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Mo, 288,
ON THE
INTERPRETATION
OF
HO
GREEK MUSIC
By CECIL TORR, M.A.

As I B B AB
OF
Washington University
S2 LOUD
ως αρχαιολογήσων υμίν και κατανίψων από γλώττης άπαντας
LUCIAN
London
HENRY FROWDE
OXFORD UNIVERSITY PRESS WAREHOUSE
AMEN CORNER, E.C.
1896
[All rights reserved]
Music
MA
169
T69
Oxford
HORACE HART, PRINTER TO THE UNIVERSITY
Slack
Eych.
Wamhaushikra
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dramfer to
Что
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A YOUTH, PLAYING AN EIGHT-STRINGED LYRE, WITH THE PLECTRUM IN HIS
RIGHT HAND AND HIS LEFT HAND ON THE STRINGS.
From a Greek vase in the Author's collection.
ON THE INTERPRETATION OF
GREEK MUSIC.
1.
THE ancient notation was arranged for one-and-twenty notes
within the octave, each of the seven original notes being
followed by two supplementary notes. And as the compass
was three octaves and a third, there were seventy notes in all.
Two letters were assigned to every note, one set of letters
being prescribed for instruments and another set for voices.
GREEK MUSIC.
This is the lettering for voices, taking the notes in order of
ascent :-
p> * 49-зьце и уух – пH7 EP 1ву
24,XO YTCP NO : NMAK TO H ZE ALBA
on * All O'' N'M' NKVD'H'Z'E' A' T' B' A'
&
And this is the lettering for instruments :-
(+**T&WZH HAHIHEw 3 - 1 + ELE
NIMF 4F cu) k vymGA <V >EUNII
z kt477 K' x' x VGA<'V'>'E''='N'IV
Z'
Thus, in the lettering for voices, the common letters of the
alphabet are taken in their natural order from A to 12. Next
beyond the common 12 comes V, a modified A, beginning the
alphabet again with every letter modified. And next before
the common A comes V, a modified. 12, finishing the alphabet
with another set of modifications. But here the alphabet
omits the letters s, p, , and passes on from I to 0', E', N', &c.,
for the sixteen highest notes. This departure from the natural
order must be connected with the fact that in the lettering for
instruments, as well as here, the letters for these sixteen notes
are distinguished only by an accent from the letters for the
notes an octave lower down. And this all looks as though
the sixteen highest notes were added at some later time.
In tue sitering for instruments the modifications are in-
troduce: in groups of three, the object being to associate each
pair of supplementary notes with the original note to which
they properly belonged. No doubt, Alypios 1 is mistaken in
1 Alypios, eisagoge, describes the letters passim in setting out the scales. Thus,
he speaks of E, Α, Ε, 1 as ημίαλφα, either αριστερόν οι δεξιόν and κάτω νεύον or
ävo vellov: of 4 as îra åpenntikdv kabelAnvouévov: of, and as očela and Bapela :
of 7, A and < as tê kabelNuVO Jévov, quídelta Kad elluvouévov and guidenta
Thaylov: also of N, M, 2 as opéuv, åplotepóv or detróv or ŰTTIOV.
GREEK MUSIC.
deriving Hand A from / instead of H, and also in deriving 7
from I, as this would naturally be 7, a modification of A
that matches the adjacent < and A. And clearly / and
are modifications of u and N, not accents, as he says. But
apparently and t must be derived from A, and F and y
from V, though the first pair is attached to Z, and the second
pair to h, which appears to be derived from H. And these
anomalies make one suspect that the notes originally ended
here at Z, an octave under Z'. The lettering, moreover, is
anomalous in the lowest notes as well, ending there with T,
3, -, *, ¢, C, just as the lettering for voices ends with 7,
,-, *, 4, 5. And that looks as though the notes
originally stopped at & , two octaves under Z. But this
and the r and C above coincide with the 3 and 7 and C in
the lettering for voices. So these three letters and the six
below may be survivals from a time when both the letterings
were alphabetical. But this is all a barren ground for specu-
lation, as nothing can be proved.
The letterings have been transcribed as follows in all the
best-known versions of the music found at Delphi' and else-
where within the last few years:-
babolil W-m 7 F 12 V YCPO
** Un vvs TRVX I
Hilo v
oto o oto-oto-oop
τ 3ε ου Η Η Η Ε ΔΕΙΓΜΣ FL Cυ Κ
€*T 3 A IH PLEMIST)
MALOZr Bo o od O'M'N PO'ZIB' U'
ENK HEAA * TE'N'K' H' E'A'A'
oto-oo-o eto-oto-oto-o-oo-oto o ose
16<V ENI Z ELEKT<'<'V' C'N'IZ'
xX4 > U I
4 x XA >U' '
A 2
GREEK MUSIC.
This system of transcription produces only forty notes in
place of seventy, so that thirty of the modern notes are used
for two Greek notes apiece. And this gives rise to three
objections.
In the first place, the transcript is ambiguous. Thus, for
example, the group 10 too may stand for any
one of sixteen different things, CO Z, > NII, &c. And
this ambiguity does actually lead to blundering. Quite re-
cently, in dealing with a piece of ancient music, an excellent
critic took an
in the transcript to mean an N in the

original, and proceeded to argue that the piece was written in
a certain key; when in reality the original had u, a note
belonging to another scale 1.
Secondly, in making one modern note do duty for two
ancient notes, the transcript is giving the same pitch to notes
that were of different pitch.
In a degenerate form of ancient music there were indeed
such things as homotones, or notes of equal pitch. Thus, in
setting out a scale of semitones, Gaudentius a puts 4,4,.,
b,9,11| in the scale with +, L1, n as Quótova, while Aristeides 3
puts 4, 7, 2, 0,9, in the scale with , b, ll as ouupwviai,
But the origin of these homotones is plain. The old notation
had provided for a pair of supplementary notes after each of
the original notes, putting < (u) and + (c) between 4 (°) and
: (s), b (e) and U (T) between 3 (s) and 9 (o), and so forth.
But when the scale was limited to semitones, there was only
one supplementary note between 0 and s, only one between
s and 0, &c.; and as these supplementary notes were placed
about half-way between, it did not really matter whether they
See the Classical Review for December last, vol. ix. p. 468, and the author's
note thereon in the Atheneum of December 14.
