amaya garcía - ptolemy, pipes and shepherds

Proceedings of Crossroads Conference 2011 – School of Music Studies, A.U.Th. / I.M.S.

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Ptolemy, pipes and shepherds

Amaya Sara García Pérez Departamento de Didáctica de la expresión musical, plástica y corporal, Universidad de Salamanca, Spain

[email protected]

Abstract. The diatonon homalon is one of Ptolemy’s tetrachordal divisions. Due to its mathematical properties and to the special treatment it deserves in Ptolemy’s treatise, it has looked to many scholars just like an arithmetic speculation of the author. It has been related, by others, to some older descriptions of actual musical tunings, to be found, for example, in the writtings of Aristoxenus. But Ptolemy himself makes remarks about it that may show he actually heard it in existent musical practice. In this paper we will discuss all previous theories about the origin of Ptolemy’s diatonon homalon, and we will present a new hypothesis that would connect it with real musical praxis. As we will show, this tuning system could be related to tuning schemas used nowadays in Iberian traditional three-‐hole pipes. This connection can help us to better understand the real place of the diatonon homalon in Ptolemy’s treatise and in the music of his time.

1. Ptolemy’s homalon diatonic. A mathematical speculation?

In the 2nd century AD, the great Hellenistic astronomer and music writer, Ptolemy, wrote a famous music treatise in which he describes various musical tunings. To do so, he explains different possible divisions of the tetrachord (the perfect forth), which are described in the most common way of the time: through mathematical ratios between string lengths.

Among the tetracordal divisions of Ptolemy we can find a special type of diatonic genus called homalon diatonic1 (translated by different authors as “even”, “equal” or “equable” diatonic), whose mathematical ratios are (from low to high): 12/11, 11/10, 10/9. According to Ptolemy, this genus can be used in two disjunct tetrachords, forming an octave system. Thus, the whole homalon diatonic octave system would be (table 1): String lengths Notes Ratios Cents2 18 nete 10/9 182 20 paranete 4/3 11/10 165 22 Trite 12/11 151 24 Paramese 9/8 9/8 204 27 mese 10/9 182 30 lichanos 4/3 11/10 165 33 parhypate 1 2/11 151 36 hypate

Table 1. Homalon diatonic as described by Ptolemy (ratios and string lengths) and put into cents.

1 Ptolemy, «Harmonics», in Greek Musical Writings: Harmonic and acoustic theory, vol. II, trans. Andrew

Barker (Cambridge: Cambridge University Press, 1989), 311–312. 2 A cent is the 1200th part of an octave, the 100th part of a tempered semitone.


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In this shade of diatonic, the ratio of the perfect forth (4/3) is directly divided by two means of an arithmetic proportion, which results in three musical intervals close in size. The name homalon (even), given by Ptolemy to this system, arises precisely from this characteristic of its intervals.

Ptolemy introduces the discussion of the homalon diatonic when he is talking about the evenness and pleasentness of the tense chromatic. This chromatic genus arises from the division of the tetrachord with an arithmetic mean into two almost equal intervals (8/7 and 7/6) and then further dividing the pyknon (the smaller interval 8/7) into two new intervals. According to Ptolemy, the evenness that arises from the arithmetic division of the forth is what gives the tense chromatic its specially agreeable quality. Ptolemy continues explaining that the sweetness that arises from the evenness of the tense chromatic made him investigate whether a diatonic made out of three almost equal intervals would also be appropriate, in other words, it gave him the idea of directly dividing the tetrachord into three almost equal intervals, inserting two arithmetic means in the original 4/3 ratio. Doing so, he gets the homalon diatonic, where that evenness is found in the three ratios of the intervals of the tetrachord and is extended further when the ratio of the disjunctive tone (9/8) is placed above them.

Let us see how Ptolemy presents it:

In the segmentation of the whole tetrachord into two ratios, it [the tense chromatic] is defined by the ratios that are nearest to equality and are consecutive, that is, by the ratios 7:6 and 8:7, which divide in half the whole difference between the extremes. For the reasons given, then, this genus seems most agreeable to the ears.

It [the evennes and pleasentness of the tense chromatic] also suggests to us another genus, when we set out from the melodicness that is consituted in accordance with equalities, and investigate the question whether there is any appropriate ordering of the tetrachord when it is initially divided into the three nearly equal ratios, again in equal excesses. The ratios comprising this sort of genus are 10:9, 11:10, 12:11, [...]. There arises a tetrachord close to the tense diatonic, and more even than it, both in itself and still more in association with the filling out of the fifth. For when the disjunction, which makes an epogdoic ratio [9:8], is conjoint with the “leading” note, the characteristic of equality is no longer produced only in the three excesses, but in the four that are contained by the succesive ratios from the epogdoic to the ratio 12:11. The first numbers that make this kind of octave, when the disjunction is placed in the middle, are 18 and 20 and 22 and 24 and 27 and 30 and 33 and 36. When a division is taken in strings of equal pitch on the basis of these numbers, the caracter that becomes apparent is rather foreign [xenikoteron] and rustic [agroikoteron], but exceptionally gentle, and the more so as our hearing becomes trained to it, so that it would not be proper to overlook it, both because of the special character of its melody, and because of the orderliness of the division. Another reason is that when a melody is played in this genus by itself, it gives no offensive shock to the hearing, which is true, pretty well, only of the intermediate one of the other diatonics, the others being attuned by forcible constraint when taken by themselves, but capable of being succesful in a mixture with the diatonic just mentioned [...] So let us call this genus the homalon (even) diatonic, from the characteristic it has.3

3 Ptolemy, «Harmonics», 311–312.


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This explanation of the origin of the homalon diatonic has made several scholars think that, as Ptolemy himself recounts, this genus is just a mathematical speculation which, when confronted to the ear, brings out an agreeable sensation.4 But in order to be able to evaluate Ptolemy’s homalon diatonic we must take into account some further considerations:

The first thing we must consider is that Ptolemy’s procedure to present his different divisions of the tetrachord follows a strict method. In fact, we could talk of Ptolemy’s using a scientific method to “discover” all the possibilities of tetrachordal divisions. An acceptable tetracordal division must follow certain rational principles (hupotheseis) and, at the same time, must be audibly acceptable.5 Reason and perception are in fact two faces of the same reality and must agree. Perception gives more general considerations, while reason is much more accurate and is the only way to give certainties. But, and here is the interesting thing for us, the homalon diatonic does not follow Ptolemy’s regular procedure to get his tetrachordal divisions. It does not follow his hupotheseis, although, as Ptolemy himself exposes, it is mathematically beautiful and perception approves it.

