[88d] “What argument shall we believe henceforth? For the argument of Socrates was perfectly convincing, and now it has fallen into discredit.” For the doctrine that the soul is a kind of harmony has always had (and has now) a wonderful hold upon me, and your mention of it reminded me that I had myself believed in it before. Now I must begin over again and find another argument to convince me that when a man dies his soul does not perish with him.
[88d] pros emauton eperchetai: “tini oun eti pisteusomen logôi; hôs gar sphodra pithanos ôn, hon ho Sôkratês elege logon, nun eis apistian katapeptôken.” thaumastôs gar mou ho logos houtos antilambanetai kai nun kai aei, to harmonian tina hêmôn einai tên psuchên, kai hôsper hupemnêsen me rhêtheis hoti kai autôi moi tauta proudedokto. kai panu deomai palin hôsper ex archês allou tinos logou hos me peisei hôs tou apothanontos ou sunapothnêiskei hê psuchê.
[86a] divine in the well attuned lyre, but the lyre itself and its strings are bodies, and corporeal and composite and earthy and akin to that which is mortal. Now if someone shatters the lyre or cuts and breaks the strings, what if he should maintain by the same argument you employed, that the harmony could not have perished and must still exist? For there would be no possibility that the lyre and its strings, which are of mortal nature, still exist after the strings are broken, and the harmony,
[86b] which is related and akin to the divine and the immortal, perish before that which is mortal. He would say that the harmony must still exist somewhere, and that the wood and the strings must rot away before anything could happen to it. And I fancy, Socrates, that it must have occurred to your own mind that we believe the soul to be something after this fashion; that our body is strung and held together by heat, cold, moisture, dryness, and the like,
[86c] and the soul is a mixture and a harmony of these same elements, when they are well and properly mixed. Now if the soul is a harmony, it is clear that when the body is too much relaxed or is too tightly strung by diseases or other ills, the soul must of necessity perish, no matter how divine it is, like other harmonies in sounds and in all the works of artists, and the remains of each body will endure 86d] a long time until they are burnt or decayed. Now what shall we say to this argument, if anyone claims that the soul, being a mixture of the elements of the body, is the first to perish in what is called death?”
Then Socrates, looking keenly at us, as he often used to do, smiled and said: “Simmias raises a fair objection. Now if any of you is readier than I, why does he not reply to him? For he seems to score a good point. However, I think
[85e] kai ho Sôkratês, isôs gar, ephê, ô hetaire, alêthê soi phainetai: alla lege hopêi dê ouch hikanôs.
tautêi emoige, ê d' hos, hêi dê kai peri harmonias an tis kai luras te kai chordôn ton auton touton logon eipoi, hôs hê men harmonia aoraton kai asômaton kai pankalon ti kai [86a] theion estin en têi hêrmosmenêi lurai, autê d' hê lura kai hai chordai sômata te kai sômatoeidê kai suntheta kai geôdê esti kai tou thnêtou sungenê. epeidan oun ê kataxêi tis tên luran ê diatemêi kai diarrêxêi tas chordas, ei tis diischurizoito tôi autôi logôi hôsper su, hôs anankê eti einai tên harmonian ekeinên kai mê apolôlenai--oudemia gar mêchanê an eiê tên men luran eti einai dierrôguiôn tôn chordôn kai tas chordas thnêtoeideis ousas, tên de harmonian [86b] apolôlenai tên tou theiou te kai athanatou homophuê te kai sungenê, proteran tou thnêtou apolomenên--alla phaiê anankê eti pou einai autên tên harmonian, kai proteron ta xula kai tas chordas katasapêsesthai prin ti ekeinên pathein--kai gar oun, ô Sôkrates, oimai egôge kai auton se touto entethumêsthai, hoti toiouton ti malista hupolambanomen tên psuchên einai, hôsper entetamenou tou sômatos hêmôn kai sunechomenou hupo thermou kai psuchrou kai xêrou kai hugrou kai toioutôn tinôn, krasin einai kai harmonian [86c] autôn toutôn tên psuchên hêmôn, epeidan tauta kalôs kai metriôs krathêi pros allêla--ei oun tunchanei hê psuchê ousa harmonia tis, dêlon hoti, hotan chalasthêi to sôma hêmôn ametrôs ê epitathêi hupo nosôn kai allôn kakôn, tên men psuchên anankê euthus huparchei apolôlenai, kaiper ousan theiotatên, hôsper kai hai allai harmoniai hai t' en tois phthongois kai en tois tôn dêmiourgôn ergois pasi, ta de leipsana tou sômatos hekastou polun chronon paramenein, [86d] heôs an ê katakauthêi ê katasapêi--hora oun pros touton ton logon ti phêsomen, ean tis axioi krasin ousan tên psuchên tôn en tôi sômati en tôi kaloumenôi thanatôi prôtên apollusthai.