2 Gaudentius, eisagoge, 22.
3 Aristeides, de musica, i. 11. The reader may be cautioned against the version
of this passage in Jahn's edition of Aristeides, p. 17. It is printed there with
Bellermann's emendations; and he 'emended the reading of the MSS. to make it
suit a theory of his own.
GREEK MUSIC.
were known as v or T, as p or 7, &c. Gaudentius, however, has
selected the letters that come first, v after 0, p after s, É after
o, &c., while Aristeides has selected the letters that come
second, T, 77, v, &c.
That is exactly what has happened to the sharps and flats
in the modern scale of equal temperament. There is only
one supplementary note there between F and G, only one
between G and A, &c.; and as these supplementary notes are
placed half-way between, they are known indifferently as
F sharp or G flat, as G sharp or A flat, &c. In some keys,
however, the modern musician follows Gaudentius in taking
the first term in each pair, and speaks of all the supplementary
notes as sharps; whereas in other keys he follows Aristeides,
takes the second term, and calls them flats.
But this method of transcribing does not make homotones
of the notes that actually were such. It attaches + () to(s)
in place of < (u), u (T) to 9 (o) in place of b (b), and so forth.
In fact, it follows the system of the tempered scale in giving
the same pitch to F double shårp and G natural, to G double
sharp and A natural, &c. And there is not any precedent
for that.
The lettering for instruments does indeed suggest the use
of sharps and double sharps in place of sharps and flats, as
the modified letters are both placed after the original letter
in every group. For example, the letters F, 4, 7, C, U, ),
K, V, X, 7 seem to answer better to G, G sharp, G double
sharp, A, A sharp, A double sharp, B, B sharp, B double
sharp, C, than to G, G sharp, A flat, A, A sharp, B flat, B,
B sharp, C flat, C. And there is no harm in speaking of the
supplementary notes as sharps and double sharps. But when
the transcript gives the same pitch to 7 and C, to ) and K,
to x and 7, &c., it is assuming that the same pitch may be
given to G double sharp and A, to A double sharp and B, to
B sharp and C, &c.—a thing that is impossible without the
tempered scale.
And that provokes the third objection to this system of
transcription, to wit, that it involves the modern scale of equal
temperament.
GREEK MUSIC.
There are some passages in Aristoxenos which look as
though he was acquainted with a tempered scale; but when
these passages are taken with their context, that notion is
dispelled.
He defines a Fourth as the smallest of the intervals that
make a concord, and a Fifth as the next smallest of those
intervals; so that his Fourth is clearly a true Fourth, and his
Fifth a true Fifth. Having made these definitions, he states
that a Tone is the amount by which a Fifth exceeds a Fourth,
and elsewhere he defines a Tone as the difference between the
two smallest of those intervals that make a concord; so that
his Tone can only be a major Tonel. And thus, when he
proceeds to say that a Fourth contains two Tones and a half,
he must be using a half' quite roughly, to indicate a quantity
that was not really more than 442475.
Several of the ancient writers call attention to the fact that
what was termed a Semi-Tone was not exactly half a Tone.
And they explained the fact by saying that the term was not
to be compared with terms like Semi-Cubit, which meant
a measurement of half a Cubit, but only with such terms as
Semi-Vowel, which meant a letter not unlike a Vowel ? No
doubt this explanation may be incorrect; but the fact itself
is incontestable.
In face of those definitions of a Tone and Fourth, it is
absurd to argue that Aristoxenos must be speaking of a mean
Tone and a tempered Fourth, when he says that a Fourth
contains two Tones and a half, or when he uses language to
· Aristoxenos, p. 45, ed. Meibom, čoTW 87 Tậv ouupuvwv ØKTÙ Meyéon. édáx
OTOV Név, rò did Teoopw ... DEÚTepov 8é, TÒ Slà névte ... róvos 8° ¢OTIV, TÒ Sid
πέντε του διά τεσσάρων μείζον. Cf. p. 21, έστι δη τόνος ή των πρώτων συμφώνων
κατά μέγεθος διαφορά.
3 Gaudentius, eisagoge, 13, Tò se quitóriov Karoúuevov OỦk ČOTLV årpißws
Multóvlov. Proclos, in Timæum, p. 191 D, E, Td deyóuevov qutóvlov, où Hupíws
hutróvov. Theon, de musica, 8, Tò xétToi hutbviov oỦx ds mulơi TOYoo AeYeTai,
4.7.1. Censorinus, de die natali, 10, abusive hemitonion appellat. Macrobius, in
somnium Scipionis, ii. 1, non ita accipiendum est (semitonium) ut dimidius tonus
putetur. Boetius, de musica, ii. 27, videntur enim semitonia nuncupata, non quod
vere tonorum sunt medietates, &c.' Theon and Macrobius proceed to a comparison
with the guidavov or semivocale.
GREEK MUSIC.
the same effect in other passages. And outside his writings,
and those of his disciples, there is nothing to imply that any
tempered scale was known in ancient times.
The third objection to this system of transcribing is there-
fore very serious. Not only are the notes within the octave
reduced from twenty-one to twelve by making nine of them
of equal pitch with others, but the remaining twelve are
forced into the tempered scale; the result being that, which-
ever of the twenty-one is selected as a tonic, the other twenty
must all be more or less inaccurate in pitch.
These errors would be avoided by transcribing the ancient
letters into modern letters in the following way :-
5+*Axtab UO!!WWVY-MHZF VIR V
e ez ez ff1f28 8182 a az az b by ba C C7 C7 d d, d, e ez ez
&*03T OUH HAHIHEW 3 EITTLE
12 V XO Y TCP MOENMAKTOH ZE ALBA
f fif288182 a a, b b; bz ¢ ¢z ca dd, d, e ez ez f fifa
NEMFLE CUDK xx 734 <V >CUNII
on toit
8 882 a azaa
attery
b
O'E'N'M'N KľO'H' Z'E'AT' B' A'
b b c a a d d dz e e la ţ fi fa
K'x'X'N'G'A'<'V'>'c'u''N'I
a
N:09
The notes that are transcribed as a, b, c, d, e, f, g have
enough in common with the modern notes A, B, C, D, E, F, G
to make these letters useful as a guide to memory. A suffix
1 or , distinguishes the first and second supplementary notes,
while the punctuation . and .. and ... distinguishes the octaves.
