The measurement of
Aristoxenus's Divisions of the Tetrachord
© 1999-2004 by Joe Monzo
PREFACE
I have been pondering the mathematics of Aristoxenus's
tetrachord
intervals
for a long time. This is the most rigorous mathematical
analysis I have been able to make of them.
Aristoxenus described the various different sizes of intervals
in his classification of shades and
genera of tetrachord scales.
But he never used any mathematics in his theory beyond simple
geometric subdivisions of the
tone. This was, in fact, the aspect
of his music-theory that was distinctively new, a result of the
formulation of classic geometry around this time by Euclid.
Aristoxenus adamantly maintained that all larger musical intervals
could and should be determined soley by ear. Both of these
basic tenets were in sharp contrast to
Pythagorean theory
and other rational harmonic theories.
Aristoxenus's theory was forgotten in Western Europe after the
Germanic invasions (400s AD), then rediscovered during the
Renaissance. The earliest surviving manuscript of his treatise
was written in Constantinople [Istanbul] around 1150, so his
theory seems to have remained known the whole time in the
Greek-speaking areas.
When his work was rediscovered by the Italians around 1450, the idea of
12-tone equal-temperament
(the oridinary scale in everyday
use today) was a topic of much debate, and many theorists
who advocated that tuning interpreted Aristoxenus's approximations
of small intervals as a justification for it. They were wrong,
or at any rate not entirely correct,
but the idea has persisted for over 500 years.
The key thing I discovered about Aristoxenus's theory is that
he was dividing *two* different sizes of semitone: the
limma
(about 90 cents) and the
apotome
(about 114 cents). Therefore,
his smaller divisions
(1/4-tones, etc.) also come in two sizes.
No one else ever realized this; they all assumed that his
theory used one size of semitone and therefore had only one
size of each of the small intervals. But I'm fairly certain that
his 'basic scale' used the usual (in his time)
Pythagorean
tuning, and thus had two different
semitones.
There are features to Aristoxenus's theories that go far beyond
mere description of intervals, and indeed features that go beyond
music alone and into philosophy; some of this has been treated
in Barker 1978 and Litchfield 1988. The 'General Introduction'
to Barker 1989 (p 1-27) provides a good overview of all the
musical aspects of Aristoxenus's theory. I will focus here
only on his intervallic divisions of the
tetrachord.
Thanks to Paul Erlich for asking a few simple questions which
inspired me to finally put all of this together.
NOTE ON MATHEMATICAL NOTATIONAL CONVENTIONS
The caret sign (^) indicates exponentiation: thus 2^2 means
'two squared' = 4, 3^3 = 'three cubed' = 9, etc.
Whenever Aristoxenus uses terminology which implies an equal
division of an interval, that will be expressed mathematically
by raising the interval to the power of that division: thus,
'half a tone' will be expressed by raising the 9/8 'tone' to
the power 1/2: (9/8)^(1/2), etc.
In the tabulations of intervals, the exact mathematical calculation
will be on the left, and the geometric expression of it (where
'^' is generally rendered as 'times', 'times' rendered as 'plus',
'divided' rendered as 'minus', and so on) on the right.
NOTE ON CITATIONS OF ARISTOXENUS'S BOOK
As with most other recent authors, my citations follow Meibom's
page and line numbers, which run consecutively from Book 1 thru
Book 2; thus 1.29 (the last page of Book 1) is followed by 2.30
(the first page of Book 2). They simply appear in square brackets
at the appropriate point in my text, in the format: book.page.line.
My transliterations of the Greek are based on the text given
in Macran 1902 (and I assume full responsibility for their
shortcomings); the text has been arranged according to Meibom's
page and line breaks, and the punctuation is from Meibom. Text
given by Meibom but omitted by Macran is indicated in {}.
The English translations are from Barker 1989 unless otherwise
indicated; there are bound to be discrepancies with Macran's text,
because for his text Barker used da Rios 1954, a source unavailable
to me.
|
ABOUT ARISTOXENUS'S TREATISE
Aristoxenus's treatise _Harmonika stoicheia_ (usually known by
its Latin name _Elementa harmonica_, 'the Elements of Harmonics')
appears to have been assembled from two different treatises
written at different times (see the comments in the introduction
on Aristoxenus in Barker 1989, p 119-125), and it seems to me to
have been drawn from notes that were taken down by students as
he lectured.
The treatise is divided into three books, of which both 1 and 2
cover more-or-less the same material presented in the same order,
with some differences in both content and presentation. Accordingly,
I will frequently cite parallel passages from both books of the
treatise simultaneously; I have not had the need to quote from
Book 3.
[Barker 1989, p 122]
Book II ... uses as a key concept the notion of melodic 'function'
(_dynamis_), implying in particular that notes are to be understood
as _dynameis_, not simply as pitches, and that the character of
melodic relations in general is to be understood 'dynamically',
by reference to the melodic role or character in which they are
perceptually grasped, not just through representations of the
sizes of intervals they involve. Book I says nothing of _dynamis_,
and pursues its quantitative analyses without any hint that they
will need reinterpretation. It explicitly defines a note as a
pitch. The books also differ markedly in style, the second being
much more leisurely, reflective and methodologically self-conscious.