Secondly, after presenting all his possible divisions of the tetrachord (divisions that match both reason and perception), Ptolemy discusses which ones are actually used in real musical practice. He gives diferent kinds of octave tunings which are actually used in lyres and kitharas, the only instruments discussed by Ptolemy. In these instruments, most octave tunings are made out of two diferent types of genera. In other words, most genera cannot be used by themselves to get a complete octave system in kitharas or lyres. The exception is the tense diatonic, which is used by itself. The homalon diatonic does not appear among the possible practical tunings of kitharas and lyres treated by Ptolemy. But at the same time, as we saw earlier, Ptolemy recognises that the homalon diatonic can be used by itself in an octave system.

As we said earlier, these evidences have made several scholars think that the homalon is a rational, theoretical possibility, which perception approves, although it does not exist as an actual practical tuning. This idea has been clearly defended by Barker, who thinks that the homalon diatonic did not exist in actual musical practice, nor was it described in any earlier treatise, and therefore it was just a theoretical invention of Ptolemy’s mathematical thinking. Barker’s arguments are the most consistent and representative of this doctrine about the genus, so let us present them. In Barker’s words:

4 The idea of the homalon diatonic being just a mathematical speculation was suggested by R. P.

Winnington-‐Ingram, «The Spondeion Scale», The Classical Quarterly 22, no. 2 (1928), 83-‐91, http://journals.cambridge.org/action/displayAbstract;jsessionid=865B14190F5D2C5CCFD07A353A18E8AA.tomcat1?fromPage=online&aid=3573076., and more recently has been very cleverly presented by Andrew Barker, Scientific method in Ptolemy’s Harmonics (Cambridge: Cambridge University Press, 2000). We can find the same idea in Pedro Redondo Reyes, La Harmónica de Claudio Ptolomeo: edición crítica con introducción, traducción y comentario, Ph. D. (Universidad de Murcia, Spain, 2002). I myself had that opinion when I first encountered Ptolemy’s theories, as I wrote down in my doctoral dissertation, published as: Amaya Sara García Pérez, El concepto de consonancia en la teoría musical: de la escuela pitagórica a la revolución científica, Biblioteca Salmanticensis 289 (Salamanca: Publicaciones Universidad Pontificia, 2006).

5 Here we will not present the whole argument. A thorough study of Ptolemy’s scientific procedure is made by Barker, Scientific method in Ptolemy’s Harmonics.


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If Ptolemy’s form of exposition is to be relied on, it [the homalon diatonic] was “suggested” to him initially by purely theoretical considerations. When presented to the ear it is found to have a certain charm, and pleasing melodies can be played in it, even if its character is not strictly Greek. At the end of the paragraph he does not say: “and this is what is called the “even diatonic”, as though he had identified the form of another, generally recognised attunement. He says: “so let us call this genus the “even diatonic”, from the characteristics it has”. The implication seems to be that his reflections had led him to a new variety of division, one that the ear enjoys, but not one already found in practical music-‐making, or represented, accurately or otherwise, in the theoretical textbooks.

Ptolemy claims, then, to have devised this division on the basis of “rational” considerations suggested to him by another case. He tried it out on his strings, and was apparently so intrigued by the results that he persisted until his hearing had “become trained to it”. But on this occasion he had no special axe to grind. If the division had proved audibly unacceptable the fact would in no way have undermined his hupotheseis, since it is not derived from them in the regular way, nor would it have conflicted with anything he goes on to say about the music of practice. The passage has all the appearence of being a report of an unbiassed piece of experimentation, designed straightforwardly to test a theoretical possibility. To Ptolemy’s ears at least, the experiments showed that it was equally an aesthetic possibility, something that was perceptibly agreeable and capable of being used as the basis of pleasing melodies. But its acceptability was not entailed by the theory, and if the results of the experiment had been different, no harm would have been done. The only consequence, I suspect, would have been a tactful silence; the division would not have been mentioned at all. I can see no reason to suppose that the experiment was not conducted, and conducted in good faith.6

This is what Barker thinks on the subject, but things can be interpreted in another way.

First of all, when Ptolemy is describing the audible character of this system, he does not describe it as smooth, or even, or something of that kind, which would be the appropriate thing if the mathematical evenness was the principal feature giving this system its agreeable sound. In fact, he uses the words “rustic” (agroikoteron) and “foreign” (xenikoteron) to describe it in a rather despective manner. If we take a look at Ptolemy’s paragraph cited above, we can clearly see that he argues in favor of the system in spite of its rustic and foreign character.

Secondly, let’s then take a look at the words “foreign” and “rustic”. The word “foreign” could serve to describe something never heard, but, how can something sound rustic that has never been heard before? The word “rustic” clearly alludes to an extra musical reference: to something rude, not sophisticated, not urban but rural, from a pastoral context. Musical sound, in itself, cannot be rustic, but the object or the person making that sound can be. And, as we can see, that rustic characteristic is something not much appreciated in Ptolemy’s time. In fact, if we come back to the word “foreign”, it could also mean something thought of as non hellenic, something that would not correspond to the cultivated hellenic tradition in which Ptolemy’s treatise is inserted; and therefore, in this sense, the word “foreign” would also allude to an extra musical reference. And both “rustic” and “foreign”, have a negative meaning to the eyes of Ptolemy.