So this modern lettering gives all the information that is
given by the ancient lettering, and gives it in a simpler form.
IO
GREEK MUSIC.
II.
When a musical note is produced by the vibration of
a string, its pitch. depends upon the rate at which the string
vibrates. And the rate of this vibration varies inversely with
the length of the string, so long as the tension and the
thickness of the string remain the same.
Thus, an Octave is produced by strings whose lengths are
in the ratio of 1 to 2, a Fifth by those which have the ratio
of 2 to 3, and a Fourth by those which have the ratio of
3 to 4. Suppose, then, that four strings are taken with
a length of 6 inches, 8 inches, 9 inches, and 12 inches respec-
tively. Then the 6-inch and the 12-inch strings will give an
Octave, since the lengths are in the ratio of 1 to 2. The
6-inch and the 9-inch strings will give a Fifth, and so also
will the 8-inch and 12-inch strings, the ratio being 2 to 3 in
both these cases. The 6-inch and the 8-inch strings, and
again the 9-inch and the 12-inch strings, will give a Fourth,
the ratio being 3 to 4. And the 8-inch and the 9-inch strings
will give a Tone, with the ratio of 8 to 9, since a Tone is
represented by the difference between a Fourth and Fifth 1.
These simple ratios of 1 to 2 and 2 to 3 and 3 to 4 were
used to fix the four chief notes of every octave, two near the
middle and one at either end. And generally the ratio of
8 to 9, which is involved in those of 2 to 3 and 3 to 4, was
used to fix the four remaining notes.
Between the two notes near the middle and the two at
either end there were two intervals in which the ratio was
3 to 4. Now 3 to 33 and 3 to 351 each had the ratio
of 8 to 9; or if these figures were multiplied by 64 in order
to eliminate the fractions, 192 to 216 and 216 to 243 each
had the ratio of 8 to 9, while 192 to 256 was in the ratio of
3 to 4. And thus the introduction of two intervals that had
1 This is all demonstrated in due form by Euclid, sectio canonis, propositions 6,
8, 12 and 13: also by Plutarch, de musica, 22, 23, and others.
GREEK MUSIC.
II
the ratio of 8 to 9 into an interval that had the ratio of
3 to 4 produced an interval that had the ratio of 243 to 256.
In other words, the introduction of two Tones into a Fourth
produced the interval that technically was called a Diesis or
Leimma and vulgarly a Semi-Tone.
Plato seems to have styled the interval a Leimma, though
the word itself does not occur in any of his extant writings? ;
and this name lived on beside the name of Semi-Tone, which
probably was introduced by Aristoxenosa. The earlier
Pythagoreans had used the name of Diesis 3.
When the first four notes of any octave were given by strings
whose lengths were 192 and 216 and 243 and 256, the fifth
note and the eighth note were given by strings whose lengths
were 288 and 384 respectively; since 192 to 384 was in the
ratio of 1 to 2, while 192 to 288 and 256 to 384 were in the
ratio of 2 to 3, and 288 to 384 was in the ratio of 3 to 4.
And as this interval was treated like that other interval in
which the ratio was 3 to 4, the sixth and seventh notes were
given by strings whose lengths were 324 and 3644 ; since
288 to 324 and 324 to 3641 each had the ratio of 8 to 9, while
3641 to 384 was in the ratio of 243 to 256. These figures
being multiplied by 2 in order to eliminate the fraction, the
eight notes of the octave were given by strings whose lengths
were 384 and 432 and 486 and 512 and 576 and 648 and 729
and 768 respectively.
If these eight notes were taken in order of descent, the
pitch decreasing as the length increased from 384 to 768,
the intervals were Tone, Tone, Leimma; Tone; Tone, Tone,
1 Plato, Timæus, p. 36 A, B, ruloliwv 8d Olaorágewv kål émiTpítwv kal énroydów
γενομένων εκ τούτων των δεσμών εν ταις πρόσθεν διαστάσεσι, τα του επογδόου
διαστήματι τα επίτριτα πάντα συνεπληρούτο, λείπων αυτών εκάστου μόριον, της του
μορίου ταύτης διαστάσεως λειφθείσης αριθμού προς αριθμόν εχούσης τους όρους σνς'
προς σμγ'. This use of λείπων and λειφθείσης seems to involve the use of λείμμα.
cf. Macrobius, in somnium Scipionis, ii. 1, Plato semitonium delupa vocitavit.
2 See above, page 8 and note 2.
9 Nicomachos, encheiridion, 9, 12, Stobæos, eclogæ, i. 21.7, and Boetius, de
musica, iii. 8, quote sayings of Philolaos in which this interval is styled a dieois ;
and Theon, de musica, 12, and Macrobius, in somnium Scipionis, ii. I, remark
this usage of the term among his followers.
A 3
12
GREEK MUSIC.
Leimma—there being a ratio of 8 to 9 for every Tone and
243 to 256 for every Leimma. But in the scales the Hypo-
Lydian group put e a Fourth above b and a Fifth above a,
and the Hypo-Phrygian group put d a Fourth above a and
a Fifth above g, while the Hypo-Dorian group put c a Fourth
above g and a Fifth above fl. And thus, as a Tone was the
difference between a Fourth and a Fifth and a Leimma was
the difference between two Tones and a Fourth, the eight
notes e, d, c, b, a, &, f, e were separated by the intervals of
Tone, Tone, Leimma; Tone; Tone, Tone, Leimma. So
these eight notes, or their equivalents in the octaves up
above, were given by strings whose lengths were in the ratios
just described.
A supplementary note was placed in every Tone by reckon-
ing the ratio for the Tone as 16 to 18 in place of 8 to 9, and
thereby bringing in the ratios of 16 to 17 and 17 to 182. And
commonly the name of Semi-Tone was given to both the
intervals that were created in this way, although the first
of them was rather more than half a Tone and the second
was rather less. But technically the first was known as an
Apotome and the second as a Leimma 3.
As there were five Tones in every Octave, these ratios of
16 to 17 and 17 to 18 introduced five supplementary notes.