It seems to me that there are strong grounds for supposing that
Books I and II are from different treatises, the first having
belonged, perhaps, to an earlier and rather less sophisticated
attempt at the project. Either book could have had a continuation
of roughly the sort we find in Book III, but our Book III must
in fact go with Book II, if only in view of its liberal use of
the concept of _dynamis_.
Aristoxenus was alive around the mid-300s BC in Athens,
but the earliest surviving manuscript of his treatise,
known as 'M', was written in Constantinople [now Istanbul]
around 1150, in Greek.
A synopsis of Aristoxenus's theory was given in the _Eisagoge_
of Cleonides around 100 AD. Note, however, that Cleonides
interpreted Aristoxenus's intervals as specific numerical
quantities that I think are incorrect. Cleonides's interpretation
has been widely misconstrued as the actual theory of Aristoxenus.
The classic early printing of Aristoxenus is Meibom 1652, which
presents the Greek text and parallel Latin translation, and includes
several of the other famous Greek music-theory treatises.
Barker 1989 [p 125] calls Laloy 1904 'the classic work on' Aristoxenus.
The standard English translation of Aristoxenus has long been
Macran 1902, and the most recent one is in Barker 1989, which is
comprehensively annotated and was my main source.
CONCEPTS BASIC TO ARISTOXENUS'S THEORY
Aristoxenus sought to separate the study of music from that of both
physics and mathematics. He refused to make use of any ratios in
his treatise, distinctly contra the Pythagoreans, and he also spoke
of the raising and lowering of pitch in terms of tension and relaxation,
which in his day would admit of no exact numerical measurement, rather
than in terms of string-length, from which ratios could be calculated.
[Landels 1999, p 131]
... ancient acousticians never solved the riddle of the relationship
between tension and pitch. ... they seem not to have attempted to
assign a numerical value to the tension (in terms of units of weight),
or to relate such a numerical value to the pitch. In fairness it
should be said that the true formula, by which the pitch varies
according to the square of the tension, is not one that would easily
be arrived at by chance.
Aristoxenus therefore chose to use the newly formulated methods of
geometry to pinpoint examples of his divisions of the tetrachord.
Thus, his music-theory was radically different from anything that had
come before in ancient Greece.
As Andrew Barker says [Barker 1989, p 119-120]:
In harmonics his work was truly revolutionary, as he would have
been the first to agree. It sets aside the researches of the
Pythagoreans as irrelevant and misdirected, and seeks in effect
to establish a wholly new science that will study music on the
basis of principles intrinsic to itself, not borrowed from physics
or mathematics.
The lack of rational accuracy is an intentional and fundamental
aspect of Aristoxenus's theory. Rapid developments had been taking
place in Greek musical practice in his day, and he wished to describe
the wide variety of intervals, which could not be done numerically with
the mathematics then available. Scientific music-theory had up to
that time been based primarily on arithmetic manipulation of integer
ratios as described by the followers of Pythagoras [see Burkert 1972].
Another school of music-theorists called the 'Harmonists' apparently
had as their goal the illustration of all the different scales thru
the use of a small 'basic unit' of intervallic measurement, with which
they diagrammed the scales next to one another within the same overall
pitch-space, presenting a geometric view based on a common metric, in
a diagram which Aristoxenus calls _katapyknosis_. But Aristoxenus
criticizes their methods as not portraying the pitches according to
their musical *function*.
His main goal is to assemble all the different tetrachordal divisions
into a comprehensive system which allows for modulation between various
different scales, which was something that had become an important element
in Greek music of his time.
Geometry was a new concept in Greek mathematics and Aristoxenus
made use of it in his descriptions because it enabled him to illustrate
irrational divisions of musical pitch-space. Thanks to more recent
developments in mathematics in modern times, we are now able, thru the
manipulation of roots and exponents, to probe further into the calculation
of his intervals.
It seems clear to me that Aristoxenus still intended for his audience
to understand certain of the harmonic relationships as ratios, even if
the others must be rendered in another way.
[2.55.2-12]
Epei de ton diastematikon megethon ta men ton symphonon etoi holos
ouk | echein dokei topon all heni megethei horisthai, e pantelos
akariaion tina, ta de ton diaphonon pollo hetton touto peponthe
kai dia tautas tas aitias poly mallon tois ton symphonon megethesi
pi|steiei he aisthesis e tois ton diaphonon; akribestate d' an eie
diaphonou diastematos lepsis he dia symphonias.
Of the magnitudes of intervals, those of the concords appear to have
either no range of variation at all, being determined to a single
magnitude, or else a range which is quite indiscernible, whereas those
of the discords possess this quality to a much smaller degree. Hence
perception relies much more confidently on the magnitudes of concords
than on those of discords. The most accurate way of constructing a
discordant interval will therefore be by means of concords.
(We will examine his method of 'Tuning by Concords' shortly.)