diablepsas oun ho Sôkratês, hôsper ta polla eiôthei, kai meidiasas, dikaia mentoi, ephê, legei ho Simmias. ei oun tis humôn euporôteros emou, ti ouk apekrinato; kai gar ou phaulôs eoiken haptomenôi tou logou. dokei mentoi moi chrênai pro tês apokriseôs eti proteron kebêtos akousai
Plato. Plato in Twelve Volumes, Vol. 1 translated by Harold North Fowler; Introduction by W.R.M. Lamb. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1966.
The Virtual Lyre™ for Reaktor™ Version 3.06was developed for 'Hearing Greek Microtones', an audio-demonstration of harmonic and resonant phenomena, and a paper on the history of the Greek use of the lesser superparticular (i. .e. resonant) intervals—a practice subsumed within the chromatic genus—presented at 'Performing Ancient Greek Music Today', Österreichische Akademie der Wissenschaften, Vienna, September 29-October 1, 2003. It features:
* Pretty picture of Apollo's lyre from the Delphi kylix.
* Sound based on a sample of a lyre with flax strings built by Susanna Rühling; samples of the first overtone may be accessed through modulation wheel or panel switch.
* The intonation of each white key on a midi keyboard may be set, independently, to any pitch between a fifth above and below 440Hz. These may be accessed by a midi controller, being optimized for the Midiman Oxygen 8 keyboard. Individual meters display value in both cents and Hz.
* The ratio between each pair of strings is displayed on a 7x7 matrix, and is recalculated continuously as individual strings are adjusted.
* An oscilloscope which may be adjusted for amplitude level and time, and can take snapshots.
* Toggle between equal temperament and microtonal settings.
* A synecheia ('continuity') function can automatically set the pitches of the upper strings to either a fifth (diazeuxis) or a fourth (synaphê) from the lower tetrachord.
* A midi keyboard may be constrained to the same seven pitches, regardless of the octave played. Thus one may finger a melody an octave or greater in scope (as often occurs in the Greek fragments), while the actual pitches sounded will be limited to seven.
* A series of 13 switches transposes the overall tuning up or down by semitones, to simulate the Aristoxenian tonoi.
* Presets of all tunings known from ancient Greek theorists (Philolaus, Archytas, Aristoxenus, Eratosthenes, Didymus, Ptolemy).
* Lowpass filtering with emphasis control. ADSR envelope generator.
* Toggle between lyre and aulos samples. Saw and sine wave oscillators allow a more striking and audible demonstration of harmonic and resonant relations.
* A separate lamp lights as each of the seven strings are struck.