And as the Tones were e to d, d to c, b to a, a to g and g to f,
these supplementary notes belonged to d, c, d, g and f. But
the scales all show that d2, C2, , 82 and f, were the only
supplementary notes that had to stand a Fourth or Fifth
away from one or other of the original notes: so these must
represent the supplementary notes that were inserted in the
middle of each Tone.
1 The scales are all given by Alypios, eisagoge. See below, page 18.
2 Plutarch, de animæ procreatione, 18. Aristeides, de musica, iii. 1. Ptolemy,
harmonica, i. 10. Boetius, de musica, i. 16, iii. 1. Proclos, in Timæum, p. 195 A.
8 Α λείμμα and αποτομή together made a Tone, and the αποτομή was larger than
the nelupa-see Gaudentius, eisagoge, 14; Boetius, de musica, ii. 29; Proclos, in
Timæum, P. 195 D-so that, if the delupa took the ratio of 17 to 18, the åmotouń
would get the ratio of 16 to 17. And this was probably the genuine neiuua, the
interval of ratio 243 to 256 being properly a dieous. Thus in Theon, de musica, 36,
the interval of ratio 243 to 256 is distinguished as to diecialov neîupa.
1
GREEK MUSIC.
supplemen d, and is and I
Two more supplementary notes might then have been
inserted in each Tone by doubling the ratios of 16 to 17 and
17 to 18, and bringing in the ratios of 32 to 33 and 33 to 34
within the ratio of 32 to 34 and the ratios of 34 to 35 and 35
to 36 within the ratio of 34 to 361. But the notation shows
that only one more supplementary note was placed in every
Tone ; and as dy came between d, and d, cy between c2 and c,
ay between dy and Q, & 1 between 82 and g, and fi between fa
and f, the ratios of 34 to 35 and 35 to 36 were those that
fixed the five notes d1, 61, Q1, 81 and fi.
The interval that had the ratio of 35 to 36 was smaller
than the interval that had the ratio of 34 to 35; and the
whole interval of ratio 34 to 36 was smaller than the interval
of ratio 32 to 34 that was required to complete the Tone.
So these little intervals were far from being a quarter of
a Tone apiece, although they generally were reckoned as
such”. The usual name for them was Diesis, but originally
they bore the name of Diaschisma 3.
If similar methods were employed in dealing with the
intervals in which the ratio was 243 to 256, one supplementary
note was inserted there by doubling this ratio and bringing
in the ratios of 486 to 499 and 499 to 512, and then another
was inserted by doubling the latter ratio and bringing in the
ratios of 998 to 1011 and 1011 to 1024. And as c b and fe
were the intervals in which the ratio was 243 to 256, the first
of these supplementary notes was b, or e, and the second was
by or ej.
The eight original notes being given by strings whose
lengths were 384 and 432 and 486 and 512 and 576 and 684
and 729 and 768 respectively, the following list will show the
lengths required for all the supplementary notes, the figures
being multiplied by 4 in order to get rid of fracţions :-
1 Aristeides, de musica, iii. I.
2 Thus, for example, Baccheios, eisagoge, 8, and Vitruvius, de architectura, v. 4,
make a dieous a quarter of a Tone. Aristeides, de musica, iii. I, and Proclos, in
Timæum, p. 191 E, remark the error.
3 Boetius, de musica, iii. 8, quoting Philolaos, diaschisma (est) dimidium dieseos.
Cf. note 3 on page 11 for diesis in Philolaos.
GREEK MUSIC.
1536, e. 1728, d. 1944, C. 2048, 6.
1680,dy. 1890,67• 2022,64:
1632, d. 1836, cz. 1996, bą.
2304, a. 2592, g. 2916, f. 3072, e.
2240, Qz. 2520, 81: 2835,fi. 3033,61.
2176, Q2. 2448, 82: 27549f2. 2994, lg.
Having some doubts about the working of a scale con-
structed in this way, the author requested Messrs. Broadwood
to procure for him a series of ten tuning-forks to represent
the notes from e to b. The length of the strings was enough
to fix the relative pitch of all the notes, but there was nothing
to fix the absolute pitch of any one of them. For con-
venience, however, the author reckoned the pitch of e at 324
double vibrations per second, and thus obtained the following
pitch for the remaining notes, the rate of vibration varying
inversely as the length of string :-
1,324 d, 288 C, 256 b, 243
d, 2963 01, 263}} by, 24611401
d2, 30419 62,27111 b2, 24919
Or, putting the fraetions into decimals,
d1, 296.23 61, 263.31 b1, 246.12
dg, 304.94 62,271•06 b2, 249.33
The tuning-forks were made at Sheffield by Messrs.
Valantine and Carr, and were tested by the makers on
a Tonometer of their own. After they were delivered to
Messrs. Broadwood, they were tested again by Mr. Hipkins
on the late Mr. Ellis's copy of the Scheibler Tonometer,
now in the Science Department of the South Kensington
Museum? And these were the results :
, 323.43 d, 287•20 C, 255.64 b, 242.24
dj, 295033 (1, 262.45 64, 245.37
: d2, 304:45 02, 270-35 b2, 248:52
1 The author is greatly indebted to Messrs. Broadwood, and especially to
Mr. Hipkins, for taking such trouble in the matter.
GREEK MUSIC.
15
In every instance the rate of the vibrations is rather less
than was demanded. But the error is never more than 90 and
never less than .36; and being always in the same direction, it
has scarcely any effect upon the relative position of the notes.
Excepting the group of b, 61, 62, the notes all come out
clearly at intervals that can easily be recognized; and the
sequence is exceedingly melodious. But when b or by or by
is sounded by itself, there is some difficulty in saying which of
them it is; although the difference is obvious enough when
they are sounded in succession.
Now, the length of the strings for by and b, was settled on
the supposition that the intervals of ratio 243 to 256 were
treated like the intervals of ratio 8 to 9. As the lengths
87 and 8 were taken between 8 and 9, the lengths 2491 and
252 were taken between 243 and 256; and as these figures
were raised to 34 and 35 between 32 and 36 in the first case,
they were raised to 998 and 1011 between 972 and 1024 in
the second case, in order to eliminate the fractions. Suppose,
however, that the fraction was eliminated from 252 by the
simpler process of ignoring the saltogether, and treating the
length as 252. Some very curious results will follow.