6 Barker, Scientific method in Ptolemy's Harmonics, 240–241.


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Thirdly, as we can infer from Ptolemy’s treatise, the homalon diatonic was not an actual musical tuning in Ptolemy’s time, in string instruments (kitharas and lyres), which are the only ones discussed by the author. But he says nothing about other types of instruments.

Forthly, Ptolemy does explicitely say something very important to us about this genus: it can be used by itself in an octave system formed by two disjunct tetrachords. In fact, when he proposes string lengths for the system, he assumes a whole octave, something he has not done when talking of all other genera, which were always presented just in a tetrachord system. So, we could say that for Ptolemy, the natural way of using the homalon diatonic is in an octave system made out of two disjunct tetrachords.

And finally, the procedure followed by Ptolemy to get this homalon diatonic is not derived from his principles (hupotheseis) in the ordinary way, as has been correctly pointed out by Barker (2000, 239). In other words, it does not follow Ptolemy’s scientific method.7 Moreover, its presentation in chapter 16 of book I is made aside from the other genera, all of them discussed in chapter 15. Only one of the other genera does not follow Ptolemy’s regular procedure, the tonic diatonic (9/8, 9/8, 256/243),8 although it is discussed together with all other genera, and its inclusion is justifyed by Ptolemy by the fact that it is used in actual musical practice. On the contrary, two genera discussed by Ptolemy (the enharmonic and the soft chromatic) are derived by his regular procedure, but Ptolemy himself admits that they are no longer in use.9 The homalon diatonic’s case is a different one. Let us consider that, as Barker thinks, it does not respond to a tuning actually heard by our author in real musical praxis. Why would then Ptolemy follow an awkward procedure to get a non existing, never heard, musical system?

2. The homalon diatonic and 3/4 tone intervals

Before we continue our discussion one thing must be clarifyed. Even though Ptolemy describes the homalon diatonic as “even”, and, actually, the three intervals of this system are rather similar in size, to someone accustomed to think of music space in terms of tones, semitones, quartertones, etc. the intervals of this homalon diatonic would roughly sound like two lower 3/4 tone intervals and an upper interval of a tone, as has been already pointed out by many scholars, like Winnington-‐Ingram, Schlessinger or Chalmers.10

Although writers following the pythagorean tradition (and Ptolemy could be included in this group) presented musical tunings in terms of mathematical ratios, this was just the “scientific way” of doing it, but it is obvious that in the Classic and Helenistic Antiquity, music space was intuitively thought of as being constituted by tones and parts of a tone.

7 The regular procedure divides the fourth into two epimoric ratios (ratios of the form n+1/n), and then

one of them is further divided into two smaller epimoric ratios. In the homalon diatonic, the fourth is directly divided into three epimoric ratios.

8 As we can see, the lower interval of this genus has a non epimoric ratio, something that contradicts Ptolemy’s principles of melodicness. This interval is the leimma (256/243).

9 Nevertheless Ptolemy is forced to present them, both because they derive from his regular procedure and because they respond to a long Greek musical theory tradition.

10 We will discuss these authors later in our paper. Let us remember that the intervals of the homalon diatonic, from low to high, are: 151 cents, 165 cents, 182 cents, 204 cents, 151 cents, 165 cents, 182 cents.


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The aristoxenian tradition, relying on musical perception, always describes musical systems as successions of intervals measured as aliquot parts of a tone. And even followers of the “pythagorean” tradition, as Ptolemy, used the words tone and semitone in their treatises. This means that the whole homalon octave system, expressed in an aristoxenian way, would roughly be, from low to high: 3/4, 3/4, 1, 1, 3/4, 3/4, 1.

3. Evidences of the homalon diatonic in earlier music treatises

We must question ourselves if there is any evidence of the homalon diatonic in earlier music treatises. We must say that no other tuning system clearly reported by earlier authors can be directly related to the homalon diatonic, but we can find some traces of Greek systems with 3/4 tone intervals.

Curiously, all these traces have reached us through Aristoxenus. His hemiolic chromatic has a pykna of 3/4 tone (divided into two equal intervals of 3/8 tone each), and his soft diatonic has a second interval of 3/4 tone. Apart from these particular intervals, the whole structure of these two systems has nothing to do with Ptolemy’s homalon diatonic.11

Other example of 3/4 tone intervals can be found in a pasage where Aristoxenus, talking about the relative picth of the different tonoi, implicitely describes the aulos tuning system:

[...] while others again, with an eye to the boring of the finger-‐holes of auloi, separate the three lowest tonoi, the Hypophrygian, the Hypodorian and the Dorian, by three diesis from one another, and the Phrygian from the Dorian by a tone, placing the Lydian at a distance of another three diesis from the Phrygian, and the Mixolydian at the same distance from the Lydian.12

From this passage we can infer13 that auloi (or at least some of them), at the time of Aristoxenus had six holes and were tuned by the sequence: 3/4, 3/4, 1, 3/4, 3/4. As we can see, the lower part of this tuning could correspond to the homalon diatonic tetrachord (expressed in an aristoxenian way); but in Ptolemy’s system there are two disjunct tetrachords, while in Aristoxenus’s description there seems to be a lower tetrachord and the two lower intervals of a conjunct tetrachord.

Other examples of 3/4 tone intervals are found in some passages by Ps. Plutarch and Aristides on the spondeiasmos. According to Aristides,14 an ascent of three incomposite dieseis was called spondeiasmos. Likewise, Ps. Plutarch,15 talking of the spondeion scale, mentions an interval, characteristic of that scale, smaller than a tone by a diesis. Scholars agree that the information contained in both passages is likely to come from

11 The relation of the homalon diatonic to Aristoxenus’ hemiolic chromatic and soft diatonic has been

already pointed out by: John H. Chalmers, Divisions of the Tetrachord (Hanover, NH: Frog Peak Music, 1993).