"Microtonal intervals occupy an important place in ancient Greek music theory, beginning with Archytas and Aristoxenus, our earliest witnesses (early and late fourth century B.C. respectively). It is certain that their treatments reflect an important dimension of practice. Yet most performances today render the ancient Greek fragments with uniformly diatonic intonation. To be sure, many of the late pieces are in fact in the diatonic genus. And even for those which do exhibit the non-diatonic pykna—the ‘close set’ pitches at the bottom of a tetrachord—the notation system does not distinguish between enharmonic and chromatic, much less specify which ‘shade’ (chroa) is to be used. Furthermore, these microtonal shadings surely rank among the more culturally peculiar, and therefore elusive to the modern performer, elements of Greek practice. Nevertheless, Archytas, Eratosthenes (third century B.C.), Didymus (first century A.D.) and Ptolemy (second century A.D.) propose exact ratios for the intervals of non-diatonic systems, and even versions of the diatonic with microtonal modifications. It would be unfortunate if, in the current renaissance of performing Greek music, re-enactors continued to overlook this important material, which could bring greater life both to the fragments and to new music based on ancient principles."
"Now that we see what kind of conceptual thinking Archytas employs in geometry, we are in a position to begin investigating the musical ideas of the Pythagoreans, and other Classical Greeks, as well. In particular, we can raise the question: To what extent was their music polyphonic? Witness the way in which Archytas created the crucial breakthrough on his construction: He took a simple principle (the “Basic Construction”) and extended it in at least two different ways—the first and second “Extended Constructions.” Then, by juxtaposing the results of those two constructions in a unique way (by raising one semicircle to a vertical position, and rotating it) he developed a “cross voice” between the two, generating a relationship which otherwise would not be seen to exist. But that cross-voice principle, as a method of composition, is exactly the method which characterizes all polyphonic music. Therefore, since Archytas binds music and geometry so closely together, as areas of science, we cannot help but suspect that when he and his contemporaries talk of music, they mean polyphony.
Nevertheless, so little exists, in written form, of the music of Classical Greece, that there is no reliable, direct representation of exactly what that music sounded like. What representations do exist, have been subject to the prejudices of those who chose to translate and interpret the few writings that survive. I will therefore leave it as a challenge, to rediscover the true nature of the music which Archytas and Plato might have sung."
Regiomontánt sem követtem ám... GUICHE Tudós bolond! CYRANO
Szamárság valamennyi!
(Guiche-nek végre sikerült elhaladni Cyrano mellett. Egyenesen a ház kapuja felé tart. Cyrano utánamegy, hogy kellő pillanatban megragadhassa) Hatféleképp tudok az égbe menni! GUICHE (visszafordul) Hatféleképpen? CYRANO (folyékonyan)
Úgy van! Hallja ön:
Először is pőrére vetkőzöm, S a napra fekszem, ha ragyog a reggel, Harmattal töltött sok kristály-üveggel, Amit testemre aggatok. A nap, Amint járása mindig magasabb, A harmatot felszívja s véle megy A testem is. GUICHE (meglepetve tesz néhány lépést Cyrano félé)
Nem rossz! De még csak egy.
CYRANO (hátrál, hogy a grófot a színpad másik oldalára vezesse) A levegőt cédrus-ládába zárom. Gyujtó-tükörrel fölfogott sugáron Addig hevítem, addig ritkítom, Míg száll s a holdig meg sem áll, tudom! GUICHE (megint egy lépést tesz Cyrano felé) Kettő! CYRANO (egyre hátrál)
Mint gépész meg rakéta-mester,
Kemény acélból löveget csinálok... Lőport alája – aztán uccu, vesd el. Tüzes golyómmal az egekbe szállok! GUICHE (követi Cyranót, anélkül, hogy észrevenné és az ujjain számlál) Három! CYRANO
Ha egy gömb füsttel van tele,
Magasba röppen s én lengek vele. GUICHE (követi Cyranót fokozódó ámulattal) Négy! CYRANO
Rám kenek sok lágy ökör-velőt,
Mert Föbosz ezt gyönyörrel szívja! GUICHE (elhűlve) Öt! CYRANO (lassanként a színpad másik oldalára vezette, ahol egy pad áll)
Végül: felállok egy arasznyi vasra,
S mágnest dobok föl, mégpedig magasra. A mágnes röppen és mint egy bolond: A vonzott vas rögtön utánaront. S addig vetem föl mágnes-darabom, Amíg elérem holdam vagy napom! GUICHE Hat! – Nagyszerű! – A hat közűl pedig Melyik a legjobb? CYRANO
Mék?... A hetedik!