The intervals 6 by and e e, will then be determined by the
ratio of 252 to 256, or 63 to 64. The intervals ddy, CC1, a Q1,
881 and ffi are determined by the ratio of 35 to 36. And an
interval of ratio 35 to 36 becomes an interval of ratio 63 to 64
on subtraction of the well-known interval of ratio 80 to 81.
Taking 2016 in place of 2022 and 3024 in place of 3033 as
the lengths for b, and , and then examining the seven
original notes and the seven with suffixy, there is the ratio of
I to 2 in ee, bye1, ff, &c., the ratio of 2 to 3 in eb, ey bu, fc,
&c., the ratio of 3 to 4 in e a, 86, &1C1, &c., the ratio of 4 to 5 in
81b, and C1@, the ratio of 5 to 6 in €181 and bydı, the ratio of
6 to 7 in ezg and b, d, the ratio of 7 to 8 in ab, and den, the
ratio of 8 to 9 in fg, f181, 8 a, &c., the ratio of 9 to 10
in ay by and dy 61, the ratio of 15 to 16 in en fi and b1c1,
the ratio of 27 to 28 in e, f and b4c, the ratio of 27 to 32 in
eg, a c, d,C1, &c., the ratio of.35 to 36 in ffi, 881, a dj,
110
GREEK MUSIC.
&c., and the ratio of 243 to 256 in ef and bc. Hence, with by
and ey in those positions, these fourteen notes involve the
fourteen most important ratios that Ptolemy employs in his
arrangement of the scales?. But seven of these fourteen
ratios depend on by and ex.
The four original notes b, c, d, e are separated by intervals
of ratio 243 to 256, 8 to 9 and 8 to 9, and thus produce the
ordinary Tétrachord of the Pythagoreans, which Ptolemy
called the Diatonic Tetrachord of Eratosthenes, and retained
in his own system as the Double-Tone Diatonic 2. The four
supplementary notes b1, C1, dy, ey are separated by intervals of
ratio 15 to 16, 8 to 9 and 9 to 10, and thus produce the
Tetrachord that Ptolemy described as the Severe Diatonic 3.
But the four notes b1, C,61,61 are separated by intervals of ratio
27 to 28, 35 to 36 and 4 to 5, and thus produce what Ptolemy
called the Enharmonic Tetrachord of Archytas. And such
coincidences make it fairly certain that by and ey have now been
put into their proper places.
Setting aside the Tetrachords that Ptolemy ascribes to
Aristoxenos—their structure being questionable there are
sixteen other ratios employed in his arrangement of the scales.
But all these ratios may be derived from those already quoted.
Thus the ratio of 9 to 10 gives rise to those of 18 to 19 and 19
to 20, and to those of 38 to 39 and 39 to 40. The ratio of 5
to 6 gives rise to those of 10 to it and 11 to 12, and to those.
of 20 to 21 and 21 to 22. The ratio of 15 to 16 gives rise to
those of 30 to 31 and 31 to 32; and this ratio of 15 to 16 with
that of 14 to 16, or 7 to 8, produces the ratio of 14 to 15.
Again, the ratio of 243 to 256 with that of 224 to 256, or 7 to
8, produces the ratio of 224 to 243. The ratio of 15 to 19
would come from those of 19 to 20 and 15 to 20, or 3 to 4.
And the ratios of 45 to 46, 23 to 24 and 24 to 25, or 46 to
48 and 48 to 50, might be obtained from that of 45 to 50, or
9 to 1o.
As an adjustment of b, and e, brought in seven of Ptolemy's
1 Ptolemy, harmonica, ii. 14.
? Alárovov ditovialov.
3 Alátovov OÚvtovov.
GREEK MUSIC.
17
ratios in addition to the seven that already were involved in
the original notes and those with suffix 1, the probabilities are
that some of the remaining ratios could be obtained by a proper
adjustment of b, and e, and the introduction of the other notes
with suffix z. But the author has not been fortunate enough
to find a likely place for b, or ez.
The length of 252 was obtained for by and ez by ignoring
the è in 252; and this suggests that the è in 2493 might be
ignored in dealing with b, and ez. Again, the ratio for bby
and e eq has been converted into 256 to 252, or 64 to 63; and
this suggests that the ratio for b, b, and en ég may possibly be
252 to 248, or 63 to 62. But neither 249 nor 248, nor any
other number in that neighbourhood, seems to give a
satisfactory result.
If the ratio of 248 to 256, or 31 to 32, is used provisionally
for determining the intervals 6 b, and e 62, the lengths of string
will have to be reduced from 1996 and 2994 to 1984 and 2976
for b, and eg. And the lengths for by and e, have already been
reduced from 2022 and 3033 to 2016 and 3024.
Then, supposing that an octave of twelve mean Semi-Tones
is reckoned as 1200, the intervals e e, and bb, may be reckoned
as 27 apiece, eye, and b, b, as 28 apiece, and exf and byc as 35
apiece; ff1,881, a dy, CCand dd, as 49 apiece, fifa, &182,
Qy 22, C1 C2 and dyd, as 50 apiece, and f28,82 a, a, b, cyd and d, e
as 105 apiece 1. And then the ancient notes may be arranged
as follows:-
o, e. 204, d. 408, c. 498,5. 702, a. 906,8. 1110, f. 1200, e.
155, d . 359, 67• 471, 61. 653, Az. 857,81: 1061,f1. 1173, €1.
105, dz. 309, Cz. 443, 62. 603, Az. 807,82. 1011, f2. 1145, 6z.
According to Alypios, the scales were formed of eighteen
notes apiece; and there were five-and-forty scales in all, or
fifteen scales of three types each-diatonic, chromatic, and
enharmonic. But in the chromatic and enharmonic types the
notes were just the same; and in the diatonic type the only
1 The formula 38% *(log. x --log. y) x 40 gives the number of mean Semi-Tones
in an interval of ratio x to y.