This certainly means that the 'octave', '4th', and '5th' are to be
determined by ear, which means that they will be measured as their
usual Pythagorean ratios 2:1, 4:3, and 3:2, and the intervals smaller
than the '4th' (the 'discords') calculated in relation to them,
by means of a series of alternating '4ths' and '5ths'.
Because of his adamant stance against using ratios (or string-lengths,
from which ratios can be easily calculated), it is quite difficult
to ascertain Aristoxenus's interval measurements in a numerically
exact way. Clues must be found by paying close attention to his
terminology and his described methods of tuning. Our findings will
be applied to his ideas about the ranges of the _loci_ for the two
moveable notes.
THE BASIC TETRACHORD, AND THE NOTE NAMES
Aristoxenus explains [1.22.5-21 and 2.46.20-22] that he will select
one tetrachord taken from the whole system of notes, with which he
will illustrate the intervals of various tetrachordal divisions.
He describes his intervals in terms of the tetrachord _meson_,
spanning a 'perfect 4th' below _mese_, which was the reference pitch
in Greek theory, and explains that for any given genus or shade,
the same intervals should appear in all other tetrachords of the
complete system.
The _tetrachord meson_ has these note-names, from highest
to lowest:
1) mese
2) lichanos
3) parhypate
4) hypate
The interval between _mese_ and _hypate_ is, as always in Greek
theory, a 'perfect 4th'. These two notes are both fixed in pitch,
whereas Aristoxenus states that the _lichanos_ and _parhypate_
are both moveable to various postions in the tetrachord, within
their respective ranges [1.22-23, 2.46-47].
The various intervals resulting from these moveable notes determined
the different _genera_ of ancient Greek theory, of which three had
been acknowledged and described by Archytas [Barker 1989, p 46-52]:
the _diatonic_ ('thru the tones'), the _chromatic_ ('thru the colors'
or 'shades'), and the _enharmonic_ ('thru proper attunement').
Aristoxenus, no doubt trying to accomodate the proliferation of
widely divergent varieties of genera used in practice in his day,
divided the 'diatonic' and 'chromatic' genera further into 'shades'
or 'colors':
[2.50.25...] treis de chroma-[26]tikai,
he te tou malakou chromatos,
[27] kai he tou hemioliou,
kai he tou toniaiou.
Three divisions [of the tetrachord] are chromatic, those of the
soft, the hemiolic and the tonic chromatic.
. . . . . . . .
[2.51.22...] Eisi de duo diatonou di-[23]aireseis,
he te tou malakou, kai he tou synto-[24]nou.
There are two divisions of the diatonic, those of the soft
diatonic and the tense.
(Barker and almost everyone else translate _malakon_ as 'soft',
but I prefer 'relaxed', as it indicates the opposite of _syntonon_
'tense', with the pair thus illustrating an important concept in
Aristoxenus's theory.)
He stated further that these were only examples, and that an infinite
variety of other shades was possible, including those for the 'enharmonic'.
Greek theorists reckoned their scales downward (the opposite of
the way we do today), and the largest intervals were always at
the top of the tetrachord, with the smallest at the bottom.
The 'characteristic interval' of a tetrachord is the largest one
(or the 'tone' in the case of the 'tense diatonic' genus).
LOCATION OF THE FIXED NOTES
We will give a ratio of 1:1 to _mese_. As we go down in pitch
the ratios will fall between the octave bounded by 1 at the top
and 1/2 at the bottom. The cents value of _mese_ will always
be 0 cents, with cents values of intervals below _mese_ given
as negative values. Rather than relating Aristoxenus's tetrachord
pitches to our modern ideas of a scale, this will give us
a clear impression of the ancient Greek conception.
_Hypate_ is by definition a 'perfect 4th' below _mese_:
1/(4/3) = 3/4 = 0.75 = ~-498.045 cents.
Aristoxenus states at several places that 'the fourth is two and
a half tones' [1.24.4-10, 2.46.2], and gives an elaborate 'proof'
that this is so [2.56.14-58.4]. Litchfield addresses this claim:
[Litchfield 1988, p 64-65]
The crux of this proof is whether the fifth is a full perfect
fifth. In concept, there is no reason to admit this interval as
anything but a fifth; in practice, however, it does not sound like
a fifth. Had Aristoxenus performed this proof, he would surely
have heard the discrepancy, as would any other musicician used to
manipulationg the monochord.
Nevertheless, the discrepancy between theory and practice appears
not to have bothered Aristoxenus. He changed nothing in his treatise
because of it. ... Thus, the concept alone must have been Aristoxenus's
concern, and he appears not as an empiricist but as a conceptualist
or conceptual idealist.
I believe that this is an extremely important aspect of Aristoxenus's
theory, and that it in fact implies that Aristoxenus tempered all
of the 'concords' so that the final one would audibly be indistinguishable
from the rest. I feel that it is presumptuous to assume that
Aristoxenus would have been incapable of conceiving a tempered
system, because the mathematics to approximate it had already
been recorded 1000 years earlier by the Babylonians. If Aristoxenus
did in fact conceive of his system as being a closed, tempered
one, then 144-tET is indeed exactly what he was describing.