GUICHE Például! Lássuk! CYRANO
Nem találja el!...
GUICHE (félre) Ez a bolond már szinte érdekel! (széles, misztikus mozdulatokkal a hullámok mozdulatát utánozva) CYRANO Hu-üh! Hu-üh! GUICHE
Mi az?
CYRANO
Nem érti?
GUICHE
Nem!
CYRANO Ez a dagály!... A tündöklő egen Fönnjár a hold és vonja, vonja lágyan A halk hullámot fénylő magasában. Én fürdöm és lefekszem a homokra. Egyszer csak húznak a fejemnél fogva, - Mert legtöbb víz a hajfürthöz tapad És lebegek, mint egy szeráf-csapat, Halkan, szelíden, mindig jobban és E pillanatban egy szörnyű lökés Taszít alá... és akkor... GUICHE (feszült várakozással leült a padra)
Akkor?
CYRANO
Akkor -
(Természetes hangján) Lejárt a fertály és a szép lovag-kor! Mehet az úr! Asszonnyá lett a lány! GUICHE (egy ugrással felszökik a padról) Hah, ez a hang? Vagy álmodom talán? (A ház kapuja föltárul. Inasok jönnek ki karos gyertyatartókkal. A színpad megvilágosodik. Cyrano leveszi a fövegét) Nem! Ez az orr!... Cyrano? CYRANO (meghajtja magát)
Úgy van! Ő!...
És odabenn megvolt az esküvő! GUICHE Kié?
(Megfordul. – Csoportozat. – A libériás inasok mögött Roxán és Christian, kezüket egymás kezében nyugtatva. A kapucinus mosolyogva megy utánuk. Ragueneau szintén fáklyát tart a kezében. A menetet Duenna zárja be, esti pongyolában, ijedt arccal)
"A morális kategóriák létét a mechanikus materializmus sem vonja kétségbe, erősen emberi alapokra helyezi, s ez a görög filozófiával összecsengve az ember önmagára utaltságának felemelő heroizmusát alapozza - Kölcseynek életreszóló magatartást sugallva. Archytas ismertetéséhen ilyen önzetlen eszményt olvashatunk: "A' legfőbb jó a Virtus, melyet magáért kell kívánni"
"Pythagoras Samos szigetérõl, görög filozófus, a krotoni állam- és vallásbölcsészeti iskola megalapítója, 582. Kr. e. Elhunyt a 6. század végén. Zeneelméleti tanai, melyeket csak tanítványainak, a kanonikusok-nak (Archytas, Didymos, Eratosthenes, Euklides) írásaiból ismerünk (l. Kanonikos), alapvetõi az antik görög zene matematikai, számviszonyokon nyugvó elméletének. A húrhosszúságra, rezgésszámokra, hangviszonyokra vonatkozó kanonikus megállapítások - melyekhez a monochordon végzett kísérletek útján jutottak - máig alapvetõ jelentõségûek. A P.-féle matematikai elmélet a hangviszonyokat a quint-intervallum alapján számította (innen ma is pythagoreusi hangközöknek nevezzük azokat az intervallumokat, melyeket a quint-távolságok, a quintkör alapján állapítunk meg; pythagorasi komma pl. az a differencia, mely a 12 quint-intervallum-adta hang és az oktáv között fennáll [l. Komma] stb.); ez a számításmód egyébként nemcsak az ókori, de jóformán az egész középkori zeneelméletnek is alapjául szolgált. A iskolájával és annak sokszor absztrakt számspekulációkba veszõ matematikai módszerével szemben Aristoxenos (l. o.) és követõi, a harmonikusok a zeneelméleti kutatást tapasztalati és akusztikai alapokra (hangmagasságok különbségei) igyekeztek fektetni s elvitatták a számviszonyok elhatározó fontosságát (l. Harmonikos). Pythagoras zakynthosi, a Kr. e. 6. században mûködött görög zenész. Nevéhez fûzõdik a hármas, azaz három hangrendszerben (dór, fríg, lyd) hangolt kittharának, a triposnak megszerkesztése. "