18
GREEK MUSIC.
variation was in notes 4, 7, 10, 14 and 17. Putting the dia-
tonic notes for 4 and 7 and 10 in brackets, and omitting the
notes from 12 to 18, as these are only a repetition of the
notes from 2 to 8 an octave higher up, the entire system
is reducible to this :-
1 2 3 4 (4) 5 6 7 (7) 8 9 10 (10) 11
HYPO-DORIAN f & 8182 (az) c czcz (dz) f fifa (52) az
HYPO-IASTIAN fa 82a az (6) cad dg (e) f2&& (a)
HYPO-PHRYGIAN g a aj az (c) d dyd, (f) & $1&2 (@) Ç
HYPO-ÆOLIAN 82 a, b bę (cz) dz e ez (fa) &2a az (b) 42
HYPO-LYDIAN a b 6, 62 (d) e zez (8) a dyan (c) d
DORIAN z C G7C2 (da) F fi fz ($2) az 5 ba (@a) da
IASTIAN b czd d, (e) f2&?(a) b b1 b2 (a) ?
PHRYGIAN c d d dą ($) { $1&2 (29) ¢ ¢1 C2 (da) f
ÆOLIAN (z dne ep (fa) &2a az () çad d, (e) fa
LYDIAN de ezez (8) a aqaz () d d d, (f) &
HYPER-DORIAN. d, f fif2 (82) az 5 b2 (92) de ez (£2) 82
HYPER-IASTIAN ? fa$ 82 (a) by 62 (d) e zez (f) a
HYPER-PHRYGIAN f & &182 (an) C 4162 (dy) f fifa (82) az
HYPER-ÆOLIAN $28za az (6) çad d, (e) 128 E2 (a) 5
HYPER-LYDIAN & a az az (?) d d d, (f) & 8182 (az) !
I 2 3 4 (4) 5 6 7 (7) 8 9 10 (10) 11
YGIAN
LIAN
DIAN
LAN
IAN
Of course, these scales all differ in pitch, like the modern
scales of different keys. But they do not differ, like the
modern scales of different modes, in the order of the inter-
vals alone. The difference here involves the nature of the
intervals as well.
GREEK MUSIC.
19
For example, in the Phrygian scale of c the only supple-
mentary notes with suffix î are d, and 81 and C1, with dy and gi
in the octave up above; and these are all determined by the
ratios of the Tone. But the Lydian scale of d has e, and ev,
which are determined by the ratios of the Leimma, as well as
Qy and d, and an, which come from Tones. And thus the
Lydian scale admits the well-known intervals of 231, or ratio
7 to 8, in de, and den, of 267, or ratio 6 to 7, in eng and 218,
and finally of 182, or ratio 9 to 10, in d, e1, which is thus
a minor Tone. And in the Phrygian scale these intervals are
not to be obtained.
Thus, in passing from the Phrygian to the Lydian scale
there was a change of mode as well as key; and a change of
mode had more effect in ancient than in modern music, since
it varied the selection of the intervals.
No doubt the Lydian music differed from the Phrygian
in many things besides this raising of the pitch from cc to dd
and the admission of these intervals of ratio 6 to 7 and 7 to 8
and 9 to 10; for the Phrygian was accounted fit for warriors
on the field of battle, while the Lydian was consigned to lazy
banquetting 1. But presumably the change of mode and key
together helped to make these different types of music arouse
such different emotions.
In several of these scales some curious intervals are formed
by the connexion of the Leimma with the segments of the
Tone, such as that of 195, or ratio 243 to 272, in azc at the
beginning of the Dorian scale. As a rule, the scales began
with major Tones of 204, or ratio 8 to 9. But in a scheme of
Ptolemy's for seven scales to suit the seven notes within an
octave, the seven intervals of ratio 7 to 8, 8 to 9, 9 to 10, &c.,
are taken in turn for the beginning of the scales2: so this rule
was not imperative. And there is really no justification for
altering the place of ag to make coc a major Tone, or for
tampering with the other notes that incidentally produce
abnormal intervals.
1 Plato, politeia, pp. 398 E, 399 A.
Ptolemy, harmonica, ii. II.
20
GREEK MUSIC.
A couple of examples will suffice to show the nature of the
music that was extracted from the ancient notes.
In the first of the Hymns from Delphi the passage about
Klurà ueyarómodis ’Abols has the notes f, g, 81,8, C2,C1, C, b1.
Here the voice ascends 204, ascends 49, descends 49, descends
603, descends 50, descends 49, descends go. The transcript
gives F, G, A flat, G, D, D flat, C, B. And there the voice
ascends 200, ascends 100, descends 100, descends 500, descends
100, descends 100, descends 1oo.
Again, in the second of these Hymns the words de čáuevos
åußpóray have the notes d, b, b, d, e, e1,e². Here the voice
descends 294, ascends 27, ascends 267, ascends 204, ascends
27, descends 27. The transcript gives D, B, C, D, E, F, E.
And there the voice descends 300, ascends 100, ascends 200,
ascends 200, ascends 1oo, descends 1oo.
In fact, the charm of ancient melodies was in the subtle
variation of the intervals through which they rose and fell ;
and all their charm is sacrificed when they are forced into
a modern scale.
III.
The duration of the notes was indicated by the syllables
to which they were attached. And thus, if syllables in verse
are classed as long and short, and a short is reckoned as half
a long, this mode of reckoning must be adopted for the notes
as well.
For example, take the music for the twelfth of the Homeric
Hymns 3. The notes may not be quite so ancient as the
words; but they can scarcely be condemned as spurious,
although their pedigree is incomplete.
1 Bulletin de Correspondance Hellénique, vol. xvii. plate 21 bis, and vol. xviii.
plate 25.
; Ibid., vol. xviii. plates 12 bis and 20.
3 Published in 1724 by Benedetto Marcello in his Parafrasi sopra li primi
venticinque Salmi, vol. iii. p. 132.
GREEK MUSIC.
21
d d deed d by bei & ei eg ei ea
Ahuntphúkopuov, oeuvNu beóv, õpxou' đeldelv,
a & & & & &t@zed d d && b1
Αυτήν, και κούρην, περικαλλέα Περσεφόνειαν.
61 62 661 61 by b1 €1&1 een eeee ea
Χαίρε, θεά, και τήνδε σάω πόλιν άρχε δ' αοιδής.
