source file: mills2.txt Date: Thu, 14 Nov 1996 09:00:53 -0800 Subject: Post from Brian McLaren From: John Chalmers From: mclaren Subject: Technology drives tuning -- Having made this claim, it behooves me to offer proof. We read in Boethius that Pythagoras discovered the relationship of fifth and octave by listening to a blacksmith's hammers. The weights of the hammers purportedly produced different pitches when the blacksmith smote the anvil. This is of course pure fantasy. Hammers of different weights striking an anvil give off the same tone at different volumes. It is the bell, not the clapper, which produces a tone, and the pitch of the tone is not determined by the mass of the clapper but by the mass and shape of the bell. (Bells have recently been designed by computer to sound a major chord rather than a minor chord.) In any case, simply doubling the weight which hangs at the end of a string would produce a pitch in the ratio of 1.414 to the original note (Pythagoras was supposed to have rushed home and performed this experiment; obviously he did not, nor did Boethius. Galileo's father did, and published the results.). Doubling anvil weights produces an interval not of an octave but a tritone. As it happens, vibrating metal bars behave differently from strings, and so the tale about Pythagoras is a myth. However, this tall tale shows pretty clearly the level of technology available in classical Greece. Pythagoras didn't rush home and try out his theory on a keyboard instrument or a zither with 88 different strings at high tension because the Greeks didn't have the technology to build such instruments. For one thing, you have to know a good deal about metallurgy to produce tough wires as would be needed in a harpsichord or zither or piano; for another, the Greeks didn't have the technology to machine metal as is required to produce metal screws, iron frameworks for pianos, etc. What sort of intonation system is most appropriate for a technology limited to lyres with only a few gut strings and some wind instruments? Because gut changes its tension with humidity and because it must be kept at low tension lest the gut strings break, the intonation best suited for such technology would use only the simplest and most obvious members of the harmonic series. Plucked gut strings produce harmonic series timbres, so the second and third members of the harmonic series would be most useful in tuning such instruments. Because of the extremely low tension of a tortoise-shell lyre (try and add a lot of strings or tune them to high tension and either the back of the lyre will collapse or the gut strings will snap), the sounds will be faint and only the very lowest harmonics will be audible. The 5th harmonic and higher harmonics would probably have been extremely faint on a plucked gut string, too faint to tune to. Thus the technology of classical Greek music lent itself primarily to Pythagorean tuning. The tuning of auloi is not so limited; however, it remains unclear what the Greek auloi were. Some authors claim they were similar to oboes, in which case they would demand a tremendous amount of air. Other authors claims the Greek auloi were more flute-like, in which case the sound would have been much louder and the players less sorely taxed. However, even today it's impossible to calculate precisely the position of tone holes for simple wood instruments theoretically; the theory never matches the actual position. In classical times, the position of tone holes would have been a matter of guesstimation, and this would have necessarily limited the intonational complexity of Greek auloi. Net result? Pythagorean tuning was a technological necessity for the Greeks, as it was for the bablyonians and the Sumerians. These latter recorded a Pythagorean tuning on tablet U7/80 (Side 2) in the British Museum; it is almost certain that the tall tales about Pythagoras mask an intonational tradition which drifted from the older civilizations of Egypt and the Euphrates valley to the newer civilizations of the Mediterranean. -- Inventions were common in the classical era of ancient Greece: Ktesibos of Alexandria build a device to produce "intermittant bird song" around 270 B.C. It worked by regulating the flow of water into a closed chamber. But such devices were very limited in their musical utility because the air pressure was low and so the "bird songs" would have been extremely faint--and when the chamber designed to catch the streamof water filled up, the sound would have stopped. Ktesibos solved this problem by using a double-barreled water pump he had devised to fight fires--he modified this pump to create a continuous source of compressed air. The roman author Hero reports around the first century B.C. that Ktesibos used three components to build his hydraulis (water organ): a single-cylinder air pump, a large cistern filled with water, and a smaller vessel attached to the bottom of the cistern to act as a regulator to keep constant the rate of flow of water out of the cistern. This organ used sliders moving in and out of slots below each pipe; the sliders were controlled by the keys of the keyboard and when a player pressed a key, the hole in the slider aligned with the opening in the corresponding pipe. By spring action, the keys recoiled, dragging each slider back to its closed position. Notice several problems with this organ. First, it must have required at least as much force to depress a key as would be required to drag each slider back into its closed