12 Aristoxenus, «Elementa harmonica», in Greek Musical Writings: Harmonic and Acoustic Theory, vol. II, trans. Andrew Barker (Cambridge: Cambridge Unviersity Press, 1989), 154.

13 As does, for example, Martin Litchfield West, Ancient Greek music (Oxford University Press, 1994), 97. 14 Aristides Quintilianus, «De musica», in Greek Musical Writings: Harmonic and Acoustic Theory, vol. II,

trans. Andrew Barker (Cambridge: Cambridge University Press, 1989), cap. 430. 15 Andrew Barker, Greek Musical Writings: The Musician and his Art, vol. I (Cambridge: Cambridge

University Press, 1984), 116.


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Aristoxenus, and that “diesis” refers to the enharmonic diesis of a quarter of a tone. The spondeiasmos, as mentioned by Aristides and Ps. Plutarch, would then be an interval characteristic of the spondeion scale, and it would be a 3/4 tone interval.16 From these evidences Winnington-‐Ingram reconstructed the Spondeion as a pentatonic default scale used in wind instruments, and in a haphazard way he tries to relate it to the homalon diatonic. He says:

Still more interesting is the homalon diatonic of Ptolemy (Harm. I 16), of which the lowest interval is 12/11 -‐i.e. roughly a three-‐quarter tone-‐ the complete tetrachord being 12/11x11/10x10/9. It does not occur in the lyre scales he describes in Harm. II 16. In fact, the account suggests that Ptolemy had invented it himself. It may, however, be a more or less concious reflection of the Spondeion, which may not have become completely obsolete by then. If so, the original intervals of Olympus’ scale [the spondeion scale] were E 12/11 F 11/9 A 9/8 B 12/11 C.17

As we can see, the first thought of Winnington-‐Ingram is that the homalon diatonic was invented by Ptolemy (as we said earlier). But then he opens the door to the existence of a practical tuning, the spondeion scale, which Ptolemy would be describing. The spondeion (as described by Ps. Plutarch) is a pentatonic default scale. In Winnington-‐Ingram’s version of the spondeion, there is a first interval of 12/11 (which would correspond to the first interval of the homalon scale), a second interval of 11/9 (which would correspond to the second and third intervals of Ptolemy’s homalon, undivided), a third interval of 9/8 (which would correspond to the disjunction tone) and a fourth interval of 12/11 (which would correspond to the first interval of the upper tetrachord). The upper part of the scale does not appear at all.

The resemblance of the spondeion scale with Ptolemy’s homalon is not evident, and less so if we consider that in the original description of the spondeion scale by Ps. Plutarch all we can find is a reference to the use of a 3/4 tone interval as the bottom interval of the upper tetrachord. In fact, other scholars have interpreted the Ps. Plutarch and the Aristides passages in a different manner. Barker,18 for instance, has proposed a rather different version of the spondeion scale, a version which has nothing to do with Ptolemy’s homalon.

Apart from these references to 3/4 tone intervals, we have found no other hint in ancient texts that could help us find the origin of Ptolemy’s homalon diatonic. In any case, it is worth noting that most references we have found of 3/4 tones appear in the context of wind instruments: on the one hand, the spondeion scale, which presents intervals of 3/4 of tone, was used in wind instruments; on the other hand, at the time of Aristoxenus auloi were tuned using 3/4 tone intervals.

From these evidences we agree with Barker in the assumption that the homalon diatonic had not before been named or described by any earlier author. According to our view, the homalon diatonic is then an heterodox diatonic. We don’t think its origin can be related to earlier described systems. But that does not necessarily mean that it did not

16 For further discussion on the subject, see: Winnington-‐Ingram, «The Spondeion Scale». Barker, Greek

Musical Writings: The Musician and his Art, I: 255–257. Andrew Barker, Greek Musical Writings: Harmonic and acoustic theory, vol. II (Cambridge: Cambridge University Press, 1989), 430.

17 Winnington-‐Ingram, «The Spondeion Scale». 18 Barker, Greek Musical Writings: The Musician and his Art, I, 255–257.


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exist as an actual tuning system in Ptolemy’s time. In fact, from Ptolemy’s words, we can infer that it could correspond to a type of music that would be thought of as rustic, non sophisticated, of foreign origin and, therefore, not taken into acount by earlier writers, commited themselves only with cultivated music and with the long tradition of Greek musical theory.

4. The homalon diatonic in actual musical practice

That Ptolemy’s homalon could be describing an actually used musical system has been already pointed out by some scholars. Appart from Winnington-‐Ingram’s problematic connection with the spondeion scale, at least two other scholars have discussed the subject.

The first one is Kathleen Schlesinger. In 1936 she participated in a discussion on the origin of the modern major and minor scales. Talking of Ptolemy’s homalon as the origin of the modern minor scale, she said:

The name of the scale is the Homalon or Equal diatonic. The ratios of that pentachord are 12/11, 11/10, 10/9, 9/8, which produce on the tonic a minor third and a perfect fourth and fifth in just intonation. That scale has been traced in use in the Greek Church in Asia Minor up to 1870, so one reads from Joh. Tzetzes, who was born in Asia Minor and brought up there. He states that the scale was still being played in many of the Greek Churches in Asia Minor in that year. That would give the first tetrachord of our minor mode in just intonation. The only note that is different is the intermediate note of the minor third, the d, if you take the scale of c minor. The d is less than the meantone d. It is of 151 cents, practically a 3/4 tone. That scale continued from those Greek times and through the Middle Ages. Here and there it may be traced right through the centuries. It may be found also in the music of the Folk. I have it on a flute made by a peasant in Sicily, and I have it on flutes from Java, India and other parts of the world. It also appears in the vibration frequencies that have been taken from phonographic records of scales in all different parts of the world. So I suppose that that would give an indication of how our minor scale came to birth.19

As she correctly points out, the interval formed between the first and third degrees of the scale is a minor third in just intonation (of ratio 6/5), and the first interval is not a tone, but 3/4 of a tone. The interesting thing is that she refers to have listened to this type of intonation in flutes and actual music all over the world. It is curious that she does not mention the whole homalon octave (as Ptolemy himself presents it), but only the lower pentachord, as though it was only this pentachord which would be present in those refered folk flutes and recordings.

Further on, Schlesinger presents an interesting theory about the homalon diatonic in her book from 1939, The greek aulos. The ideas she presents in this book are quite peculiar and aside from the main stream and have therefore not received much attention of later scholars. But some of the theories presented in this book can help us understand a little better the question of intonation in wind instruments; and, as we have seen, most

19 James Swinburne, «The Ideal Scale: Its Ætiology, Lysis and SequelÆ», Proceedings of the Musical

Association 63 (Enero 1, 1936), 60.


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references we have discussed up to now are, in some way or another, linked with wind instruments.

Schlesinger’s hypothesis is that the old Greek harmoniai derive from the intonation of wind instruments, particularly auloi. Her point of departure is that, as ethnomusicologists have found, most pipes in most musical cultures of the world have equidistant fingerholes. Then, she presumes that Greek auloi had, at least at an early stage, equidistant fingerholes, and these determined the intonation of the different harmoniai.

From a pipe divided in 12 equal parts, having a hole in each one of the divisions (up to the division 6 which marks the octave), she derives what she calls the “old Phrygian harmonia”, which would correspond to the ratios: 12/11, 11/10, 10/9, 9/8, 8/7, 7/6. The scale she gets is a defective scale, of only 6 sounds per octave. As we can see, the lower pentachord of this scale corresponds to the homalon tetrachord plus the tone of disjunction. The upper part is a default tetrachord (with only one infix instead of two) and does not correspond to Ptolemy’s homalon octave system. Schlesinger follows that, in a later evolution of ancient Greek music, an extra infix was inserted dividing the originally undivided pyknon of the upper tetrachord and getting a complete scale of seven sounds per octave.20 This division was made by inserting an extra fingerhole between the two holes bordering the upper pyknon, again at equal distances:

12/11, 11/10, 10/9, 9/8, 16/15, 15/14, 7/6.

As we see, Schlesinger’s Phrygian scale of seven sounds per octave does not correspond to Ptolemy’s homalon octave system. Only the lower pentachord corresponds. This gives us a hint of why in her discussion of 1936 she only talks of the lower pentachord, and not of the whole octave. The whole homalon octave system cannot be obtained by seven or eight equidistant fingerholes –or by adding extra fingerholes dividing in equal parts the distances already presented, as she does to get her Phrygian harmonia-‐, and therefore it surpasses Schlesinger’s point of departure.

A section of Schlesinger’s book is devoted to scales of folk music traditions, and there she argues that the Phrygian harmonia (with six or seven sounds per octave) can still be found in folk pipes from Sicily and Greece.21 She also recalls to have identified a scale consisting of two conjunct Phrygian tetrachords in records of gamelan orchestras from southeastern Asia and of African marimbas.22

The other scholar who has related the homalon with actually used musical scales is Chalmers in his book Divisions of the tetrachord. As he says:

The equable diatonic has puzzled scholars for years as it appears to be an academic exercise in musical arithmetic. Ptolemy's own remarks rebut this interpretation as he describes the scale as sounding rather strange or foreign and rustic. Even a cursory look at ancient and modern Islamic scales from the Near East suggests that, on the contrary, Ptolemy may have heard a similar scale and very cleverly rationalized it

20 Kathleen Schlesinger, The Greek Aulos (London: Methuen, 1939). 21 She describes a pipe from Sicily which is probably the same one she referred to in the discussion

previously commented, and a modern Greek pipe. Both follow, according to her, the Phrygian harmonia in the way she presents it, not the homalon octave. Schlesinger, The Greek Aulos, 456.

22 Schlesinger, The Greek Aulos, 311–312.


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according to the tenets of Greek theory. Such scales with 3/4-‐tone intervals may be related to Aristoxenos’s hemiolic chromatic and may descend from neutral third pentatonics such as Winnington-‐Ingram’s reconstruction of the spondeion or libation mode, if Sachs's ideas on the origin of the genera have any validity.23

Further on he devotes a section to the medieval Islamic theorists. Both Al-‐Farabi (ca. 950) and Avicenna (ca. 1037) present tetrachordal divisions with 3/4 tone intervals, which, according to Chalmers,24 resemble the homalon diatonic. He even places this fact as evidence of the actual use of 3/4 tone intervals in Ptolemy’s time:

The resemblance of these to Ptolemy's equable diatonic seems more than fortuitous and further supports the notion that three-‐quarter-‐tone intervals were in actual use in Near Eastern music by Roman times (second century CE). These tetrachords may also bear a genetic relationship to neutral-‐third pentatonics and to Aristoxenos's hemiolic chromatic and soft diatonic genera as well as Ptolemy's intense chromatic.25

Any scientific book on Arabe scales, from the Middle Ages to the present, discusses the use of 3/4 tones intervals. According to Touma, for example, the interval of 3/4 tones is characteristic of Arabe music throughout history and it is still common today.26 In fact, three of the eight modes (which constitute the general material from which the different maqams are made of) of today’s Arabe music divide tetrachords in a way similar to Ptolemy’s homalon (that is, with two intervals of 3/4 tones and one interval of a full tone). So is the case of mode Rast, mode Bayati, and mode Sikah.27

5. Shepherds and pipes in Ptolemy’s time

Let us come back to Ptolemy’s sentence: “the character that becomes apparent is rather foreign [xenikoteron] and rustic [agroikoteron], but exceptionally gentle, and the more so as our hearing becomes trained to it, so that it would not be proper to overlook it, both because of the special character of its melody, and because of the orderliness of the division”. To our eyes, this sentence makes it clear that Ptolemy had already listened to the scale when he described it and gave it the name homalon. But it also makes clear that, to the eyes of Ptolemy, the normal thing to do when confronted with this intonation schema, would be to overlook it because it was a rustic, foreign, musical system. Nevertheless, Ptolemy, fascinated by its mathematical properties, tried it out on strings and found that it did not sound that bad after all. In fact, it sounded rather agreeable in spite of its rustic origin. His earlier remark on the evenness as the feature giving it its agreeable sound would just be a justification of the author to convince himself to include it in the treatise. Its evenness is just a mathematical one.