"A mérések közül Archytas-nak van legnagyobb érdeme. Látszik, hogy Aristoxenos szimplex hangközelméletéhez képest pontosabb mérésnél (és a gyakorlatban is) pl. a normális diatonikus tetrachord valóságos osztása ingadozó volt, a két egészhang váltakozó nagyságú volt, maga a félhang sem maradt eszerint mindig egyazon értéknek, továbbá, hogy a második egész hangköz volt többnyire a nagyobbik (8:7). "
’ArcutaV, Tarentumból, Pythagorast követő bölcsész, jeles matheamtikus, kiről Aristoteles és Aristoxenus külön műveket írtak. Athen. XII, 545. a. Diog. Laërt. V, 25. Élt a Kr. e. IV-ik század első felében 400–365 tájt. Hétszer volt hazájának hadvezére. Érdekes Platóval való összeköttetése, melyre vonatkoznak a Diog. Laërt. III, 22. VIII, 80. idézett apocryphus levelek. Horatius egyik ódájában (Carm. I, 28) motivumul használja fel azt a mondát, hogy az apuliai vagy calabriai hegyfok Matinum közelében hajótörést szenvedett és itt lenne eltemetve. Önuralmát magasztalja Val. Maximus IV, 1. Cicero Cat. m. XII, 39. tusc. d. IV, 78. A tudományos mechanika megalapítója, több geometriai tétel feltalálója (két félhenger viszonya a koczka kétszereséhez); a zenében a három hangnemben a hangok mathematikai viszonyát pontosan megállapítá. A gyakorlati gépészetben való jártasságának bizonyítékául hozzák föl egy automata repülő galamb feltalálását. Gell. X, 12. Irodalom: Töredékeit a többi pythagoreuséval együtt kiadta Mullach (fragm. philos. Graecorum I, 2. 1860–1868). Hartentstein: de A. Tar. frg. philosophicis, Lips. 1833. F. Beckann: De Pythagoreorum reliquiis Berlin, 1844. 1850. A. E. Chaignet: Pythagore et la philosophie pythagoricienne, Paris 1873. H. I.
Archytas of Tarentum was a mathematician, statesman and philosopher who lived in Tarentum in Magna Graecia, an area of southern Italy which was under Greek control in the fifth century BC. The Pythagoreans, who had at one stage been strong throughout Magna Graecia, were attacked and expelled until only the town of Tarentum remained a stronghold for them. Archytas led the Pythagoreans in Tarentum and tried to unite the Greek towns in the area to form an alliance against their non-Greek neighbours. He was commander in chief of the forces in Tarentum for seven years despite there being a law that nobody could hold the post for more than a year. Plato, who became a close friend, made his acquaintance while saying in Magna Graecia. Heath writes in [4]:-
... he is said, by means of a letter, to have saved Plato from death at the hands of Dionysius.
In fact Plato made a number of trips to Sicily and it was on the third of these trips in 361 BC that he was detained by Dionysius II. Plato wrote to Archytas who sent a ship to rescue him. For more details on the relationship between Archytas and Plato consult the interesting article [8].
Given the above story and the conclusion that Archytas came after Socrates, it may seem strange to include him in works on pre-socratic philosophers as is done in [3]. This is done, however, because of the style of Archytas's philosophy rather than the strict chronology.
Archytas was a pupil of Philolaus and so was a firm supporter of the philosophy of Pythagoras believing that mathematics provided the path to the understanding of all things. Although Archytas studied many topics, since he was a Pythagorean, mathematics was his main subject and all other disciplines were seen as dependent on mathematics. He claimed that mathematics was composed of four branches, namely geometry, arithmetic, astronomy and music. He also believed that the study of mathematics was important in other respects as a fragment of his writings that has been preserved shows (see [3] or [6]):-
Mathematicians seem to me to have excellent discernment, and it is not at all strange that they should think correctly about the particulars that are; for inasmuch as they can discern excellently about the physics of the universe, they are also likely to have excellent perspective on the particulars that are. Indeed, they have transmitted to us a keen discernment about the velocities of the stars and their risings and settings, and about geometry, arithmetic, astronomy, and, not least of all, music. These seem to be sister sciences, for they concern themselves with the first two related forms of being [number and magnitude].
This fragment comes from the preface to one of his works which some claim was entitled On Mathematics while others claim that it was entitled On Harmonics. Certainly, coming after this quote, there is a discussion of pitch, frequency and a theory of sound. It does contain some errors but it is still a remarkable piece of work and formed the basis for the theory of sound in the writings of Plato.
Archytas worked on the harmonic mean and gave it that name (it had been called sub-contrary in earlier times). The reason he worked on this was his interest in the problem of duplicating the cube, finding the side of a cube with volume twice that of a given cube. Hippocrates reduced the problem to finding two mean proportionals. Archytas solved the problem with a remarkable geometric solution (not of course a ruler and compass construction).
One interesting innovation which Archytas brought into his solution of finding two mean proportionals between two line segments was to introduce movement into geometry. His method uses a semicircle rotating in three dimensional space and the curve formed by it cutting another three dimensional surface.
We know of Archytas's solution to the problem of duplicating the cube through the writings of Eutocius of Ascalon. In these Eutocius claims to quote the description given in History of geometry by Eudemus of Rhodes but the accuracy of the quotation is doubted by the authors of [10].
Another interesting mathematical discovery due to Archytas is that there can be no number which is a geometric mean between two numbers in the ratio (n+1) : n. The most interesting thing about his proof is that it is close to that given by Euclid many years later, and also that it quotes known theorems which would later appear in Euclid's Elements Book VII.
The arguments just given led van der Waerden to claim (see for example [5]) that many of the results which appear in Book VII of the Elements predate Archytas. Clearly, he claims, there were some works, written many years before Euclid wrote the Elements, which covered the same material. Archytas built on this earlier work and his discoveries are then largely those presented by Euclid in the Elements Book VIII. Following these arguments of van der Waerden it is now widely accepted that Euclid borrowed Archytas's work for Book VIII of the Elements.
Archytas is sometimes called the founder of mechanics and he is said to have invented two mechanical devices. One device was a mechanical bird [2]:-
The bird was apparently suspended from the end of a pivoted bar, and the whole apparatus revolved by means of a jet of steam or compressed air.
Another mechanical device was a rattle for children which was useful, in Aristotle's words (see for example [4]):-
... to give to children to occupy them, and so prevent them from breaking things about the house (for the young are incapable of keeping still).
This does seem a remarkably modern thought for an inventor in 400 BC! In fact this interest in applying mathematics is in contrast to the pure mathematical ideas of Plato and this contrast formed the basis for a poem written by the Polish author C K Norwid (1821-1883). This fascinating poem is discussed and given in French translation by Marczewski in [9].
Simplicius, in his Physics, quotes Archytas's view that the universe is infinite (in Heath's translation [4]):-
If I were at the outside, say at the heaven of the fixed stars, could I stretch my hand or my stick outward or not? To suppose that I could not is absurd: and if I can stretch it out, that which is outside must be either body or space (it makes no difference which it is as we shall see). We may then in the same way get to the outside of that again, and so on, asking on arrival at each new limit the same question; and if there is always a new place to which the stick may be held out, this clearly involves extension without limit. If now what so extends is body, the proposition is proved; but even if it is space, then, since space is that in which body is or can be, and in the case of eternal things we must treat that which potentially is as being, it follows equally that there must be body and space extending without limit.
When it came to a philosophy of politics and ethics, again Archytas based his ideas on mathematical foundations. He wrote (see for example [3] or [6]):-
When mathematical reasoning has been found, it checks political faction and increases concord, for there is no unfair advantage in its presence, and equality reigns. With mathematical reasoning we smooth out differences in our dealings with each other. Through it the poor take from the powerful, and the rich give to the needy, both trusting in it to obtain an equal share...
Finally we quote again from the writings of Archytas about his theory of how to learn. The fragment appears in [3] or [6]:-
To become knowledgeable about things one does not know, one must either learn from others or find out for oneself. Now learning derives from someone else and is foreign, whereas finding out is of and by oneself. Finding out without seeking is difficult and rare, but with seeking it is manageable and easy, though someone who does not know how to seek cannot find.
Georgiosz Lykouras, igen erdekfeszito kutatasokat vegez a gorog zenematematikarol.
"The Musical Geography of Delphoi G.Lykouras
In this article the ancient musical rations, that is the musical fractions stil alive in the musical tradition of Greece, the Balkans and Eastern Mediterranean, are related to the geographical data of ancient sites. Thus, the musical intervalic constans also from geographical constans and represent the marginal relation between day and night during the summer solstice, which was determining the geographical latitude of each location. For example, for Babylon this relation was 3/2 (there parts day and two paerts night during tehe solstice of June 21st), for Cyzicus 5/3 (15 hours day and 9 hours night), while for Alexandria it was 7/5 ( 14 hours day and 10 hours night). Furthermore, it is quite probable that already since the second milleneum BC it was known the ratio of the perfekt 'fourth' 4/3, wich reperesents the geographical latitude of the Egyptian Thebes. The ratio between day and night for Delphi, having a geographical latitude 38[a fok jelet nem lelem ide!] 35', coincides with the harmonic ratio of the golden mean, wich justifies the name of the site 'omphalos". In Delphoi the duration of day on June 21st is 14 hours and 50' and that night 9 hours and 10'. This atio, which also represents the symbolism Apollo-Dionysus, is the golden ratio (1.61808...). An analogous interpretation can be applied to the meridian important ancient sites, on the basis of the difference of sunrise and sunset in two locations. In this way the position of Delphi in the omphalos of Europe is warranted, since its location is in reality the golden mean of the solar distance between the Atlantic of the Hesperides and Atlas and the Caucasus of Prometheus.( Archeologia kai technes 86.)
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The Virtual Lyre™ for Reaktor™ Version 3.06was developed for 'Hearing Greek Microtones', an audio-demonstration of harmonic and resonant phenomena, and a paper on the history of the Greek use of the lesser superparticular (i. .e. resonant) intervals—a practice subsumed within the chromatic genus—presented at 'Performing Ancient Greek Music Today', Österreichische Akademie der Wissenschaften, Vienna, September 29-October 1, 2003. It features:
"Microtonal intervals occupy an important place in ancient Greek music theory, beginning with Archytas and Aristoxenus, our earliest witnesses (early and late fourth century B.C. respectively). It is certain that their treatments reflect an important dimension of practice. Yet most performances today render the ancient Greek fragments with uniformly diatonic intonation. To be sure, many of the late pieces are in fact in the diatonic genus. And even for those which do exhibit the non-diatonic pykna—the ‘close set’ pitches at the bottom of a tetrachord—the notation system does not distinguish between enharmonic and chromatic, much less specify which ‘shade’ (chroa) is to be used. Furthermore, these microtonal shadings surely rank among the more culturally peculiar, and therefore elusive to the modern performer, elements of Greek practice. Nevertheless, Archytas, Eratosthenes (third century B.C.), Didymus (first century A.D.) and Ptolemy (second century A.D.) propose exact ratios for the intervals of non-diatonic systems, and even versions of the diatonic with microtonal modifications. It would be unfortunate if, in the current renaissance of performing Greek music, re-enactors continued to overlook this important material, which could bring greater life both to the fragments and to new music based on ancient principles."
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