position. Second, the organ can play only as long as the cistern contains water. Third, the size of the cistern and its height determine the maximum available air pressure and thus the total number of pipes and the maximum volume of the sound. But the biggest problem is that as more and more keys were depressed, the air pressure would drop because the regulator at the bottom of the cistern would prohibit water from flowing at more than a certain maximum rate from the reservoir. This means that if more than one key was depressed, the overall air pressure of the organ would drop and the overall pitch of all notes would fall. It would not have been possible to remedy this by eliminating the regulator, since its purpose is to maintain even air pressure--otherwise there would initially be high air pressure as a great mass of water started to press down on the pump at the start and the air pressure would continually fall as the mass of water in the cistern continually lessened. Thus Ktesibos' organ would not have been useful for performing with other instruments, since its pitch changed as more keys were depressed. Also, it could only play for a short time, and the creaking of the wooden pump and the burble of water pouring out of the cistern would have made the instrument hard to hear. As a result, the hydraulis was only a novelty item. Even so it impressed contemporaries: Athenaeus describes a feast at which the hydraulis was discussed: "The sound of the hydraulis was heard close by. So pleasant and charming was it that we all turned towards the sound, fascinating by the harmony." The reaction here hints at the surprise and shock Alexandrian citizens must have felt at hearing sustained chords. This was clearly alien to their experience. It is reasonable to assume that the hydraulis used Pythagorean intonation; remember that the pitch changes as more keys are depressed. This would make an elaborate tuning system very hard to tune up. Even Pythagorean was probably only roughly approximated on such organs. The Romans used such instruments at the arena; the oldest archaeological remains of an organ were unearthed at Aquincum (near currest-day Budapest), dedicated in A.D. 228 to the college of weavers there. Having listened to the hydraulis, the Pythagorean Philolaus proclaimed: "The nature of number and harmony admits no falsehood... But in fact number, fitting all things into the soul through sense-perception makes them recognizable and comparable with one another." This is a fine statement of the Pythagorean conception of the universe as an expression of pure theoretical math. Music was unpopular with the early leaders of Christianity. Divine revelation was preferred to the study of nature. Early Christian thinkers did not have much interest in the application of logic and the study of physical evidence (an attitude represented on this forum by Greg Taylor); instead, they preferred the mystic contemplation of sacred verse, from which music proved an unwanted distraction. Saint Augustine wrote in the early 5th century that he found music in any form suspect, but allowed as how "now when I hear sung in a sweet and well-trained voice those mleodies...I do, I confess, feel a pleasurable relaxation. But this bodily pleasure to which the mind should not succumb without enervation, often deceives me.... In these matters I sin without realizing it." The message is clear: to the early Christians, beautiful music was a sin. (This is an attitude remarkably similar to that of many academic music theorists of the modern day.) -- The first organ to reach Western Europe after the sack of Rome in 476 was a gift from the Byzantine emperor Constantine V to the Frankish king Pepin in the year 757. The gift excited amazement because its like had not been seen for hundreds of years--a clear indication of how much knowledge and scientific thought could be lost forgotten and how badly the capacity for clear thinking could erode during the Dark Ages (a fate which awaits us all if we follow the prescriptions of the Eric Lyons and the Greg Taylors of the world). Ermold le Noir wrote an epic in which he proclaimed "Even the organ, never yet seen in France, which was the overweening pride of Greece and which in Constantinople was the sole reason for them to feel superior to us--even that is now in the palace of Aix." The organ was slowly transformed into an engine of divine worship in the churches--this took a while, given the attitude of Augustine: "Whatever knowledge man has acquired outside of Holy Writ, if it be harmful it is there condemned; if it be wholesome, it is there contained." (An attitude remarkably similar to that of contemporary music professors, save that their Holy Writ is Schoenberg's "Harmonielehre" and John Cage's "Silence.") Organs were increasingly optimized for volume. By the 990s this led to what Wulstan described as "Like Thunder, the strident voice assails the ear, shutting out all other sounds than its own; such are its reverberations, echoing here and there, that each man lifts his hands to stop his ears, unable as he drawn hear to tolerate the roaring of so many different and noisy combinations." Clearly volume came at the price of intonational precision--the "noisy combinations" surely describe the effect of an unsteady air-flow on the pitches of the individual pipes. This organ (like most around the 900s) did not seem to have been used for music so much as to amaze and shock the crowd and entice them into attending church services. By the 12th century, organs had been accepted into the church in a feat of intellectual jiu-jitsu similar to Thomas Aquinas' introduction of Aristotle. By this time the organs clearly had worked up high air pressure, though the steadiness of their intonation was probably still poor: Saint Aelred, abbot at Rievaulx in Yorkshire, wrote "What use, pray is this terrifying blast from the bellows that is better suited to imitate the noise of thunder than the sweetness of the human voice..." This quote indicates that the organs were now using bellows--in fact banks of them, one for each pipe, with serfs treading on them in time to the music. This would have greatly increased air pressure, but it required the serfs to tread in lockstep and more to the point the air pressure would still change over the course of a note as the bellows emptied. The initial higher air pressure would, ironically, have produced a more drastic drop in the pitch of each note while it sounded. Moreover, the notes could not sound for a very long time--only as long as it took the bellows feeding air to that pipe to empty. The overall effect would have been of a set of notes which dropped in pitch as they were sounded and which would have had to be played in strict robotic meter; however, the problem of polyphony changing the overall pitch of the organ had been solved, and the organs of the 12th century would have sounded much louder than that of Ktesibos. Moreover, these 12th-century organs still didn't have keyboards. They were played by ramming blocks of wood forward and back to open up and cut off the flow of air into each pipe. Given the size of the pipes, this would have been a real workout. Some time between the 11th century and the 14th century, true keyboards appeared. These were spring-loaded, like Ktesibos' keys. They still had to be bashed with the fist--but they could now be played more musically. Given the persistent problems with changing pitch and lack of any kind of real keyboard, Pythagorean intonation was still used into the 12th century according to the organ- building manuals of that period--even though modern keyboards had started to evolve. However, by the 14th century small portable foot-pumped organs were starting to appear. Henri Arnaut published the best suriviving text on building medieval organs in 1450; around this time the single greatest innovation in musical technology between 100 B.C. and 1800 A.D. was introduced--the multiple-chamber bellows. Water-operated organs were clumsy because they demanded a source of water and they could only play for a limited time; bellows were better because they could be pumped relatively silently (I've played some of these portatives and you can't hear the bellows). Adding a second chamber onto the bellows produced constant air pressure. The second inner chamber of the bellows had an aperature into which air could be forced but could exit except through the organ pipes. Thus, even though the pressure of the primary bellows constantly changed as it was pumped, the secondary chamber maintained a relatively constant air flow. Around this time Napier also introduced the logarithm, making possible calculations which treated musical intervals as portions of the octave which could be added and subtracted rather than as messy complex grade-school fractions which had to be multiplied and divided. These two advances had an explosive impact on intonation. Within a few generations of the late 1400s, the Pythagorean intonation was no longer in widespread use (though it was still taught in music theory--much as 12-TET is still universally taught today even thought modern composers are using it less and less). Organs with large numbers of pipes became common. Moreover, serfs no longer needed to tread in strict time on sets of bellows. By adding a secondary chamber, all the pipes could be connected to a single bellows and as long as it was large enough, the air pressure would be sufficient that no matter how many keys were depressed (within some reasonable limit) the overall air pressure inside the inner chamber (after the secondary bellows) wouldn't change. This not only allowed composers and performers to explore much wilder and less regular rhythms, it also allowed more elaborate intonational schemes than 3-limit just, and it made possible the exploration of complex polyphony with many notes of stable pitch sounding all at once. With more notes available on the organ keyboard, the possibility of modulation is correspondingly greater. Between the early 1500s and the middle 1700s this increasing use of modulation by composers would have made various meantone systems particularly popular. Indeed, Mark LIndley claims that the early English virginal piece "Ut, Re, Mi, Fa, Sol, La" by John Bull (written in the late 1500s) used 1/3-comma meantone. Bull was a wild-eyed avant garde composer, the Stockhausen of his time, and this sounds reasonable given Bull's penchant for pushing the outside of the musical envelope. The next post concludes this examination of technology's effect on tuning. --mclaren Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 14 Nov 1996 18:24 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA08898; Thu, 14 Nov 1996 18:25:54 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA08919 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id JAA08110; Thu, 14 Nov 1996 09:25:47 -0800 Date: Thu, 14 Nov 1996 09:25:47 -0800 Message-Id: <83961114165538/0005695065PK3EM@MCIMAIL.COM> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu