INTRODUCTION
In attempting to demonstrate a variety of ancient tunings for my 10 string lyre to the layman with an interest in ancient music, I have decided to compromise and use modern equal temperament terminology ie. a 12 note chromatic scale, with alphabetic representation of specific pitch (eg. A,B,C,D,G,F,G) & modern musical terms such as sharps & flats to indicate the respective placement of whole-tone & half-tone steps.
However, in antiquity, lyres were tuned either cyclically, in perfect 5ths, the 3rds & 6ths then being fine-tuned by ear (Pythagorean tuning) or divisively (using exact mathematical ratios to precisely divide a musical string into specific pitch ratios) to achieve what is called "Just Intonation".
The Pythagoreans approached tuning from an almost mathematically religious orthodoxy, disregarding the musicality altogether. The Pythagoreans admitted only 3 intervallic concords because they adhered to the numbers in the ‘Tetraktys’:
The Tetractys is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row, which is the geometrical representation of the fourth triangular number. As a mystical symbol, it was very important to the secret worship of Pythagoreanism. There were four seasons, and the number was also associated with planetary motions and music.
It is said that the Pythagorean musical system was based on the Tetractys as the rows can be read as the ratios of 4:3 (perfect fourth), 3:2 (perfect fifth), 2:1 (octave), forming the basic intervals of the Pythagorean scales. That is, Pythagorean scales are generated from combining pure fourths (in a 4:3 relation), pure fifths (in a 3:2 relation), and the simple ratios of the unison 1:1 and the octave 2:1
The monochord provided the basis of the ratios of these concords as their tuning was the same as the divisions of the monochord: 2:1, 4:3, 3:2
In contrast, Ptolemy, was concerned with both musical sounds being truly consonant and that there be a scientific basis for their presence. He was probably influenced by Archytas of Tarentum, who replaced the virtually un-singable Pythagorean major 3rd (81:64) with a mathematically pure 3rd (5:4), who observed how musicians did this whilst tuning by ear and then sought a mathematical rationale for this aural phenomenon.
Almost all of my recordings since 2012, feature just intonation tunings based around Ptolemy's 'Intense Diatonic' scale.
The modern tuning system of equal temperament was devised to enable music to be performed in any of the 12 keys of the chromatic scale whilst keeping exactly the same equal ratio of pitch between each of the 12 notes of the chromatic scale - which sadly has sacrificed the essential purity of tone, which can only be heard in the various forms of just intonation outlined, once used in antiquity.
My album, "A Well Tuned Lyre; The Just Intonation of Antiquity", features just intonation (Ptolemy's Intense Diatonic) throughout - in comparison, the out-of-phase, 'jangly' sound of music in equal temperament, in comparison, is rather like a rose without it's scent...
Indeed, I have found the effects on the listener, of hearing music in authentic ancient just intonation, is an increase in the intensity of emotion conveyed through the music, a more serene effect, yet at the same time, more inspiring!
Here is a video featuring track 5 from this album - my arrangement of a fragment of ancient Greek music for solo lyre, in just intonation:
Below is a video featuring my performance of the "First Delphic Hymn To Apollo (c.128 BCE)" on my replica lyre, in the pure-sounding, authentic just intonation of antiquity :
ANCIENT JUST INTONATION VERSUS MODERN EQUAL TEMPERAMENT?
Joseph Ennis kindly sent me the following interesting article he wrote on all the technical aspects of "Just Intonation" in comparison to modern "Equal Temperament:
Another detailed article by Joseph Ennis can also be viewed in the link below, which describes the history of tuning from ancient Sumeria to the present system of equal temperament:
Below is a quote from Wikipedia, further explaining the basic difference between just intonation & equal temperament...
"In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of whole numbers. Any interval tuned in this way is called a just interval; in other words, the two notes are members of the same harmonic series.
Justly tuned intervals are usually written either as ratios, with a colon (for example, 3:2), or as fractions, with a solidus (3 ⁄ 2). Colons indicate that division is not done, so it is the preferred usage in music: In practice, two tones, one at 300 Hertz (cycles per second), and the other at 200 hertz is a perfect fifth (3:2).
Just intonation can be contrasted and compared with equal temperament, which dominates western orchestras and default MIDI tuning. Equal temperament starts by arranging all notes at multiples of the same basic interval, but the intervals themselves are altered slightly, relative to just intonation. Each interval possesses its own degree of alteration. The process results in a tuning system where all intervals will have exactly the same character in any key."
Just intonation was achieved in antiquity, by either Cyclical or Divisive tuning methods...
CYCLICAL TUNING
This is the most natural way to tune either a solo archaic harp or lyre - although first famously advocated in the writings of Pythagoras, cyclical tuning methods are well documented in ancient Babylonian Cuneiform texts several thousand years before this.
This tuning system entails tuning the strings of the lyre in a cyclical series of perfect 5ths. The 3rds & 6ths were then finely tuned by ear to the required sweetness of tone, to achieve just temperament.
Consequently, when a lyre is tuned correctly using a system of cyclic tuning, there should be no wavey, out of phase "wooowooowooow"- sounding beat waves heard in the pure intervals tuned using cyclic tuning (ie the 5ths, 4ths & octaves) - unlike a string instrument tuned using equal temperament - these "wooowooowooow" sounding beat waves are generated by the slight lack of symmetry in the sound wave ratios, due to the compromise in exact tuning - which is the unfortunate consequence of the equal temperament tuning system.
The 3rds & 6ths resulting from cyclical tuning, though, do sound similar to 3rds & 6ths tuned using equal temperament, as it is harder to precisely tune 3rds & 6ths by ear than it is to tune the purer-sounding 5ths, 4th & octaves - the 3rds & 6ths are fine tuned by ear, by a subjective process of "sweetening" their sound; a process which requires much experience, and relies solely on the ear of the musician.
Therefore, cyclic tuning results in what can best be described as a close approximation to the mathematically precise pitch ratios of pure just intonation - however, it is the ancient tuning system which requires the most skill on behalf of of the lyre or harp player, as the entire tuning process relies entirely on the experienced ear of the musician being able to tune their instrument "in tune with itself"...
Below is a recent experimental video I recorded, with my lyre in tuned by ear using the ancient Pythagorean cyclical tuning system - the video features my arrangement of "Hatikvah" (The Hope), the new Israeli National Anthem, played on my evocation of the ancient 10-string Biblical Kinnor (hand-Marini Made Harps in 70 CE, was Israel's National Instrument:
In most of my earlier albums, recorded prior to 2012, due to the constrictions of studio time (and the even tighter constrictions of my bank account to pay for it!), I reluctantly decided to eventually compromise, and use a digital tuner to tune my lyre into equal temperament. However, for my albums "The Ancient Egyptian Harp" & "King David's Harp" (which features my newly acquired, incredible quality, hand-crafted "Marini Made Davidic Harp") I decided to leave the the 21st century behind for good, and to take the time to tune my archaic arched harp & lyre cyclically, by ear...just as the ancients did!
I decided on attempting cyclic tuning, rather than divisive tuning in my first major project dedicated to experimenting with truly authentic ancient tuning systems, because as mentioned above, cyclical tuning requires the most skill to properly execute!
Professor Richard Dumbrill gives a fascinating talk in the video below, about how the Silver Lyre of Ur (dating back to c.2600 BCE) , according to the miraculously surviving Cuneiform musical texts, was also tuned cyclically - some 4600 years ago:
DIVISIVE TUNING
Cyclical tuning, as described above, since it relies on nothing but the ear of the trained musican, only result in an approximation of just intonation - pure just intonation can most precisely achieved using the sytem of Divisive Tuning. Divisive tuning entails precisely dividing the length of a monchord into specific mathematical ratios, resulting in the respective desired musical pitches being achieved.
Divisive tuning was the most natural way to tune the ancient lutes, or any fretted instrument. which uses frets to divide the vibrating portion of each string into the required precise ratio of pitches. Although described in the writings of Pythagoras in his experiments at dividing the scale via a musical monochord, the divisive tuning system predates Pythagoras by thousands of years and may have evolved along with the origin of the long-necked lute in ancient Babylonia some 5000 years ago, according to John Wheeler:
"The long-necked lute (according to Curt Sachs) was invented in Babylonia, and indeed thanks to that fact divisive tuning was invented there also. Cyclical tuning was also known there (and that got documented long after his death by the famous theory and hymn tablets from Babylon and Ugarit), but there is this curious fact: the Babylonians used divisive tuning as the basis for their symbolic correlation of the pillar degrees of the octave (e.g., C-F-G-C') with the four seasons, while the Chinese used cyclical tuning as the basis for the symbolic correlation of the same. This (wrote Sachs) is consistent as Babylon was the "home" of the lute and China the "home" of the harp (even though Babylon knew of harps and lyres too and China, if memory serves, also knew the lute from very early times). Divisive tuning is the "natural" tuning of the lute, as cyclical tuning is the "natural" tuning of the harp and lyre, according to Sachs. By that he meant that it's easiest and most natural to tune, and then to play, folk instruments of those genres that way - as I can vouch as a working musician"
Indeed, there is surviving contemporary account of the calculation of where to place frets on lutes from Nippur, in ancient Sumer, from the 3rd Millenium BCE:
"When I fix the frets on the lute, which enraptures my heart, I never damage its neck; I have devised rules for raising and lowering its intervals." Culgi, king of Urim
John Wheeler also recently explained to me (in answer to some questions I posed, also cited below in bold text) that divisive tuning was also used to tune the lyres & harps of antiquity, particularly when played in ensemble with other instruments (eg wind instruments & fretted string instruments) which were tuned divisively. John described the details of the program he used to derive MIDI files of divisively tuned chromatic scales:
"I use the 2011 edition of Finale (designed for writing musical scores that can then be played back or else converted to various file formats) to create the pitches in equal temperament, and then the free Scala program (which requires its own operating interface to be installed on Windows) to retune them. I import the resulting tuned MIDIs back into Finale, convert them into MP3s there, and the job is done. In your case I just sent you the Scala-tuned MIDIs, which are more than adequate for anyone's tuning needs. (Plus you can put them on servers that don't accept MP3s, as the one hosting my Web sites doesn't.) For my Earthlight Orchestra Web site I convert from MP3s to WMAs using Windows Movie Maker. I just corrected some long-standing flaws in the arrangement of my favorite original song, "Hey, Christopher Alan" (the strange-but-true story behind the character mentioned is here) - be sure to play the file at the bottom, if you have time, to hear a modern, original song based on pure divisive tuning and created by Finale and Scala! Like you, after hearing such music I don't want to hear equal temperament anymore...
I list the Scala program on several of my Web pages, including this one (scroll to the bottom of the page). If you're daring enough to take on Scala (be warned, it is a VERY sophisticated program, especially for someone who is only a beginner at music theory, and I've barely scratched the surface of its technical abilities), then I'll send you a file which doesn't seem to be present in their vast but overspecialized archives, the base file for creating the full divisive chromatic scale starting from any desired tonic.
> A point which is dawning on me, after finally becoming acquainted with the tuning methods of antiquity, is an inference of the consequence of cyclic tuning and the possible accompaniments that were possible on the lyre as a result - is due to the problems with fine tuning the 3rds & 6ths, simple pure intervals were presumably used for any form of basic harmony in authentic lyre playing techniques?
EXACTLY. There is no other way it could be done, using cyclical tuning. When we do see examples in archaeology (such as in the famous mural of Elamites receiving Asshur-bani-pal with harps, hammered harps, singing, chironomy, etc.) of harmony with pentatonicism, it's based on perfect fifths. I'd need to recheck my page on Egyptian chironomy and its pentatonic scale basis to see what the harmony used in the mastabas implies about the underlying tuning. It might have been tweaked to divisive tuning, not a surprise if diatonic flues and lutes were also used with the harps. But that depends on whether any intervals were used in the portrayals besides octaves and fifths.
Ah. I see that on this page I give examples of what appears to be harmony that could only be done properly were the pentatonic scale tweaked from cyclical to divisive tuning. (I also spotted an error I'll have to correct: B-F is a tritone, not a perfect fourth, and likely the interval was either B-G or B-G#, either way demanding divisive tuning as these intervals are functionally minor or major in harmony as opposed to melody, but that doesn't work out properly in cyclical tuning, only in divisive tuning.)
But Michael... who keeps telling you, yourself or someone else, that cyclical tuning is the only possible basis for "authentic lyre playing techniques"? Whoever's telling you that, he's wrong. Biblical chant demands divisive tuning, yet it was accompanied by lyres of two different kinds as well as by trumpets in the same tuning perforce. Cyclical tuning is the basis of authentic Greek style as Curt Sachs documented it (and as no doubt others have too) - it's not as easy to stop the strings in divisive tuning, because in cyclical tuning you can use the same proportionate distance string by string but you can't in divisive tuning. But there are ways of getting around that difficulty in divisive tuning and a lyre documented from 700 BC in Celtic France may demonstrate how the Israelites and others did it. You put a sort of fretbar on the yoke below the crossbar.
...And again, the best explanation of one of the Mesopotamian tablets is that it's a guide for adjusting cyclical tuning to divisive tuning by "sweetening the thirds and sixths" by ear.
Sachs (and others, often quoting Plato) points out that harmony was surely used by the Greeks (heterophony, actually: parallel melodies) but this would involve a fine ear (and maybe a tolerance of dissonance for effect, as some believe!) to use simultaneous notes using cyclical tuning if one went beyond fifths and fourths.
Although I presume the ancients understood a lot more about polyphony & harmony than the "urban myth" of the monotony of monophony in the music of the ancient world, the cylclic tuning method for tuning lyres & harps would have imposed limitations on exactly what harmonies could actually best be used?
