Experimental Designs
Some Further Flights of Fancy: Why Not?
Site MapThe dictates of practical reality and the economics of being a working musician mean that the goal of achieving a functional and practical work guitar was a good way to limit the scope of my imagination.
Since I have spent about 6 years obsessing about ERG design, it was inevitable that the idea should spread into other areas. Imagination doesn't cost anything, after all, until it emerges into the physical world by awakening Guitar Acquisition Syndrome, and this can be suppressed for some time by reason or by lack of dough.
However, I offer here two of the more coherent ideas that have gone off on tangents from the nine-string fanned fret classical nylon-string harp guitar which I actually had built and am currently playing. These have specific musical purposes behind them, both being explorations of microtonal harmony. I don't expect to ever use microtonal harmony in any performance context, but I have been very impressed by listening to some U-tube videos of folks playing on the archi-cembalo of Nicola Vicentino some of Vicentino's compositions. Vicentino was a genius off on a wild track.
If anybody has comments or questions, I am "jack" at this domain. This is not a blog, and there are no public comments. Furthermore, if anybody wants to build one of these instruments, please feel free! I would appreciate hearing about the results, but I am not claiming these as proprietary designs, and certainly will not stand in the way.
(1) The Greater Perfect Guitar
I don't have a drawing, but I offer a verbal description.This is a seven-string unfretted instrument with a 60 centimeter scale (that's 23.62 inches).
The seven strings are tuned
A4
E4
B3
A3
E3
B2
A2
These are the fixed tones of the ancient Greek "Greater Perfect System". The scale tones which lie between these notes were all subject, in antiquity, to microtonal variations. So, this instrument is fretless in order to permit that.
It appears that the tetrachord was the basic unit of Greek musical thinking, and the structure of their note names includes the association of the index finger with the second descending note of some tetrachords. This, in the context of a four stringed lyre, indicates that the tetrachord may have been thought of as a musical unit to be played in descending order with the thumb on the highest string and three fingers following, in a manner very similar to the most common modern guitar technique, except that the thumb would play the uppermost note as do those left-handed guitarists who play a standard guitar upside down.
The outermost notes of the tetrachord must be tuned in a perfect fourth, as an invariable rule. The two inner notes of the tetrachord may be stretched and squeezed in a variety of ways. It appears to me that the desired sound was to hear all of them sometimes ringing at once, similarly to when a sitar player strikes all of his sympathetic strings at once - it is a beautiful sound - and sometimes individually or in smaller combinations, and in rhythmic patterns. It seems to me probable that those right-hand rhythmic patterns are ancestral to flamenco; flamenco shares with ancient Greek music an obsession with descending tetrachords with the smallest intervals placed last. How that Greek elephant got into the Gypsies' pajamas I wouldn't pretend to know, but the musical resemblance appears to me to be genetic. (See all of the tables of Aristoxenes' scale tunings reproduced below, they are all of this form.) There is a great deal that we don't know about the playing of plucked stringed instruments in the middle ages.
Regarding the tuning of the two inner notes of the tetrachord, it seems very reasonable to me to suppose that Greek kitara and lyre players enjoyed the art of tuning the inner notes in different ways, which would have been subject to informed aesthetic judgement: "Oh, what an elegant tuning," or "Oh, I thought his tuning was a little crude."
I have no interest in playing a replica kitara or lyre. Being a guitarist, my mind has run to how these musical patterns might be reproduced on the guitar. Hence, all of the open strings are tuned to the fixed nodes of the Greek tetrachords, and leave the stopped notes on each string free to be played in an infinite array of microtones, but using the lineal division of the octave into 60 parts as practiced by Aristoxenes in the fourth century BC. Now, it is entirely possible that I misunderstand Aristoxenes, and I have some further study to do of him, but this instrument, even if mistaken, would test my understanding and provide some avenues for further experiment, even if wrong. The question is whether I am confusing logarithmic parts of the octave with lineal divisions on the fingerboard. From my analysis so far of the several scales of Aristoxenes given by Murray Barbour in Tuning and Temperament, it appears to me that the "parts" are in fact lineal divisions of the string into units of identical length, and the design of the inlayed position markers on this instrument which I am proposing reflects that assumption.
In short, what I want to do is to find out what it means to be able to distinguish 60 tones to the octave.
The granadillo fingerboard, while fretless, will be inlaid, from the nut to where the 12th fret would be, in sixty lineal 5-millimeter increments with fossilized mammoth ivory pseudo-frets flush with the fingerboard to serve as visual markers. However, the more important harmonic nodes will be marked with turquoise pseudo-frets in order to distinguish them.
Thus the entire open string length from nut to saddle will be divided into 120 5-millimeter units, which will be marked on the fingerboard throughout its length, and the span from the nut to the octave, that is, the midpoint of the string, can be regarded as the harmonically divisible octave 120:60. The stopping of any string with any left hand finger subtracts X pseudo-fret-units from the total string length of 120 units, as many as 60 units which would raise the pitch an octave above the open string. This corresponds, I believe, to Aristoxenes's division of the octave into sixty parts. There are some ratios used in antiquity which will not be precisely defined in base-60 integers, but I believe that Aristoxenes' opinion was that any interval smaller than one-sixtieth of an octave was too small to hear, and could be fudged to the nearest sixtieth. This seems reasonable enough, provided that a musician has such special ears as to be able to distinguish those sixty parts. We're talking about 5 millimeter increments on the fingerboard.
It would not be difficult to calculate out the positions in "parts" of intervals constructed by ratios for any possible scale tuning, and some notes of these may fall in between the 5-mm-spaced pseudo-frets on the Greater Perfect Guitar (GPG); and then to make a chart of those positions with numbers like 4.6667 parts, etc. - there in fact are some specifications like this given by Barbour, see below - and to make a lookup table of that scale perhaps in a type of tablature. In a musical system of such flexibility, the physical measuring stick provided by a sixty-part lineal scale on the fingerboard accomplishes the most basic necessary task of locating the note on the fingerboard. If notes are specified in ratios such as 13:12, how are you going to find it on the fly on the fingerboard? If you have translated the ratio into parts, then you have it nailed to the closest one-sixtieth, close enough, and if you also know that that ratio should be a third of a part flat, then maybe you can accomplish that too... wow, you have special ears!
Aristoxenes' sixty parts may be compared to the twenty-two Indian shrutis, but the shrutis are variable and there is no direct comparison. According to Barbour, the whole step between disjunct tetrachords, that is between A and B in the tables below, is given a uniform 10 parts in Aristoxenes' practice. We may note that Marchetto of Padua (fl. 1305 – 1319, not to be confused with Marsilius of Padua) divided the whole tone into 5 parts, provoking the criticism of the Pythagoreans. So, if you think Marchetto's five parts of the whole tone might be hard to hear, how about Aristoxenes' ten parts? This makes me think that Aristoxenes must have been working with a stringed instrument marked with a lineal scale, much like the monochord of the middle ages; however, it appears that the preferred performance instrument of his time was the kitara, and so, another possibility is that a monochord was kept on one side by musicians strictly for a tuning device for the inner notes of the lyre or kitara.
The first of the 60 pseudo-frets on my Greater Perfect Guitar (maybe I should call it "Aristoxenes' Guitar") plays about 14.5 cents above the open string as a matter of calculation (which could be pulled seriously sharp depending on how flat the nut is), but because the frets are lineal and the relationship of lineal increments to actual pitch is logarithmic, this 14.5 cents will increase a little bit on each fret in some kind of logarithmic series. (I haven't felt it necessary to calculate this out, I only checked the first few frets to verify the increase, but I am now uneasy about that result, and will check again to see that it doesn't decrease - this is the kind of thing I have to think about more than once, because I can't always keep the logic straight.) 14.5 cents is a fairly wide musical interval in terms of precise tuning, but as it is only 5 mm on the physical fingerboard, and the fingerboard is fretless, any adjustment could be made by ear; a little bit of vibrato would help to find the sweet spot.
There are better octaves than 120:60 with which to generate more microtones - see the works of Ernest G. McClain, which deal with the hidden harmonic structures in Plato's ideal political systems, which generate far more microtones than any practical musician could deal with in performance. (There was a musician who did a TED talk demonstrating an electronic instrument with which she could play such microtones.) Theory can generate more microtones than practice can deal with; this is why Nicola Vicentino's microtonal harmony is so rarely practiced. I chose this octave in order to pursue the harmonic method of Aristoxenes, in which scale tunings are given in "number of parts", with 60 parts to the octave, rather than in ratios. An octave of 720:360, for instance, has more harmonic divisions, and can pinpoint many harmonic ratios which cannot be precisely specified within the 120:60 octave. However, to inlay 360 increments into a 30c octave on the fingerboard would mean placing 12 units per centimeter, a little less than a millimeter apart; this might be done with a computer-controlled cutting machine but all the same it appears impractical unless one worked with a much longer scale length. 60 centimeters for the scale length both supports the A4 string well, and is easily measurable and divisible by modern metric units.
Note that in order to make the following calculations, the nut represents 120 frets, and the bridge is the zero fret. The "first" fret, in common terms, is therefore number 119.
Working with the lineal division of the octave into 60 parts, one way to organize the fingerboard would be to mark the following useful and important harmonic nodes with turquoise, yielding six (or five) just intervals and three (or four) approximations to ET half-step increments between these:
| 113 | 120:113 | closest approximation to ET semitone at 104 cents | |
| 108 | 120:108 | 10:9 | small just whole step |
| or, 107 | 120:107 | approximation to ET whole step at 198.5 cents | |
| 100 | 120:100 | 6:5 | just minor third |
| 96 | 120:96 | 5:4 | just major third |
| 90 | 120:90 | 4:3 | perfect fourth |
| 85 | 120:185 | 24:17 | closest approximation to ET tritone at 597 cents |
| 80 | 120:80 | 3:2 | perfect fifth |
| 75 | 120:75 | 8:3 | just minor sixth |
| 72 | 120:72 | 5:3 | just major sixth |
| 67 | 120:67 | closest approximation to ET minor seventh at 1009 cents | |
| 64 | 120:64 | 15:8 | just major seventh |
| 60 | 120:60 | 2:1 | octave |
This is only one solution. We could also investigate Aristoxenes' own divisions, a number of which are given in Barbour's Tuning and Temperament, as transmitted by Claudius Ptolemy. Ptolemy is not a completely reliable source, as he has been demonstrated to have used theoretical calculations to obtain results of what he claimed to be his own astronomical calculations (see R. Newton on Ptolemy). This doesn't necessarily invalidate other info transmitted by Ptolemy, but it does raise the shadow of a doubt.
Barbour's own presentation is also open to question, as he gives complete octave scales, whereas I am inclined to believe that Aristoxenes probably gave only the tuning of the tetrachord, and that Barbour duplicated this to obtain an octave scale to suit his own purposes. To verify this requires further research into other sources, which I have not yet done.
Well, whatever. We work with what we have. Here are the tunings given by Aristoxenes in parts, as transmitted by Ptolemy and as reported by J. Murray Barbour on pages 16-19 of Tuning and Temperament. Barbour gives the conversions to string lengths, which are used for the actual location of the fingers. The string lengths are what correspond to the pseudo-fret layout on the GPG. Aristoxenes' "parts", in this case, correspond precisely with the number of pseudo-frets between the intervals.
Aristoxenes' Enharmonic Scale
This would be along the length of the E4 string or the E3 string. To develop fingerings across all of the strings is a matter of some simple arithemetic or possible of memorizing a pattern.| E | 120 frets | subtract 0 parts | open string | |
| Fb | 117 frets | subtract 3 parts | 44 cents | |
| Gbb | 114 frets | subtract 3 parts | total 6 parts | 89 cents |
| A | 90 frets | subtract 24 parts | total 30 parts | 498 cents |
| (Bb) | not given | |||
| B | 80 frets | subtract 10 parts | total 40 parts | 702 cents |
| Cb | 78 frets | subtract 2 parts | total 42 parts | 746 cents |
| Dbb | 76 frets | subtract 2 parts | total 44 parts | 791 cents |
| E | 60 frets | subtract 16 parts | total 60 parts | 1200 cents |
Aristoxenes' Chromatic Malakon Scale
Here, Barbour reports Aristoxenes as having given fractions of parts, which don't fit my scheme as well.One third of a part amounts to 1.66 millimeters on the fingerboard.
However, this anomaly makes it appear quite possible that Aristoxenes gave only the lower tetrachord, and that Barbour calculated the upper tetrachord himself to fill out the octave for the purpose of his own demonstration.
It appears, logically, that we could apply the lower tetrachord to two adjacent strings, and that this might actually correspond better to what Aristoxenes had in mind. It also raises the question (further research required) of whether Greek musicians used the same or possibly different tetrachords in the lower and upper part of the octave when using either conjunct or disjunct tetrachords. Maybe that's a question for Kathleen Schlesinger.
| E | 120 frets | subtract 0 parts | open string | |
| F | 116 frets | subtract 4 parts | 59 cents | |
| Gb | 112 frets | subtract 4 parts | total 8 parts | 119 cents |
| A | 90 frets | subtract 22 parts | total 30 parts | 498 cents |
| B | 80 frets | subtract 10 parts | total 40 parts | 702 cents |
| C | 77.33 frets | subtract 2 2/3 parts | total 42 2/3 parts | 761 cents |
| Db | 74.67 frets | subtract 2 2/3 parts | total 45 1/3 parts | 821 cents |
| E | 60 frets | subtract 14 2/3 parts | total 60 parts | 1200 cents |
Aristoxenes' Chromatic Hemiolion Scale
Here the fractional parts are in the lower tetrachord. This seems weird. Someday I will find a good source of whatever is known about Aristoxenes, and clarify this. Also, I note that the lower two intervals of the lower tetrachord are given in equal numbers of parts: but that because of the logarithmic pattern of the pitches in relationship to the lineal parts, the upper interval is in fact larger, as is generally true of the ratio-based tunings given by Barbour on the same pages in Tuning and Temperament.| E | 120 frets | subtract 0 parts | open string | |
| F | 115.5 frets | subtract 4.5 parts | 66 cents | |
| Gb | 111 frets | subtract 4.5 parts | total 9 parts | 135 cents |
| A | 90 frets | subtract 21 parts | total 30 parts | 498 cents |
| B | 80 frets | subtract 10 parts | total 40 parts | 702 cents |
| C | 77 frets | subtract 3 parts | total 43 parts | 768 cents |
| Db | 74 frets | subtract 3 parts | total 46 parts | 837 cents |
| E | 60 frets | add 14 parts | total 60 parts | 1200 cents |
Aristoxenes' Chromatic Tonikon Scale
This one has no fractional parts.| E | 120 frets | subtract 0 parts | open string | |
| F | 114 frets | subtract 6 parts | 89 cents | |
| Gb | 108 frets | subtract 6 parts | total 12 parts | 182 cents |
| A | 90 frets | subtract 18 parts | total 30 parts | 498 cents |
| B | 80 frets | subtract 10 parts | total 40 parts | 702 cents |
| C | 76 frets | subtract 4 parts | total 44 parts | 791 cents |
| Db | 72 frets | subtract 4 parts | total 48 parts | 884 cents |
| E | 60 frets | subtract 12 parts | total 60 parts | 1200 cents |
Aristoxenes' Diatonic Malakon
| E | 120 frets | subtract 0 parts | open string | |
| F | 114 frets | subtract 6 parts | 89 cents | |
| G | 105 frets | subtract 9 parts | total 15 parts | 231 cents |
| A | 90 frets | subtract 15 parts | total 30 parts | 498 cents |
| B | 80 frets | subtract 10 parts | total 40 parts | 702 cents |
| C | 76 frets | subtract 4 parts | total 44 parts | 791 cents |
| D | 70 frets | subtract 6 parts | total 50 parts | 933 cents |
| E | 60 frets | subtract 10 parts | total 60 parts | 1200 cents |
Aristoxenes' Diatonic Syntonon
| E | 120 frets | subtract 0 parts | open string | |
| F | 114 frets | subtract 6 parts | 89 cents | |
| G | 102 frets | subtract 12 parts | total 18 parts | 281 cents |
| A | 90 frets | subtract 12 parts | total 30 parts | 498 cents |
| B | 80 frets | subtract 10 parts | total 40 parts | 702 cents |
| C | 76 frets | subtract 4 parts | total 44 parts | 791 cents |
| D | 68 frets | subtract 8 parts | total 52 parts | 983 cents |
| E | 60 frets | subtract 8 parts | total 60 parts | 1200 cents |
************* END OF SECTION ON THE GREATER PERFECT GUITAR *************
(2) Avicenna's Guitar
A word or two about Avicenna. Known in Arabic as Ibn Sina, he was one of the great Islamic philosophers of the medieval period. His interpretations of Aristotle were a contribution to the same thread for which Thomas Aquinas is famous, that is, the attempted reconciliation of Aristotle with monotheistic thought, which was a consideration of Islamic, Christian and Jewish thinkers in the middle ages; Maimonides (c. 1135 - 1204) is the Jewish philosopher in this line. Avicenna was born about 980 CE (?) and died in 1037 CE, which makes him a contemporary of Guido of Arezzo, living about 2 and a half centuries before Thomas Aquinas (1225 – 1274).Now, Avicenna, being a polymath and one of those who would have liked to draw all branches of knowledge together into a unified system, also was interested in music, and wrote some treatises on it.
I am going to quote, below, from The New Oxford History of Music, Volume I, Ancient and Oriental Music (1957). This history is no longer new, but it contains many valuable pieces of information which are not in the more recent Oxford Histories. Chapter XI of Volume I is entitled The Music of Islam, and is by Henry George Farmer.
For those who don't know, Farmer wrote a book called Historical Facts for the Arabian Musical Influence, in which he collected all of the tidbits of information he could find on possible or real influences on European Christian music practice and theory flowing from Islamic sources. This led him into a famous controversy with the scholar of Greek Music, Kathleen Schlesinger, who said "Say it ain't so!", preferring the story that medieval European music theory was derived entirely from the classical tradition of Boethius.
Farmer writes (page 461):
"In Persia there appeared the works of the famous Avicenna, i.e. Ibn Sīnā (d. 1037), The Cure (al-Shifā), and The Deliverance (al-Najāt), which contain full information on the state of musical theory in Iranian lands, as does the Book of Sufficiency in Music (Kitāb al-kāfī fi'l-mūsīqī) of Ibn Zaila (d. 1048). Ibn Sīnā does not appear to have accepted Al-Kindī's solution of the difficulty of the 'anterior' (mujannab) fret at 90 cents by duplicating it at 114 cents.Farmer the presents the diagram which I reproduce below. He says nothing more about Avicenna's tuning and fretting scheme, and goes on to discuss others; he is just presenting a series of "historical facts". There are two important pieces of information that appear to have been suppressed or omitted in Farmer's account.
[Note by Jack: That Farmer does not give the ratios which Ibn Sīnā probably gave originally is annoying; Ibn Sīnā certainly didn't give his fret placements in cents! Note that in the tables given by Barbour as attributed to Aristoxenes by Ptolemy, the quantity 89 cents occurs frequently; Barbour defines this in his appendix on super-particular ratios as the "approximation to the Pythagorean diatonic semitone"; and that the quantity 114 cents is quite close to the 112 cents which Barbour gives for the large just semitone 16:15. The difference of 2 cents is about the same as that between the perfect and ET fifth.]
"Further, he assigns 343 cents [this appears to be the Arabic "neutral" third, about midway between the just major third, 386 cents, and the just minor third, 316 cents] for Zalzal's second finger (wustā) fret, with its 'anterior' fellow at 139 cents [ratio 13:12, a large whole tone]. Nor does he admit the 'anterior' (mujannab) fret at 90 cents, but furnishes the just semitone of 112 cents in its place. Yet he realized that the normal tuning of the lute in fourths would not produce Zalzal's second finger fret in the second octave, and to remedy this he suggested an alternative accordatura (taswiyya). By tuning the 1st string (zīr) a major third (408 cents) [i.e. a Pythagorean major third] higher than the 2nd string (mathnā), instead of a fourth (498 cents), Zalzal's intractable notes were regularized as follows: [footnote 3]
Footnote: 3 Farmer, Studies, ii, pp. 54-57; R. d'Erlanger, op. cit. ii, pp. 234-6; M. el-Hefnī, Ibn Sina's Musiklehre (Berlin, 1930), pp. 71-73 [The "opus cit." of R. d'Erlanger (1872-1932) is La Musique Arabe, 1935]
The first and most obvious is that he has given the fret placings in cents. This was probably a concession to the academic fashion of the middle 20th c., when ratio theory was not highly valued in classical music education and "cents" were considered more scientific. The loss is ours, but I have re-translated the cents values into ratios without too much difficulty; still it would be nice to know what Avicenna's own terms were.
The second (probable) omission, if it is so, requires an investigation of the original source: did Avicenna define the pitches of the strings, or even one string, in terms of the Greek Greater Immutable System? (The pitches in the graphic below A2 - D3 - G3 - B3 - E4, were added by me to illustrate the correspondence with the tuning pattern of the guitar.) This would certainly be informative.
Most secondary accounts of Ramos's 5 course "lyra" tuning of 1482 (of which I have read a number of references in standard secondary histories of music) omit the technical information which I have reproduced below, which sparse as it is, is nevertheless critical to a complete understanding of the meaning of the text and one of the few data that we have on 15th century lute practice, and I could obtain this only through Fose's dissertation, quoted below.
Farmer's Diagram of Avicenna's Tuning and Fretting Scheme:
Now, it would not surprise me if the reader doesn't recognize the significance of this. But it is the core tuning of the entire European guitar-lute-vihuela family. A slightly different version of this tuning appears in European theory for the first time as far as I am aware (and I would appreciate being informed of any earlier mention) in the Musica Practica (1482) of Bartolomé Ramos de Pareja (c. 1440–1522). Ramos described this as the tuning of a "lyra". Since he was writing in latin, he didn't have the term "lute" or "laud" (which are derived from the Arabic), and the latin "cytara" [= Gr. kitara = Sp. guitarra] was not what he had in mind, so the only instrument we can imagine matching his description is a five course lute. Graphic representations of five-course lutes are common in European art of the 15th century. Although Ramos's tuning is of a five course instrument, it puts the "wrinkle" in the tuning (the major third interval) between the 3rd and 4th courses (considering the highest-pitched string to be the first course); thus Avicenna's tuning is closer to the modern guitar than Ramos's, as Avicenna puts the wrinkle between the 2nd and 3rd courses. This is relevant to speculations about the circumstances of the birth of the six-course lute sometime after 1482.
Here I offer the relevant passage from Ramos de Pareja:
(See The “Musica practica” of Bartolomeo Ramos de Pareia: A critical translation and commentary, by Fose, Luanne Eris, Ph.D., a dissertation written at the University of North Texas, 1992, pp. 228-230)
Sunt et alia, quorum chordae sunt contrario modo dispositae, quoniam quanto digitus superpositus ad locum, in quo torquetur, appropinquat, tanto sonos reddunt graviores et e contra, ut lyra. ... |
There are also other [instruments], such as the lyre,... |
The tuning that Ramos describes first is G2 - C3 - E3 - A3 - D4. (Compare with the six-course vihuela tuning which Thomás de Sancta Maria gives, Arte de Tañer Fantasía, Book II, f123r: D2 - G2 - C3 - E3 - A3 - D4. Earlier in the Arte, Book I, f56v, TSM describes the more well-known tuning G2 - C3 - F3 - A3 - D4 - G4, but he does not mention the A-tuning used by Milan and Mudarra.) The partial tuning that Ramos describes secondly has A2 - D3 - A3 for the lowest strings, and then he omits the tuning of the two upper strings. It is interesting that compared to the common later six-course lute tunings of the 16th c., Avicenna's pattern describes the lower five strings of the lute, which are the same as the five upper strings of the modern guitar (which would indicate that the sixth course of the 16th c. was added in the treble), but here, Ramos describes the tessitura of the lower five strings of the six-course lute, but the tuning pattern of the upper five strings, indicating that the 6th string may have been added at the bass end. Avicenna's pattern is the same as that of the Baroque guitar, but Ramos's pattern is not the main tuning of the 16th century; it looks somewhat as though Ramos's pattern was moved up to the next string set when the six course lute came into use, because if you added a low D2 to Ramos's pattern, you would have the standard pattern of the lute and vihuela; furthermore that tuning beginning with the low D is given as a vihuela tuning at the end of Thomás de Sancta Maria's Arte De Tañer Fantasía of 1565 (although TSM gives a higher-pitched tuning earlier in the book.) This all begs the question of whether there was any fixed pitch then, and it is generally supposed that there was not; but there was a fixed conceptual pitch in two related forms, that of Guido's Gamut and that of the Greek Greater Immutable System of antiquity (seen on the left above), and the relationship of lute tuning to this conceptual Gamut is what we have to work with, and I don't see that it is insignificant.