If Ptolemy did actually hear such a scale, it must have sounded foreign and rustic to him because of the foreign and rustic origin of the music that used it. But, which music, in the time of Ptolemy, could have sounded “foreign” and “rustic”? Obviously, it could not be the cultivated music of string instruments such as the lyre or the kithara. “Rustic”, in the

23 Chalmers, 11–12. 24 Chalmers, 21. See also “The Main Catalog” in Chalmer’s book where he presents a collection of all the

discussed tetrachords (pp. 164-‐203). 25 Chalmers, 14. 26 Habib Hassan Touma, La musique arabe, Les traditions musicales (Paris: Buchet/Chastel, 1996), 34–35. 27 Touma, 41–49.


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Ancient Greek and Roman times, refers to rural, pastoral, in other words, shepherd music. As for “foreign”, it may reflect anything outside the Hellenic tradition of string instruments.

The main pastoral instrument in Greek Antiquity was the flute or syrinx. There were two types of flutes: the more common syrinx polykalamos and the syrinx monokalamos. The former (a pan flute) was made up of various pipes of different lengths bound together; the later was a single, simple pipe with fingerholes. Both of them had no place in serious “art” music. They were used exclusively in pastoral and folk settings.28 Moreover, the flute, like the aulos, was generally seen as a foreign instrument in Greek Antiquity. As Mathiesen says:

While the aulos assumes a central place in the Greek’s high culture, the syrinx remains a simple pastoral instrument. [...] The term syrinx, as already noted, refers simply to a little whistle made of reed, and it can be applied to a single pipe, a group of reeds of graduated length bound together –the panpie-‐ or an aulos mouthpiece, which is made from the same type of reed, though cut and prepared in a different manner. The syrinx in one form or another is an instrument of considerable antiquity, and like the aulos, it tended to be viewed by the Greeks as a foreign instrument, if not as the invention of one of the gods.29

There are not many references to the syrinx monokalamos in Greek texts, at least not as many as to the syrinx polykalamos.30 In fact, among the Ancient civilizations of the Near East, only in Egypt was the flute a relatively common instrument, and in the Greek culture the flute is not attested before the Hellenistic period.31 On the other hand, most references to the flute in the Greek Hellenistic culture come from authors related in some way or another to Egypt, like, for example, Pollux (Egyptian writer of the IInd century AD), Theocritus (III century AD, native of Sicily, is thought to have spent some time in Alexandria) or Athenaeus (Egypt, ca. 160 AD).

The ancient Egyptian flute, according to Sachs, was a so called vertical flute. It had no mouth piece and it was blown across the open upper end holding it slightly sideways. The Egyptian flutes were made out of a simple cane; they were long and narrow, and, according to Sachs, had from two to six fingerholes near the lower end. The nay or qsaba, so popular nowadays in the folk music from North Africa, is supposed to directly derive from these ancient flutes32.

28 Barker, Greek Musical Writings: The Musician and his Art, I, 16. 29 Thomas J. Mathiesen, Apollo’s lyre: Greek music and music theory in antiquity and the Middle Ages (U of

Nebraska Press, 1999), 222. 30 But, as different scholars have pointed out, the words aulos, monaulos, plagiaulos or even photinx were

sometimes used to refer to a syrinx monokalamos. As in Pollux II 100, 108 and in Theocritus, Idylls 5.7, 6.42-‐6. West, 113: “Where plagios aulos or plagiaulos appears as the source of a soft wind-‐like sound, or as a rustic instrument in settings appropriate for the panpipe, we must again interpret it as a flute. [...] Another term that probably designates a flute is photinx. [...] The photinx was current at Alexandria.” See also: Barker, Greek Musical Writings: The Musician and his Art, I, 92. Most probably, monaulos refers to a vertical flute, while plagiaulos and photinx refer to a traverse flute.

31 See: West, 112, 113, and Curt Sachs, The History of Musical Instruments (W W Norton & Co Inc [Np], 1940).

32 Nay is the Persian name, used nowadays in the eastern part of North Africa. Qsaba is the Arabic name and it is used in the western part of North Africa. Sachs, 90.


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One extant Hellenistic written source on the flute is specially clarifying for us: the text by Athenaeus, Deipnosophistae (“Experts at dining”). This author was born in Egypt about 160 AD, which makes him Ptolemy’s contemporary. As Barker describes it, this text is an example of “the genre of ‘table talk’, presenting an encyplopaedic assortment of facts and opinions related to the art of convivial eating and drinking”.33 In a moment of the conversation an hydraulis (a water organ) is heard. One of the men involved in the table talk is Ulpianus, who is described by Barker as “urbane”. He is a cultivated man who defends the urbane and cultivated music of the hydraulis against the unrefined monaulos. He says, refering to the hydraulis they are hearing: “Do you hear that fine and beautiful sound [...]? It’s not like the monaulos so common among you Alexandrians, which gives its hearers pain, rather than any musical delight.” The monaulos of this passage can be identified with the monokalamos syrinx, a vertical flute, as Barker does.34 Further on in the text, the Alexandrian musician Alcides replies:

But since you disparage us Alexandrians as unmusical, and constantly mention the monaulos as endemic amongst us, listen to what I can tell you [...]. Juba says that the Egyptians call the monaulos an invention of Osiris, as they do also the plagiaulos known as the photinx.

Here again Barker interprets Plagiaulos as a flute, but this time a traverse flute.35 From both passages two things are evident:

-‐ The flute was a common instrument in Alexandria in the second century (therefore in Ptolemy’s time) and it seems that both in its vertical and in its traverse form.36

-‐ The flute was seen as an unrefined, unmusical instrument to the eyes of cultivated men.

Obviously the flute was a rural, pastoral instrument in the 2nd century, and Ulpianus’ contempt only reflects the general opinion of cultivated men. These passages give us some light on Ptolemy’s words. They provide a point of departure to reinterpret his description of the homalon octave system. Two possible reinterpretations arise:

-‐ The first one would be: This type of tuning is used in shepherd flutes in Ptolemy’s time and, therefore, it sounds rural and foreign to the Hellenic cultivated tradition. The normal thing to do, when confronted with this type of tuning, would be to ignore it. But its outstanding mathematical properties (although not derived from Ptolemy’s hupotheseis) are worth the trial. Ptolemy got to the ratios of this type of tuning by measuring lengths, although originally these were not string lengths, but pipe lengths, and, as nobody had previously described it, he gave it the new name, homalon, because of its mathematical properties.

33 Barker, Greek Musical Writings: The Musician and his Art, I, 258. Some interesting fragments of the text

on musical instruments are translated by Barker in chap. 16 of this book. 34 “It is probably the so called monokalamos syrinx or aglotos aulos, a simple tube of reed with finger

holes, sounded by blowing across the end. [...] In the Greek world this instrument was generally confined to the rustic music of shepherds etc. [...] which would explain the urbane Ulpianus’ contempt for it.” Barker, Greek Musical Writings: The musician and his art, I, 259.

35 Barker, Greek Musical Writings: The Musician and his Art, I, 264. 36 The probable Egyptian origin of the flute in Hellenistic times is reported by many authors, like the

already cited Barker and West.


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-‐ The second possibility would be: As Ptolemy says, fascinated by the evenness of the tense chromatic, he tried out the division of the tetrachord in three almost equal intervals, getting the ratios 12/11, 11/10, 10/9. He named his invention, homalon diatonic, because of its mathematical properties. He tried this division in a monochord and found that, in an octave system made up of two disjunct tetrachords, it sounded very similar to the system used in some shepherd flutes. Therefore, it sounded rather rustic and foreign to the cultivated Hellenic culture, but Ptolemy, fascinated by its mathematical properties, decided to include it in his treatise.

We are not able to determine which interpretation is more likely to be correct, but, nevertheless, both possibilities would lead us to the same hypothesis: shepherd flutes in Ptolemy’s time (or at least some of them) were probably tuned in a system very similar to the homalon diatonic octave.

6. Shepherds and pipes in contemporary Spain

Simple, three-‐hole pipes,37 have a long pastoral tradition in Europe. In fact, this type of flutes were still played by Spanish and Portuguese shepherds until recent times (figures 2-‐3). This instrument is still in use in some places of the Iberian Peninsula,38 where it is played by semiprofessional “tamborileros” who provide entertainment for folk festivals and accompany folk dances, although nowadays it is highly improbable to find a shepherd able to play it. Three-‐hole flutes are specially important in folk traditions along the Spanish-‐Portuguese border. One of the places from this border area where the three-‐hole flute is the most important folk instrument is the Spanish province of Salamanca.

Figure 1. Three-‐hole flute from Salamanca (Fundación Joaquín Díaz).

37 A three-‐hole pipe is a fipple flute with two finger holes above and one below. It is played with one

hand, while the other hand usually plays a tabor. 38 Similar three-‐hole flutes can also be found in Ibiza, the Canary Islands, Provence, Great Britain, etc.


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Figure 2. Spanish shepherd from Ciudad Rodrigo (Salamanca) playing a three-‐hole pipe. Photograph by Agustín Pazos, 1920.

A recent study has been made on the tuning of the flute from Salamanca and its conclusions –in spite of time and spatial distance-‐ may help us understand Ptolemy’s homalon diatonic.39 According to this study, traditional flutes from Salamanca present a particular tuning system very different from the nowadays standard equal temperament. These flutes are made following traditional receipes, which include equal spacing between finger holes, something relatively frequent in wind instruments all over the world, as we have already mentioned.40

The uppermost finger hole is carved to produce the interval of a perfect forth in relation to the whole length of the flute. The other two finger holes are placed to equally divide the space between the lower end of the pipe and the uppermost finger hole.

Approximately, these flutes have the following morphology (figure 3):

Figure 3. Approximate morphology of the three-‐hole flute from Salamanca.

39 Amaya Sara García Pérez and Álvaro García Pérez, «La afinación de la flauta tradicional salmantina de

tres agujeros», Revista de Musicología XXXII, no. 2 (2009), 343-‐361. 40 See: Sachs,, 181. Schlesinger, The Greek Aulos, 222.


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This flute has only four fundamental sounds, corresponding to the four positions which arise from succesively uncovering the three finger holes.41 These fundamental sounds are usually not employed in music making. In fact, all sounds usually used when playing this instrument are harmonic overtones obtained by overblowing. The playable range of the flute is little more than an octave (depending on the instrument), not counting the lower register, which, as we said, is not employed in music making. Two reasons explain why this lower register is not employed:

-‐ First of all, there is a gap of a fifth between the lower and the medium registers.

-‐ And secondly, the bore of the flute is so narrow compared to its length that it easily overblows to the first harmonic. The lower register can be obtained only with very low air pressure and therefore it sounds at a very low volume. The moment you raise the air pressure a little, the sound jumps to the first harmonic.

This means that the flute can easily produce two disjunct tetrachords, both of them obtained by overblowing the lower register, forming an octave.42

In ideal conditions, these morphological-‐organological conditions of the flute would give rise to an octave intonation schema that has been confirmed by empirical studies, as we can see in table 2.

Ideal conditions

Empirical evidence

Frecuency ratios

or

pipe lenght ratios

Cents

Cents

10/9 182 199,3

11/10 165 163,9

12/11 151 145,8

9/8 204 203,8

10/9 182 178,7

11/10 165 170,7

12/11 151 150,2

Table 2. Tuning schema of the three-‐hole flute from Salamanca. Acoustically, the impresion given by this intonation schema is that each tetrachord is composed of two 3/4 tone intervals and an upper tone, as the qualitative descriptions of

41 Traditionally cross-‐fingering is not employed, although more developed specimens (like the basque

txistu, from the ninetenth century adapted to equal temperament) do use them to get chromatic sounds.

42 The best instruments can also produce a third conjunt tetrachord, also obtained by overblowing the lower register.


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folklorists prove.43 In other words, it roughly sounds: La-‐Si -‐Do-‐Re-‐Mi-‐Fa -‐Sol-‐La. As we can see in table 2, this intonation schema seems to match Ptolemy’s homalon diatonic.

7. Making connections

We have seen how the intonation schema of the three-‐hole pipe from Salamanca could match Ptolemy’s homalon diatonic. Could it then be possible that a type of three-‐hole flute did exist in Ptolemy’s time and that he actually heard it? If it did, most probably it was a vertical flute, with no mouth piece, and therefore different from the European three-‐hole flute. Or, is it a coincidence? Can this tuning system arise in any other musical context? In other words, what instruments could be easily tuned to the homalon diatonic?

As it has been shown, the three-‐hole flute can be very easily attuned to it. It only needs three equidistant finger holes in the lower end, carving the top one to get a perfect forth in relation to the lower sound of the whole pipe. Of course, in Ptolemy’s time flutes had no mouth piece, they were either vertical or traverse flutes, but that does not alter the point: a vertical flute or a traverse flute, long and narrow, with three equidistant holes in the lower end, would also fairly match Ptolemy’s homalon diatonic.

Of course, there are other possibilities. A string instrument with two strings tuned to a fifth, and with equidistant frets, could also give rise to such an octave. But this is not as simple, and, on the other hand, would a string instrument sound “rustic” to Ptolemy? A reed pipe, like an aulos, with three equidistant finger holes and a conical bore,44 could serve too, although most auloi, had cylindrical bores, not conical ones45. And besides, a conical bore is not as simple to make as a cylindrical one. A cylindrical bore is obtained by simply using a reed. Conical bores presuppose a more refined construction method.

A three-‐hole flute is one of the most simple, rustic, and unsophisticated instruments, and, on the other hand, its simplest construction (finger holes equally spaced) gives rise to this intonation schema. The flute hypothesis seems, then, more plausible than others, more so if we consider Athenaeus’ passages.

If our hypothesis is right, Ptolemy’s homalon diatonic does not descend from any default spondeion scale, such as Winnington-‐Ingram suggests, nor is it a mathematical speculation, nor is it the old Phrygian harmonia. It is simply a heptatonic scale resulting from an easily made, “rustic” instrument, which was “endemic amongst Alexandrians” (as the urbane Ulpianus says) in Ptolemy’s time. Obviously, such a system had never before been described, nor was it named in any manner. Shepherds were not cultivated music writers, they did not discuss the intonation used in rustic pipes; they just made them in the simplest way.

43 For examples of qualitative descriptions of the flute’s intonation schema, see Alberto Jambrina Leal and

José Ramón Cid Cebrián, La gaita y el tamboril (Salamanca: Centro de Cultura Tradicional, Diputación de Salamanca, 1989), 21–25.

44 A reed instrument is a pipe closed in one end. Closed pipes have a special acoustical feature: they only produce odd harmonics. But if a reed pipe has a conical bore, it functions like an open pipe (like a flute) and it produces all harmonics. To get a complete range of an octave, a three-‐hole pipe must be a flute or function like one.

45 West, 83.


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8. Further discussions

This explanation of the homalon diatonic can give us hints about other musical systems which present 3/4 tone intervals. As we have seen, equidistant fingerholes in wind instruments easily produce the homalon tetrachord. In cylindrical auloi the whole homalon octave cannot be obtained by equidistance. Cylindrical auloi are closed pipes and therefore they do not produce all harmonic partials by overblowing.46 This means that a three-‐hole aulos (a three-‐hole closed pipe) could not produce a whole heptatonic octave by overblowing, as do open pipes.47 But the lower homalon pentachord can surely be obtained this way, as Schlesinger correctly explained. To obtain the homalon octave in an aulos, two sets of equidistant holes would be required, one for the right hand and another one for the left hand. The right hand would cover four equidistant finger holes, to get, with the uppermost, the interval of a fifth; the left hand would cover another three equidistant finger holes, but their relative distances would be different from the distances between the holes of the right hand (they would be closer).

We can conclude that it seems likely that many musical systems using 3/4 tones arise from the tuning of wind instruments in which equidistant finger holes are used. In fact, we believe that Schlessinger’s point of departure is much more interesting than what it may seem, although the later development of her theory is rather problematic. We must agree with her in the idea that many musical tuning systems may have arisen from wind instruments. Wind instruments are not easily retuned once they are built, so the placing of their fingerholes highly determines their tuning. On the other hand, string instruments can be easily retuned by varying the tension of the strings, and, therefore, their tuning is not definetly determined once they are built. We can then suppose that, whenever in a musical culture wind instruments have a important role, the tuning of these wind instruments can be decisive in the tuning system used by that culture. And, furthermore, the tuning of wind instruments is condicioned by their morphology. In any case, we think Schlessinger’s theories should be reexamined and reevaluated in order to place them in the position they deserve.

46 They only produce odd harmonic partials. 47 This is why no three-‐hole “closed” pipes exist, while many cultures have three-‐hole “open” pipes.

amaya garcía - ptolemy, pipes and shepherds

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