Yes, HARMONOGRAPH points out that cyclical tuning is best adapted for monophony (as in plainchant) and accompaniment by drones (we know the Greeks and many others, including Middle Easterners, used such techniques at one time or another), while divisive tuning is best adapted for polyphony and chordal harmony. Heterophony is possible with either but how one handles it differs according to the strengths and weaknesses of each tuning; frankly divisive tuning is by far the superior for heterophony.
Are there any records of divisive tuning actually being used to tune lyres & harps in antiquity, or was this primarily used just for fretted lutes?
Yes, it appears that Egyptian chironomy, pentatonic though it was, implies divisive tuning. Again, one of the Mesopotamian tablets is best explained as a guide to retuning a lyre from cyclical to divisive tuning by ear. Anne Kilmer and Jane Smith did a paper on that, although Dr. Kilmer couldn't find a basis for inferring what temperament resulted. I pointed out to her in Jerusalem that the method makes sense (it passes a close shave with Occam's Razor) if one is seeking to sweeten the thirds and sixths as would be required in the only other documented tuning from Mesopotamia, divisive tuning. And once again, the biblical chant implies divisive tuning was used for the kinnor and nevel. Beyond that I'd have to see more than I've yet seen what's documented in artwork from the various countries and periods of history. My knowledge of such things is actually severely limited - more than you know. Don't rely overmuch on my knowledge of art history!
Unless ancient instruments had a higher string tension than often thought (and with a kinnor, nevel or kithara, that was very probably true - the kinnor and nevel played LOUD music, says your Bible), tremolo would be hard to do on them. All right: loud music implies high string tension coupled with light construction. However, in antiquity, lyres were tuned in line with top-line modern Celtic harps, the idea was to make the instrument perpetually just this side of self-destruction in order to give it maximum resonance). Ancient instrument builders and certainly what remains have been found in the Egyptian tombs (I mean actual instruments, now in the Louvre and elsewhere!) bears this out. The native Egyptian lyres were over-engineered and were soft-toned as a result. The imported Semitic kinnorot were of much lighter construction and (as Lise Manniche points out) befit the louder music that apparently became popular after their arrival."
FREE DOWNLOADS OF DIVISIVELY TUNED CHROMATIC SCALES!
As described above, John Wheeler kindly sent me these MIDI files of mathematically precise, divisively tuned chromatic scales, from which my reconstructed Biblical lyres can be tuned.
1. The Marini Made "Davidic Harp" is naturally tuned to D as the tonic note (on the second to the bottom of the 10 strings) - a divisively tuned chromatic scale starting on D can be used to tune this exquisite little lyre precisely, into any musical mode used in antiquity - the download link is here
2. The Mid East Kinnor is naturally tuned with E as the tonic (on the second to the bottom of the 10 strings) - a divisively tuned chromatic scale starting with E as the tonic note can be download here
3. For all instruments with a natural tuning as C as the tonic, a divisively tuned chromatic scale can be downloaded here
Below is a video featuring my performance of a piece of traditional Jewish Klezmer music on replica Biblical lyre, using the mathematically precise divisive tuning generated from the SCALA program which John Wheeler kindly provided me with:
As can be heard in this video, the sound of the lyre tuned in pure just intonation achieved by the mathematically precise divisive scales generated by the SCALA program, is certainly superior to the approximation of just intonation achieved via the system of cyclically tuning by ear - besides my latest release "A Well Tuned Lyre; The Just Intonation of Antiquity" I have also used divisive tuning to finally achieve the "Holy Grail" of pure just intonation, for my albums, "Ode Ancient Rome" & "Nero's Lyre".
Finally, here are two videos I put together of how I tune my own lyre into just intonation. The first video uses the just intonation tuning tones generated by SCALA for Ptolemy's intense diatonic scale:
Free MP3s of the tuning tones heard in this video can be downloaded from here
The final video features just intonation tuning tones, this time generated by the 'Cleartune' App, available from the Google Play Store, for use on any Samsung Android device, iPod or iPad. The "Pythagorean Just Intonation" setting is so far the closest match I have so far been able to find for tuning a lyre with intervals jsut about as pure as the intervals generated by SCALA, but with the advantage that with Cleartune, it is also possible to precisely tune to any mode in any key, with the option of also adjusting the reference pitch to any frequency required, be it modern concert pitch with A at 440 Hz, or any reference pitch higher or lower than this. The video demonstrates how to tune a lyre with A at 432 Hz in the Cleartune App's "Pythagorean Just Intonation" setting: