i.e. 75 N) (*1a)
H/2nd: 6.5 daN (i.e. 65 N) (*1a)
G/3rd: 6.8 daN (i.e. 68 N) (*1a)
D/4th: 7.5 daN (i.e. 75 N) (*1b)
A/5th: 7.4 daN (i.e. 74 N) (*1b)
E/6th: 7.4 daN (i.e. 74 N) (*1b)

Series 725 (Goldin)
Tension: Medium/High
E725 MHT Set (*2)
E7251 MHTC E/1 (*3a)
E7252 MHTC H/2 (*3a)
E7253 MHTC G/3 (*3a)
E7254 MHT D/4 (*3b)
E7255 MHT A/5 (*3b)
E7256 MHT E/6 (*3b)

Comments:

- (*1a), (*1b), (*2), (*3a) and (*3b) have been added by me for reason of reference

-It would be logical to deduce from *2, that the tensions for the strings are according to *1a/*1b

-This contradicts with <MHTC> as a tension code at the *3a-entries. The definition of this code is missing.

-Furthermore, the package leaflet contains the following data:
E-1: kp 8.0 (i.e. 78.5 N) (*4a)
H-2: kp 6.8 (i.e. 66.7 N) (*4a)
G-3: kp 6.8 (i.e. 66.7 N) (*4a)
D-4: kp 8.0 (i.e. 78.5 N) (*4b)
A-5: kp 7.8 (i.e. 76.5 N) (*4b)
E-6: kp 7.5 (i.e. 73.5 N) (*4b)
Comparing the *4b-entries to the*1b-entries reveals another inconsistency.


Unevenness of frequency response

As James stated, in terms of transposed tuning, the overtones of standard tuning of the alto basses primarily support the <white notes>.

But IMO this statement cannot be generalized to the other convenient tunings.
As tunings of the alto guitar mostly are key-conform, the ringing alto basses together with their overtones mostly emphasize tones of the musical scale of the key of a piece. I.e. if a musical scale contains <black notes>, then these <black notes> are emphasized by the ringing basses, too.

I wonder whether it is desirable to achieve total evenness of frequency response (in terms of supporting every note equally independent of the music played) by using reentrant tuning, if we have the chance to create a variety of differentiated spaces of frequency response, each tailor-made for a key signature (by means of scordatura).

These spaces of frequency response have a little deficiency, though, if generated by an 11 string alto. The bass sequence of an 11 string alto is incomplete in so far, as one note of the musical scale is missing. A 12 stringer eliminates this deficiency. This is one more argument to go for a 12 or 13 string alto.

By the way: Does anybody know factors representing the felt intensity of say the first 10 overtones? My internet research was not yet successful in this respect.


Weak lute basses

In Michel Cardin's treatise <The Late baroque lute Seen through S.L.Weiss> we find some interesting details about stringing of the baroque lute. Here are some citations from this treatise:

<While made exclusively from gut at the beginning of the Baroque, it was quite common by the middle of the seventeenth century, to find, for the solo lute, examples of strings made of overspun material>.
<The bass strings made of wound metal gave added brilliance and sustain to the sound, causing an inevitable change in the conception of works written for the instrument>.

< ...it is quite clear that Weiss and his Germanic contemporaries were to favour the models in the style of Johann Christian Hoffmann's ...>.
< The preponderance of this particular model was possibly due, like the change of musical style, to the wound metal strings ...>.
<However, it is sometimes necessary to use a current model lute ...>.

<One notices while comparing the two instruments that the basses of the theorbo-lute are quite bright while those of the standard lute have a sombre but warm quality>.

I deduce from these statements, that during the baroque period there was a tendency for lutes to become less bass-weak.

Playing Weiss on my alto with E6-basses, and comparing the sound to some Barto interpretations on lute, I sometimes felt, that it would be fortunate, if some of the basses of my alto had more brilliance.


A compromise

Peter, thanks for discussing bridge stability with the luthiers you meet. The answers of the luthiers you have already interviewed and the fact that I don't live in one of the critical regions mentioned by James, have dispelled my concerns to a certain degree. The fact, that with Ramirez guitars the extent of the contact area with the top is not increased proportionally to the number of strings and that they have never had bridges coming unstuck, indicates that that there must be a considerable safety margin.

But as my Hannabach Goldin trebles possibly have more tension than I assumed and I have a 13 stringer (i.e. with 2 more strings that pull) with a 57 scale, I decided to start with a bass stringing with total tension somewhere between E6-stringing and your favourite stringing:

String 7: E6
String 8: Hannabach Black D7 (Medium Tension)
String 9: Hannabach Black D7 (Medium Tension)
String 10: Hannabach Black C8 (Medium Tension)
String 11: Hannabach Black C8 (Medium Tension)
String 12: Hannabach Black H9 (Medium Tension)
String 13: Hannabach Black H9 (Medium Tension)

Silvanig,
Just a couple of points.
1) Although supporters of Yepes re-entrant tuning are usually enthusiastic, sometimes to the point of fanaticism, over this system intended to achieve uniform response in the treble, the effect is variable with guitar. In any case, most "well-made" guitars are not seriously non-uniform in treble response.
You make a valid point about the effect of the tuning of the alto diapasons. A change of tuning of half a tone will reinforce that semitone. Supporters of Yepes would, of course, point out that it will primarily affect only that semi-tone. Even without any Yepes tuning, fretting , e.g., c# on the 5th string, will support not only c#', but also g#'. Only in music where there are a large number of unsupported accidentals is the tuning of potentially general value. Thus, it is probably appropriate for atonal music for those guitars that can be noticeably improved with this tuning.
2) There is some evidence that at least some bass strings for lutes in the early 17th century may have been "loaded" to increase their density. In a number of paintings they are shown coloured. When made from pure gut, they had to be quite thick, which resulted in a dull tone. Doubling the bass string with an octave higher one lightened the tone but created voice confusion. This strategy was condemned by Dowland, Ballard, etc. One answer was to lengthen the fret board, but then the tension needed to tune the top course to g', even half a tone lower than modern g', could result in breakage. Consequently, it would be necessary to lower pitch. Hence, lute music for 10c lute, depending on bass strings and length of fret board, may well have sounded in some cases nearer to modern E or F, rather than F#. Another alternative was to lengthen the bass strings, but keep the treble at original lengths. This latter strategy eventually led to the archlute.
I seem to recall hearing at one Lute Society meeting that wound strings were being made in some places in the early 17th century, i.e., earlier than Cardin states, but I have no special knowledge on this subject.
3) The energy in the overtones/harmonics does drop off quite quickly. The distribution of energy over the modes depends on where the string is plucked. Thus, plucking near the bridge puts more energy into the higher modes and decreases that in the lower modes resulting in a "thinner" tone. There is a formula for the modal energy based on the classical theory of a vibrating string.
James.

James.
In the limited time I have been a member of the forum you always offer excellent points to ponder and rigorous tech argument to back them up. As we say in Australia, "Good on ya!" I am going to wade back in a bit here and hopefully I wont be toppled mercilessly into the cold water.

Point 1 raised. Hmmm... most 'fanatics' of the Yepes tuning would not agree that even 'well made' 6 string guitars have an even treble response. I have had the luxury of being able to compare side by side two superb hand made guitars on exactly this basis. One a 10 string Yepes tuned and the other a 6 string. Both guitars made at approximately the same time almost side by side by the same luthier using pretty much same materials (probably cut from the same log) and exactly the same body, top and bracing - only the neck, headstock and fingerboard differing. I talk of course about the fine Australian luthier Lance Litchfield. This was not purposeful but the circumstances just perchance happened when he made my 10 string for me late last year and I had access to two of his 6 strings. I have had one of his recent 6 strings here with me for about 2 months as a favour so that prospective purchasers in Sydney can assess the instrument. The rare ideal opportunity for a highly relevant comparison. The 6 string is a very good instrument and I recently had the opportunity of a concert by the superb New Zealand guitarist and educator Prof Mathew Marshall playing a similar model. Sitting at the back of the hall it was the equal of Smallmans in prior concerts by other superb guitarists. In particular the bass response and resonance of the Litchfield was somewhat superior whilst the 3rd string response of the Smallman 3rd string was marginally crisper than the Litchfield. Clearly a fine 6 string instrument and comparable to the best.
Back in my studio playing the 6 and 10 string instruments side by side with the same strings fitted (Goldin on the trebles and Hannabach 200’s on the basses) the difference in the sound (volume, projection, clarity, bass / treble response, resonance etc) of the two instruments hits you square on in no uncertain terms. There is no doubt what "the Yepes fanatics" are on about! As much as I have done the tech analysis and produced resonance map comparisons etc, there is nothing like a side by side comparison actually playing a variety of genres. The difference is certainly marked and complex in that its not just volume or projection but timbre in its truest sense and manifestation. One could get all technical and actually dissect this but in short the difference is significant enough for one to simply concede.
Interestingly I also have the opportunity of comparing different 10 string tunings on the same model Milagro 10 string guitars made by Neris Gonzalez as I have a few of these in stock. Comparing the Yepes tuning with the "Baroque or Romantic" tuning as its commonly called, is also chalk and cheese. The Baroque tuning is similar to the Alto 11 but a minor 3rd lower in pitch. A side by side comparison in this way is really a 'no brainer'(sorry another Aussie expression) playing a wide variety of styles Renaissance to Modern. Overall the Yepes tuning ensures a much more even response and easier attainment of required timbre. However in some pieces particularly Renaissance and Baroque, the ability for added diatonic bass accompaniment is somewhat of an advantage using the Baroque tuning. Personally I don’t like the Baroque 10 string guitar tuning as I think its too uneven with a dissonant mix of deficient and exaggerated sympathetic resonances. It is interesting however (and I have no rigorous reasoning for it other than pure gut feel (no pun intended)) that I absolutely love and relish the Alto 11 tuning just a minor 3rd up. At this pitch it somehow JUST works as a delightful mixture between guitar and lute. The Baroque tuning on 10 string guitar doesn’t! So I love both 10 string Yepes and 11 string Alto with the occasional playing of my beloved 1972 Ramirez 1a 6 string. As an aside somewhat relevant in context and another topic for another day - there is in my opinion a very distinct difference in "the traditional Spanish sound" and the modern powerful "Smallman sound" of a 6 string - both different and both pleasing in different ways without needing to say one is better than the other.
Just as an after-thought, some fellow guitarists when listening to me playing the 10 string reckon there is a difference in its sound as one moves further and further away (back in the audience). The “Yepes tuning” effect as it were seems to decrease the further away one is from the player. Certainly one is very aware up front and playing but it stands to reason as the fuller more complete overtones / harmonics and sympathetic resonances are more subtle than struck notes, the effect will decrease the further away one is. This is aptly backed up when one records and the 10 string Yepes seems to have its own magical reverb without having to add any artificial means.

Points 2 and 3
This is very interesting history. Thank you these insights. Certainly on your point 3 I have found that in our ensemble we have one good fellow that unconsciously insists on plucking notes way up close to the fingerboard when we are tuning to each other. This often results not being able to clearly hear whether he is in-tune with the ref guitar or not. The “out of tune” error is much more difficult to hear than the fuller sound if plucked at the edge of the rosette. It of course stands to reason and supports what you are saying about the distribution of energy across the modes.

Regards, Peter. www.laudarra.com.au.