In the first and last lines the long syllables at the end are
sung to two notes each, and must therefore be reckoned as
pairs of short syllables in dealing with the music. And thus,
with the ordinary signs for long and short, the music is
ă ă ţ ă ě ă ţ bu ba e 8 e e a e a
:21. Bu
> : 10:
su: 15:
is. Do
*: 1
*20:10
not
set
1054
1&o: in
D04:
Top 100: DU:
so: io. 1
do: 10:
30
10:
But this is only a succession of long notes with short notes
of exactly half their length; and although a succession of
Minims and Crotchets, or Crotchets and Quavers, may be
suitable enough in music that has nothing but the Tones and
Semi-Tones of equal temperament, some refinement is
required here in music that involves the Diesis, Apotome, &c.
In all probability the duration of the notes was governed
by some such rules as those that governed the pitch, for
Hexameters were curiously like Octaves". Roughly speak-
ing, there were six Tones in an Octave, just as there were six
Feet in a Hexameter; a Tone contained a pair of Semi-
Tones, or one Semi-Tone and a pair of Quarter-Tones, just
as a Foot was either a Spondee of two long syllables or
a Dactyl of one long and two short; and the Fourth and
Fifth divided the Octave into two Tones and a half and three
and a half, just as the Cæsura divided the Hexameter into
two Feet and a half and three and a half. But, strictly
· The comparison is suggested by Aristeides, de musica, i. 14, 23.
22
GREEK MUSIC. '
speaking, the Octave was divided into two Tones and
a Leimma and three and a Leimma in place of two and
a half and three and a half, since the whole was rather less
than six entire Tones; and in every Tone the first of the
so-called Semi-Tones and Quarter-Tones was rather larger
than the second. Hence a Spondee would have made the
first note rather longer than the second, while a Dactyl would
have made the first note rather longer than the second and
third together, and also made the second rather longer than
the third. The long note before the Cæsura and the long
note at the end would each have been abbreviated to repre-
sent a Leimma, and likewise the pairs of short notes that
occasionally were used in place of them. But obviously the
analogy is somewhat strained.
No doubt the duration of the notes was modified to some
extent by their position in the verse ; and the signs for the
Cæsura and the Foot should perhaps be added to the ordinary
signs for long and short, as follows:-
dal d ça < | 1 | 2 | 3z õz ši şi či či eě ă
a šlo | || či čel & ă a š š li
or by die 56le ő őſlé čě le ě
:09,
100:
1
:
But possibly it would be better to give the words themselves,
for this formidable array of signs only touches the duration of
the notes so far as that was settled by the rhythm of the
verse.
A system of dots and dashes is employed in two of the
surviving bits of ancient music to determine the duration of
the notes or their intensity. There is a fragment of papyrus
at Viennal with a few notes of the music for the Orestes of
Euripides. The verses are-
κατoλοφύρομαι, κατoλοφύρομαι,
ματέρος αίμα σας, και σ' αναβακχεύει ;
Mittheilungen aus der Sammlung der Papyrus Erzherzog Rainer, vol. v. p. 66.
GREEK MUSIC.
23
.
ο μέγας όλβος ου μόνιμος έμ βροτοίς:
åvà dè daîços os
τις ακάτου θοάς τινάξας δαίμων
κατέκλυσεν δεινών πόνων ως πόντου κ.τ.λ.
.
In the papyrus the verses are divided by a species of
Kopwvis in the shape of Z; and there is always a dot above
this Z and a dot behind the note that follows. Thus there
is a Ž to divide karolopúpouai from matépos, and a po to give
the note for μα: s0 also a Z to divide αναβακχεύει from ο μέγας,
and an l: to give the note for ó; and again a Ź to divide
Bpotoîs from åvá, and an 1. to give the note for å But
evidently the scribe was puzzled by the half-verse dvà dè
daidos ós and completed it with tus åkátov doâs, putting a Ż
above the line to divide Doâs from Tiváčas and a v. to give
the note for ti, as though this were the beginning of another
verse. And having thus lost the guidance of the metre, he
makes a long scrawl after katék\voev, and tries to start another
verse at ós móvrov by adding ż and l. above the line.

.
!
...
.
.
.
:
.
1.
.:.
.
:::
9
.*.*
1.
.
.
.
..
СЕМВРоос
i.
a
- 9
CP
,
2
TOYON
*
.
.
ekspēlow 2733
.
.:.:
..
279 Le Kom
.
?
.
S
kt:
:
:
.
:
Si
Apparently there was a dot and dash - to mark the first
note of the fourth Foot in every verse. This is distinctly
1 Verses 338-343.
24
GREEK MUSIC.
visible above the note for ću in šui Bporois at the end of the
third verse, above the note for tov in åkátov doâs, which is
treated here as the end of the fourth verse, and above the
note for σεν in κατέκλυσεν, as though the scribe were trying to
make the fifth verse end with katÉKAVOEV móvov in order to
begin the sixth with ós móvTOV. There is also a trace of it
above the note for pup in katolopúpouai at the end of the first
verse; but nothing can be seen above the note for Bak in
åvaßakx evel at the end of the second?. In this verse, however,
the penultimate is long, whereas the other verses make it
short; and possibly the sign was not required here.
The fragment is not large enough to give the first note of
the second or third Foot in any of the verses, except the note
for kat in katékNVOEV, which must be treated here as the
beginning of the third Foot in the fifth verse. But there is
not any sign above this note.
The system is varied in a song inscribed on stone in honour
of a man named Seikilos. His monument was found at
Tralles, and is now at Smyrna.
ał ě code ă cad é diz o ā be a
όσον ζής, φαίνoυ μηδέν όλως σύ λυπού προς
Coed zd cz a big a czb d e ca a a afe.
ολίγον εστί το ζήν το τέλος ο χρόνος απαιτεί.
Excepting the first three words, this can be arranged in
verses of one type, as follows:-
undèvolws où AUTOû. -U - v- -
Trpos ólíyov ¿OTÈ TÒ Šñue vuu U- - -
Tà TếÁos ở xpopos datreĩ. 0 0 0 0 0 0 0 - -
The sign / is used here, like the sign Ż in the papyrus,
as a species of kopwvís for marking off the verses; and there is
always a dash-above the last note of the third Foot, just as
CCC
? The author has examined the papyrus.
2 Bulletin de Correspondance Hellénique, vol. xviii. plate 13.
GREEK MUSIC.
25
the papyrus has a dot and dash - above the first note of the
fourth Foot. Thus the words ocov Ss must form a verse
apart, as they answer to the endings of these other verses, où
AUTOŮ, TL TÒ Côv, åtaltel. And then the word palvov will also
form a verse apart. There is a dot underneath the kopwvis at
the end of this little verse •, and a dash – above the first
note in the verse that follows, although no dash or dot is
given to the first note in the other verses. And some dots
may be observed above the three notes of the second Foot in
the longer verses at the end, whereas the papyrus has only
a dot behind the first note of each verse.
There is altogether such a difference between the system in
the papyrus and the system on this stone, that no valid rules
can be established for the distribution of the dots and dashes.
But obviously the system is only meant to show the rhythm
of the verses, for the position of all these signs is determined
by the metre. In reality no signs were needed for this
purpose; and none were used in the remaining bits of ancient
music. Of course, such signs might possibly have been omitted
in the copying of pieces that are only known through manu-
scripts; but the Hymns discovered at Delphi are originals on
stone, and they have not these dots and dashes.
While the metre may have varied the duration of the notes
that normally were classed as long or short, it must certainly
have varied their intensity; for no rhythm can have its full
effect unless some stress is laid upon the syllables that come
at certain stages of the verse. So this lengthening or shortening
of the notes would be attended by an increase or a decrease in
their strength. Apart from this, the entire melody might
grow louder or softer in successive parts of an elaborate piece,
and the time might likewise grow faster or slower; but there
is nothing to indicate such changes.
Thus it is impossible to fix the absolute pitch, duration or
intensity of any of the ancient notes. The evidence only goes
to prove that certain notes were higher or lower, longer or
shorter, louder or softer than certain other notes. And in
dealing with their relative pitch, duration and intensity, the
pitch alone can be determined with precision.
26
GREEK MUSIC.
There is consequently a serious fault in much that has been
written of late about the music of the ancients. Transcripts
have been made upon a plan that necessarily gives the
wrong pitch to twenty notes out of every twenty-one and
does not necessarily give the right pitch to the twenty-first,
while it probably is just as much at fault in the duration of
the notes and their intensity. Arguments have been founded
on these transcripts, as though all originals had now been
superseded; and the conclusions are astonishing. In fact,
some ancient music has actually been compared to works of
Beethoven and Wagner, from which it differs fundamentally.
But this was perhaps to be expected when so many scholars
and musicians have plunged into a higher criticism of ancient
music without taking the trouble to make themselves acquainted
with the notes.

'
.
A FLUTE-PLAYER, TAKING A DRINK, WITH HER FLUTE-CASE SLUNG OVER HER LEG.
From a Greek vase in the Author's collection.
BY THE SAME AUTHOR.
ANCIENT SHIPS, Illustrated. Demy 8vo. 1894. Ios. 6d.
* Independent research, erudition without pedantry, and a respectable know-
ledge of modern seamanship and nautical terminology are conspicuous merits
of this treatise.'— Times.
,
Parla egli delle navi a remi ed a vela, delle loro dimensioni, dei materiali
impiegati, degli alberi, delle vele, della pittura, del timone, dei dipinti di
prora e di poppa, delle ancore, gomene, bandiere, fanali, scandagli, battelli,
ecc., ecc., in una parola, di tutti gli attrezzi e le particolarità inerenti alla
forma ed armamento marinaresco d' una nave a remi ed a vela, corroborando
le sue speditive affermazioni con un ricco e sovratutto ben scelto e corretto
materiale di citazioni originali, desunte dagli storici e poeti greci e latini.'
Rivista Marittima.
Das Verdienst des Verfassers liegt in der selbständigen und umfassenden
Sammlung und Verzeichnung der auf seinen Gegenstand bezüglichen literar-
ischen Uberlieferung aus dem Alterthum. Sie reicht von Homer bis auf die
Byzantiner und Kirchenväter und bietet beträchtlich mehr auch als die
ausführlichsten älteren Werke.'-Literarisches Centralblatt.
On y trouvera un exposé un peu sec et dogmatique, mais précis et appuyé
sur un solide échafaudage de textes et de monuments, de tout ce qui touche
à la structure proprement dite et à l'équipement des navires antiques : rames,
dimensions et tonnage, matériaux, coque, ancres, câbles, gouvernail, mâture,
signaux, tous ces sujets sont traités successivement avec une singulière maîtrise
des documents et un esprit critique fort aiguisé.'-Revue des Études Grecques.
His book contains the results of long, laborious, and careful research.
It is a scholarly work, and brings to the surface a vast amount of useful
information hitherto scattered on the bottom of the ocean of ancient history.'-
New York Times.
London: C. J. CLAY AND SONS,
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
AVE MARIA LANE.
Glasgow : 263 ARGYLE STREET.
BY THE SAME AUTHOR.
RHODES IN ANCIENT TIMES. Demy 8vo. 1885. Ios. 6d.
This is one of those thorough and exhaustive monographs on classical
archæology in which we English are too poor.'-Guardian.
· Eine recht fleissige Monographie über die Insel Rhodos, bei der die in den
Ausgrabungen zu Tage gekommenen kunstarchäologischen Fundstücke sowohl
wie die massenhaften Inschriften verarbeitet worden sind.'-Berliner Philo-
logische Wochenschrift.'
RHODES IN MODERN TIMES. Demy 8vo. 1887. 85.
"This book is a worthy sequel to the author's former work.'--Academy.
*Le sujet était intéressant et nouveau ; il a été traité avec soin et une connais-
sance peu commune des historiens de Byzance.'-Revue Critique.
MEMPHIS AND MYCENÆ, an Examination of Egyptian
Chronology and its application to the Early History of
Greece. Demy 8vo. 1896. 58.
“Though the results arrived at are chiefly negative, the book will be found
to contain much useful matter about Egyptian chronology, which the author
deals with in a learned, but withal highly businesslike, manner.'— Times.
'Egli ha il merito d' aver messo il dito sulla piaga, e di avere scosso la cieca
fede in una cronologia, che oramai minacciava di essere considerata siccome
canonica.'- La Cultura.
London: C. J. CLAY AND SONS,
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
AVE MARIA LANE.
Glasgow : 263 ARGYLE STREET.

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