Avicenna wrote of his tuning sometime before 1037 CE, and he lived in Baghdad; we find this similar one in Europe, given by a Spanish theorist in 1482, before the conquest of Granada and at a time when many Muslim musicians were undoubtedly still active in parts of Spain. Now, the time distance between Avicenna's death and Ramos's publication was 445 years. Where was this tuning hiding all that time? Hmm, maybe in the mountains of Andalusia; but the Gypsies didn't get there until Ramos's time, so it wasn't them. This is not the only odd chronological mismatch between Muslim and Christian-European music theory and philosophy; if you will read Farmer's Historical Facts for the Arabian Influence, there are quite a number of them, although their significance may not be clear until you put them all together. I will address that subject in another paper.
But now returning to Avicenna's tuning and fretting scheme, which is the real object of interest, I would like to point out again that the tuning is the same as that of the Baroque guitar, minus the Baroque guitar's re-entrant tuning. Now let's look at the fretting scheme: The first five frets of the modern guitar, which are set in a theoretical equal temperament, are included in Avicenna's fretting scheme, but in Pythagorean ratios, which are not too different in the case of the Pythagorean fourth compared to the equal tempered fourth (our fifth fret), a matter of only two cents, but in the case of the second and fourth frets (in the modern scheme) Avicenna uses Pythagorean ratios resulting in the a Pythagorean major third of 408 cents on the fourth (in our terms) fret, and a Pythagorean major second of 204 cents on our second fret, which provides a painfully sharp major third for those A and D major chords in first position. OK, OK, they didn't play chords back then. Yeah, yeah.
Now contemplate this diagram with me a little bit. We see that between the first and second fret of the modern fretting system, Avicenna has placed an additional fret at the position of the Arabic "neutral third" (as it appears on the 2nd string relative to the open A string, as in the modern A Major chord form), and that he has placed another similar fret between the third and fourth modern fret placements. Now, we all know that on many modern guitars the common garden-variety first-position A-major and D-major chords are out of tune, because the equal-tempered major third occurring on the 2nd fret in both of these chords is too sharp to begin with, and is often sharper than that on a guitar due to the stretching of the string against the nut. (And take note that with Pythagorean fret placements, they would be painfully sharp.)
So, I ask you (I assume you are a guitarist; if you are not, you won't know what I'm talking about): What if you had an extra fret between the first and second frets, that would make those two chords play much sweeter? Do you think you might like that? The same would apply to B-Major and E-Major chords barred on the second fret. You might have to squeeze your fingers together to get it; this might be addressed by adjusting the string spacing and / or the string length.
So this is why I would like to have such an instrument built once in my life, just to hear what those chords sound like.
I am not sure that I care to discuss the historical question of whether Avicenna played them. But take note: the first thing a little kid does when he picks up a guitar is to whang all of the open strings at once. Guitars were made for strumming. Is it possible that any thinking musician could work with this tuning for any length of time and not experiment with chords? A thousand years ago nobody had ever thought of whanging all the strings at once, some little kid anyway?
I had assumed, in my previous imaginative reconstruction of the history of the lute, that the instrument imported from Islam into Europe was tuned in all fourths (as are all of the other tunings given by Farmer in the same section as the Avicenna tuning) and that the change to the "wrinkled" tuning with a major third in the middle was an adaptation made by European musicians in response to the evolving European system of harmony. This seemed like an eminently sensible story, until I found this discussion of Avicenna by Henry George Farmer. Now: Historical Twilight Zone! (Twilight Zone Theme plays in background...)
I might also mention that to follow this line of thought brings up the question of doubling all the frets on the guitar with similar intermediate frets, yielding a scale of quarter-tones similar to the modern Arabian system. They might have to be color-coded, like the black keys on the piano - we'll make them out of Evo, so there will be alternating silver and gold frets. I shudder to think of the harmonic complexities.