But let us proceed on the assumption that his 'concords' were
the Pythagorean ratios. So
without getting entangled into this argument, and considering
Aristoxenus's claim that the 'discords' and their related 'remainders'
could be found by means of concords (to be discussed shortly), we can
here simply assume that he means that a '4th' may be divided into
2 Pythagorean 'tones' and 1 Pythagorean 'lesser semitone' or 'limma'.
Calculating these intervals downward from _mese_:
ratio decimal ratio cents
1/(((9/8)^2)*(256/243)) 0.75 ~-498.045
results in the same ratio of 3/4 for _hypate_.
These notes may be rendered in letter-names which make them
easier to understand to modern readers. We shall call _mese_
by the first letter 'A'; the _hypate_ will thus be the 'E'
a 'perfect 4th' below 'A'.
THE _PYKNON_
The _pyknon_ indicates 'compression', and refers to the grouping
together of small intervals at the bottom of the tetrachord.
[1.24.11-14]
Pyknon de legestho to ek duo diastematon synestekos ha
syntethenta elatton diastema periexei tou deipomenou
diastematos en toi dia tessaron.
Let us call _'pyknon'_ that which is composed of two intervals
which, when put together, cover an interval smaller than that
which makes up the remainder of the fourth [i.e., 'perfect 4th'].
. . . . . . .
[2.50.15...] Pyknon de legestho mechri toutou,
[16] heos an en tetrachordo
dia tessaron [17] symphonounton ton akron,
ta duo dia-[18]stemata syntethenta,
tou henos elatto [19] topon kateche.
Let us use the term '_pyknon_' for every case where, in
a tetrachord whose extremes form the concord of a fourth,
the two intervals put together occupy a smaller range than
the one.
Or, in mathematical terms, if the _pyknon_ is composed of intervals
x and y: x * y < (4/3)^(1/2).
[1.24.15-17]
Touton houtos horismenon pros toi baryteroi ton menonton
phthongon eilephtho to elachiston pyknon;
let us take the smallest _pyknon_, placed next to the lower of
the fixed notes [i.e., _hypate_].
This indicates that the _pyknon_ occurs at the bottom of the tetrachord,
with the larger interval at the top.
ARISTOXENUS'S BASIC UNITS OF MEASUREMENT
The 'tone' and its subdivisions
Aristoxenus defines his basic units of measurement:
[1.21.22-23]
Esti de tonos he ton proton symphonon kata megethos diaphora.
The tone is the difference in magnitude between the first two
concords.
. . . . . .
[2.45.35] Tonos d' estin ho to dia pente
[2.46.1] tou dia tessaron meixon;
The tone is that by which the fifth is greater than the fourth.
That is, (3/2)/(4/3) = 9/8 = ~203.910 cents. This is the standard
Pythagorean definition of the 'tone'. Aristoxenus continues:
[1.21.24-30]
Diaireistho d' eis treis diaireseis; meloideistho gar | autou
to te hemisu kai to triton meros kai tetarton; ta de touton
elattona diastemata panta esto ameloideta. Kaleistho de to men
elachiston diesis enarmonios elachiste, to d' echomenon | diesis
chromatike elachiste, to de megiston hemitonion.
It [the tone] is to be divided in three ways, since the half,
the third, and the quarter of it should be considered melodic.
All intervals smaller than these are to be treated as unmelodic.
Let the smallest of them be called the _least enharmonic diesis_,
the next the _least chromatic diesis_, and the greatest the _semitone_.
. . . . . . .
[2.46.3-8]
Ton de tou tonou meron melodeitai to hemisi, ho kaleitai hemitonion,
kai to triton meros, | ho kaleitai diesis chromatike elachiste,
kai to tetarton, ho kaleitai diesis enarmonios elachiste; toutou
d' elatton ouden melodeitai diastema.
Of the parts of the tone the following are melodic: the half,
which is called the semitone, the third part, which is called the
least chromatic diesis, and the quarter, which is called the least
enharmonic diesis. No interval smaller than that is melodic.
Aristoxenus thus calls every interval smaller than a quarter-tone
'unmelodic', meaning that they were unsuitable for use in melody
(i.e., actual music), but were useful in analyzation of the various
different shades and genera. He also takes pains to state that
more than two small intervals cannot be sung in succession in proper
melody, altho conceptually a tone may be divided into 3, 4, or more parts,
and 'that from a purely abstract point of view there is no least interval'
[2.46.19].
Dividing the 9/8 literally into 2, 3, and 4 equal parts, we get:
ratio decimal ratio cents
(9/8)^(1/2) ~1.060660172 ~101.955 'half-tone' semitone
(9/8)^(1/3) ~1.040041912 ~67.970 'third-tone' least chromatic diesis
(9/8)^(1/4) ~1.029883572 ~50.978 'quarter-tone' least enharmonic diesis
There are other statements in Aristoxenus's treatise, however, which
indicate that this general and simple description may not always be
all there is to it.
'Tuning by Concords'
Without using ratios, Aristoxenus illustrates the method of
determining 'discords' (i.e., any interval smaller than a 'perfect 4th')
by ear, by means of successive '4ths' and '5ths':
[2.55.13-23]
Ean men oun prostachthe pros to dothenti phthongo labein epi to
bary to | diaphonon oion ditonon ... epi to oxy apo tou dothentos
phthongou lepteon to dia tessaron, eit epi to bary to dia pente,
eita palin epi to | oxy to dia tessaron, eit epi to bary to dia
pente. kai houtos estai to ditonon apo tou lephthentos phthongou
eilemmenon to epi to bary.
if we have the task of constructing from a given note a discord
such as a ditone downwards ... we should construct from the given
note a fourth upwards, from there a fifth downwards, then another
fourth upwards, and then another fifth downwards. In this way
the ditone downwards from the given note will have been constructed.
This may be represented in the following diagram (with ratios,
letter-names, and cents for each note):
approx.
cents
500 4/3 D ~498
400 / \
300 (4:3) \ 32/27 C ~294
200 / (3:2) / \
100 / \ (4:3) \
0 1/1 A 0 \ / (3:2)
-100 \ / \
-200 8/9 G ~-204 \
-300 \
-400 64/81 F ~-408
-500
n^0 3^-1 3^-2 3^-3 3^-4
This results in standard Pythagorean diatonic tuning, with the
'target' pitches giving, in addition to the 'limma', the following
'discord' intervals reckoned from _mese_ (1/1):
ratio decimal ratio cents interval from _mese_ (1/1)
32/27 1.185185 ~ 294.135 'semiditone' ('minor 3rd') up
8/9 ~0.888889 ~-203.910 'tone' down
64/81 0.790123 ~-407.820 'ditone' down
[2.55.24-26]
ean d' epi tounantion prostachthe labein to diapho|non,
enantios poieteon ten ton symphonon lepsin.
And if the task is to construct the discord in the opposite
direction, the concords should be constructed the opposite
way round.
approx.
cents
500
400 81/64 C# ~408
300 /
200 9/8 B ~204 /
100 / \ (3:2)
0 1/1 A 0 / (4:3) /
-100 \ (3:2) \ /
-200 (4:3) / \ /
-300 \ / 27/32 F# ~-294
-400 \ /
-500 3/4 E ~-498
n^0 3^1 3^2 3^3 3^4
It should be quite clear that tuning by ear in this way will
result in a chromatic scale (in the modern sense of the word
'chromatic') comprised of the usual Pythagorean intervals.
The 'semitone'
Aristoxenus goes on:
[2.55.27-31]
Gignetai de kai ean apo symphonou diastematos to diaphonon
aphairethe dia symphonias kai to loipon dia symphonias eilemmenon;
aphaireistho | gar to ditonon apo tou dia tessaron symphonias;
Further, if a discord is subtracted from a concordant interval
by means of concords, the remainder will also have been found
by means of concords. For instance, let the ditone be subtracted
by means of concords from the fourth.
He then describes the procedure diagrammed above, finding a ditone
above the starting note, all over again, this time not from a single
given note, but from a given '4th', and describes the difference
between the last note found and the bottom note of the '4th' as
a remainder found by means of concords:
[2.55.31-56.12]
delon de hoti hoi ten hyperochen periechontes he to dia tessaron
hyperechei tou ditonou dia symphonias esontai pros allelous
eilemmenou; hypar||chousi men gar hoi tou dia tessaron horoi
symphonoi; apo de tou oxyterou auton lambanetai phthongos
symphonos epi to oxy dia tessaron, apo de tou le|phthentos
heteros epi to bary dia pente, (eita palin epi to oxy dia
tessaron,) eit' apo toutou heteros epi to bary dia pente.
kai peproke to teleutaion symphonon epi ton exyterou ton
hyperochen horizonton, host' einai pha|neron, hoti,
ean apo symphonou diaphonon aphairethe dia symphonias, estai
kai to loipon dia symphonias eilemmenon.
It is clear that the notes bounding the remainder by which
the fourth exceeds the ditone have also been constructed, by
means of concords, in their relation to one another. The
notes bounding the fourth are themselves concordant. From
the higher of them we find a note concordant at a fourth above,
from that note another a fifth below, then again one a fourth
above, and then from that note another a fifth below. The
last of these concordant intervals falls on the higher of
the notes bounding the remainder in question: and it is thus
clear that if a discord is subtracted from a concord by means
of concords, the remainder will also have been found by means
of concords.
approx.
cents
500 4/3 D ~498
400 / \
300 (4:3) \ 32/27 C ~294
200 / (3:2) / \
100 / \ (4:3) \
0 1/1 A 0 \ / (3:2)
-100 | \ / \
-200 (4:3) 8/9 G ~-204 \
-300 | \
-400 | 64/81 F ~-408 \
-500 3/4 E ~-498 3/4 E ~-498 / 'remainder'
This remainder is none other than
(64/81)/(3/4) 1.053498 ~90.225 cents
or the usual Pythagorean 'lesser semitone' or 'limma', which is
~11.730 cents smaller than (9/8)^(1/2). This difference, between the
semitone derived by Aristoxenus's tuning methods and the one implied by
his terminology of 'semitone' or 'half a tone', is a noticeable one to
most ears, and should probably be resolved.
[2.50.22...] Mia men [23]oun ton diaipeseon
estin enarmonios,
[24]en he to men pyknon, hemitonion esti;
to [25]de loipon, ditonon.
One of these divisions [of the tetrachord] is enharmonic, in
which the pyknon is a semitone and the remainder a ditone.
. . . . . .
[1:23.12-22]
12 ..... Ο'ι
μεν
γαρ
τηι
νυν
κατεχουσηι
13 μελοποιιαι
συνηθεις
μονον
οντες
εικο-
14 τως
την
διτονον
λιχανον
εξοριζουσι•
15 συντονωτεραις
γαρ
χρωνται
σχεδον
ο'ι
16 πλειστοι
των
νυν.
τουτου
δ'
αιτιον
το
βουλε-
17 σθαι
γλυκαινειν
αει.
σημειον
δ'
'οτι
του-
18 του
στοχαζονται.
μαλιστα
μεν
γαρ
και
19 πλειστον
χρονον
εν
τωι
χρωματι
δια-
20 τριβουσιν.
'οταν
δ'
αφικωνται
ποτε
εις
21 την
'αρμονιαν,
εγγυς
του
χρωματος
22 προσαγουσι
συνεπισπωμενου
του
ηθους.
|
....... Hoi men gar tei nun katechousei
melopoiiai synetheis monon ontes eiko-
tos ten ditonon lichanon exorizousi;
syntonoterais gar chrontai schedon hoi
pleistoi ton nun. toutou d' aition to boules-
thai glukainein aei, semeion d' hoti tou-
tou stochazontai, malista men gar kai
pleiston chronon en toi chromati dia-
tribousin, hotan d' aphikontai pote eis
ten harmonian, engus tou chromatos
prosagousi synepispomenou tou ethous.
|
....... It is to be expected that those who are used
only to the style of composition at present in vogue
rule out the ditone lichanos,
since most people nowadays use
higher ones. The reason is their
endless pursuit of sweetness: that this is
their objective is shown by the fact that
they spend most time and effort on the chromatic,
whereas when they do occasionally come to
the enharmonic they force it close to the chromatic,
and the melody is correspondingly pulled out of shape.
|
This almost certainly means narrowing the 'characteristic interval'
of the enharmonic tetrachord from a Pythagorean 81/64 ditone
[= ~408 cents] to a 5/4 'major 3rd' [= ~386 cents], as had already
been documented by Archytas. But it is obvious that in Aristoxenus's
view, the 'proper' enharmonic tetrachord contains the ditone.
The 'limma' 256/243 would be the _pyknon_ remaining in the tetrachord
after subtracting the 'ditone' characteristic interval. Aristoxenus's
terminology concerning the enharmonic genus argues in favor of accepting
the 'limma' as his definition of 'semitone'.
There is another interesting point in connection with the 'sweetening'
of the enharmonic _lichanos_ to or towards the 5/4, and the 'semitonal'
relationship of that note to others in the basic Pythagorean diatonic
and chromatic genera.
ratio decimal ratio cents intervals from _mese_
(32/27)/(9/8) 1.0534979 ~ 90.225 semi-ditone - tone
(5/4)/(32/27) 1.0546875 ~ 92.179 'sweet' ditone - semi-ditone
(81/64)/(32/27) 1.0678711 ~113.685 apotome = true ditone - semi-ditone
(5/4)/(9/8) 1.1111111 ~182.404 smaller tone = 'sweet' ditone - tone
((5/4)/(9/8))^(1/2) 1.0540926 ~ 91.202 1/2 of smaller tone
The 32/27 'semi-ditone' (what we call the 'minor 3rd') is a limma
[256/243 = ~90.225 cents] larger than the 9/8 tone. The 5/4 interval
that Aristoxenus considered a 'sweetened' distortion of the enharmonic
'ditone' is a 5-limit 'lesser semitone' [135/128 = ~92.179 cents]
larger than 32/27, whereas the true 81/64 'ditone' is an apotome
[2187/2048 = ~113.685 cents] larger. 32/27 thus almost perfectly
bisects the 5-limit 'smaller tone' [10/9 = ~182.404 cents] between
5/4 and 9/8; the exact midpoint is (10/9)^(1/2) = ~91.202 cents.
Even tho Aristoxenus described this 5/4 as being 'forced close to
the chromatic', it still functioned as the *enharmonic* _lichanos_.
Since the 9/8 and 32/27 measure the _lichanoi_ in the standard
Pythagorean diatonic and chromatic genera, respectively, the
similarity of 'step size' between these three notes would encourage
Aristoxenus to think of the progression 9/8 : 32/27 : 5/4 as a series
of nearly even 'semitones'.
Note, however, that the difference between the 'sweet' ditone [5/4]
and the true ditone [81/64] is the syntonic comma [= ~21.506 cents].
There are ~9.4814124 [= log(9/8)/log((81/64)/(5/4))] of these commas
in a tone, making the comma therefore between a 1/9- and a 1/10-tone.
Aristoxenus explicity says in his descriptions that he expects
us to perceive a 1/12-tone difference between _lichanoi_ of two
of his genera, and implies that we should perceive a 1/24-tone
difference between two others [see below].
We have already seen how he stipulated and preferred the 81/64
'ditone'. This blurring of distinctions between functionally-identical
pitches was perhaps a further reason why Aristoxenus did not use
rational values to locate his pitches. He said nothing else about
the measurement of the enharmonic _lichanos_, beyond the fact that
the upper boundary of its range is conjunct with the lowest chromatic
_lichanos_, while he meticulously located the various chromatic
_parhypatai_ at smaller distances. This seems to indicate that he
needed to blur the distinctions between the two sizes of semitone
in order for his calculations to come out right. We will explore
Aristoxenus's 'semitone' further, below.
The 'quarter-tone' and 'enharmonic diesis'
Thus, the enharmonic _pyknon_ will be a 256/243 'semitone',
and dividing this exactly in half gives (256/243)^(1/2) for the
'enharmonic diesis':
ratio decimal ratio cents
(256/243)^(1/2) ~1.026400479 ~45.112 enharmonic diesis.
This interval is ~5.865 cents smaller than (9/8)^(1/4), the ratio
implied by use of the term 'quarter-tone'. This may or may not
be a noticeable difference, depending on various factors.
Based on his meticulous description of how to tune the 'discords',
and on the way he describes the 'proper' enharmonic genus, it
makes the most sense to assume that Aristoxenus intends to imply
the ratio 256/243 for 'semitone', and its geometrically-equal
division, the irrational interval (256/243)^(1/2), for 'enharmonic
diesis' or 'quarter-tone'.
The 'third-tone' and 'chromatic diesis'
One of Aristoxenus's most brilliant discoveries is that
1.5 * enharmonic diesis [= limma^(3/4)] is almost exactly
the same as '1/3-tone' [= tone^(1/3)], with less than 1/3-cent
difference between the two:
ratio decimal ratio cents cents difference
(9/8)^(1/3) ~1.040041912 ~67.970
((256/243)^(1/2))^1.5 ~1.039860949 ~67.669 ~0.301
For now we may assume the irrational interval (9/8)^(1/3)
[= ~67.97 cents] as the definition of both 'third-tone' and
'least chromatic diesis', altho this will cause problems
upon closer analysis, which will be encountered later in this paper.
THE RANGES OF THE MOVEABLE NOTES
Aristoxenus describes the ranges for the two moveable notes in
the tetrachord as follows:
The range of the _lichanoi_
|
[1.24.22-25]
katholou gar barytatai men hai enarmonioi lichanoi esan,
echomenai d' hai chromatikai, syn|tonotatai d' hai diatonoi.
|
taken overall, the enharmonic _lichanoi_ are the lowest,
the chromatic next, and the diatonic the highest.
|
|
[1.22.27-30]
Lichanou men oun esti toniaios ho sympas topos en hoi kineitai,
oute gar elatton aphistatai meses toniaiou diaste|matos oute
meizon ditonou.
|
The total range in which _lichanos_ moves is a tone, since it
does not stand at less than the interval of a tone from _mese_,
nor at a greater interval than a ditone.
|
|
[2.46.28-33]
phainetai de syntonotate men einai lichanos he tonon apo meses
apechousa, | poiei d' haute diatonon genos, barytate d' he ditonon,
gignetai d' haute enarmonios; host' einai phaneron ek touton,
hoti toniaios estin ho tes lichanou topos.
|
It appears that the highest _lichanos_ is that which lies at
a tone from _mese_, and creates the diatonic genus; and the lowest
is that at a ditone from _mese_, which belongs to the enharmonic.
Hence it is clear that the range of _lichanos_ is a tone.
|
Since these passages make use only of the terms 'tone' and 'ditone'
to describe the intervals, we may assume a fairly straightforward
Pythagorean tuning:
total range of _lichanos_ = 9/8 tone
highest lichanos = 9/8 below _mese_
lowest lichanos = 81/64 ditone below _mese_
ratio decimal ratio cents
1/(9/8) ~0.888889 ~203.91 highest lichanos
1/((9/8)^2) ~0.790123 ~407.82 lowest lichanos
The difference between these two is ~203.91 cents, or a
Pythagorean 9/8 'tone'.
On the infinite variety of possible _lichanoi_:
|
[1.26.13-27]
Noeteon gar apeirous ton arithmon tas lichanous; ohu gar | an
steseis ten phonen tou apodedeigeenou lichanoi topou lichanos estai,
diakenon d' ouden esti tou lichanoeidous topou oude toiouton hoion
me dechesthai lichanon. Host' einai me peri mikrou ton | amphisbetesin;
hoi men gar alloi diapherontai peri tou diastematos monon,
hoion poteron ditonos estin he lichanos e syntonotera hos
mias ouses enarmoniou; hemeis d' ou monon pleious en | hekastoi
genei phamen einai lichanous mias alla kai prostithemen hoti
apeiroi eisi ton arithmon.
|
For it must be understood that the lichanoi are unlimited in number.
Wherever you arrest the voice in the range that accommodates the
_lichanos_ will be a _lichanos_, and no place in the lichanos range
is empty or incapable of receiving a lichanos. Hence the present
controversy is of no little importance. Other people argue only
about the interval in question, for instance whether the lichanos
stands at a ditone or is higher, as if there were only one enharmonic
lichanos. But we not only say that there is more than one _lichanos_
in each genus, but also add that they are unlimited in number.
|
|
[1.26.24-27]
hemeis d' ou monon pleious en | [25] hekastoi genei phamen einai
lichanous mias alla kai prostithemen hoti apeiroi eisi ton arithmon.
|
we not only say that there is more than one _lichanos_ in each genus,
but also add that they are unlimited in number.
|
|
[1.26.14-18]
ohu gar | an steseis ten phonen tou apodedeigeenou lichanoi
topou lichanos estai, diakenon d' ouden esti tou lichanoeidous
topou oude toiouton hoion me dechesthai lichanon.
|
Wherever you arrest the voice in the range that accommodates the
_lichanos_ will be a _lichanos_, and no place in the lichanos range
is empty or incapable of receiving a lichanos.
|
The range of the _parhypatai_
[1.23.27-29]
let the range of ... _parhypate_ [be agreed to be] the smallest
diesis, since it never comes closer to _hypate_ than a diesis
and is never more than half a tone away from it.
[1.23.27-29]
let the range of ... _parhypate_ [be agreed to be] the smallest diesis
[i.e., '1/4-tone'], since it never comes closer to _hypate_ than a diesis
[i.e., (256/243)^(1/2) above _hypate_] and is never more than half a tone
away from it [i.e., either (9/8)^(1/2) or 256/243 above _hypate_].
Aristoxenus's loose terminology here, using 'half a tone' instead of
'semitone', admits of the possibility of interpreting 'half a tone' to
be (9/8)^(1/2), rather than the 256/243 'limma' which is more likely.
Let us examine both:
Based on '1/2-tone' and '1/4-tone':
ratio decimal ratio cents
(3/4)*((9/8)^(1/2)) 0.795495 ~-396.090 highest parhypate
(3/4)*((9/8)^(1/4)) 0.772413 ~-447.067 lowest parhypate
The difference between these two, which is the 'enharmonic diesis',
is of course (9/8)^(1/4), or the literal '1/4-tone' of ~50.978 cents.
Based on a 'limma' semitone, and an 'enharmonic diesis' as half of that:
ratio decimal ratio cents
(3/4)*(256/243) 0.790123 ~-407.820 highest parhypate
(3/4)*((256/243)^(1/2)) 0.7698 ~-452.933 lowest parhypate
The difference between these two is the diesis we calculated above
of ~45.112 cents, which is exactly half of the Pythagorean 'lesser
semitone' or 'limma'.
We thus have two mutually incompatible explanations of the range of
_parhypate_, and must search further for clues to solving this dichotomy.
|
|
[1.23.30-24.1]
the ranges do not overlap, but their point of conjunction is their
limit, since when _parhypate_ and _lichanos_ reach the same pitch,
as the one is tensed and the other relaxed, their ranges have their
limit, the range below it being that of _parhypate_, the range above
it that of _lichanos_.
|
[2.46.34-47.8]
to de parypates diastema elatton men hoti ouk
an genoito dieseos || enarmoniou phaneron, epeide panton ton
melodoumenon elachiston esti diesis enarmonios; hoti de kai
touto eis to diplasion auxetai, katanoeteon. hotan | gar epi
ten auten tasin aphikontai he te lichnos aniemene kai he parypate
epiteinomene, horizesthai dokei ekateras ho topos. host' einai
phaneron,
|
That the interval between _parhypate_ and _hypate_ cannot be
less than an enharmonic diesis is obvious, since the enharmonic
diesis is the smallest of all melodic intervals: that it too will
increase to double the size remains to be shown. When _lichanos_
in its descent and _parhypate_ in its ascent reach the same pitch,
the range of each evidently arrives at its limit, so that it is
plain that the range of _parhypate_ is not greater than the smallest
diesis.
| |
In view of these statements, as well as what he says about the proper
enharmonic containing a 'ditone', with most contemporary musicians
explaining it as being higher ('closer to the chromatic'), Aristoxenus
must have intended the latter division presented in our table for the
range of _parhypate_, as only this _parhypate_ reaches the pitch of
the lowest _lichanos_ determined above and gives the true Pythagorean
'ditone' with the ratio 64/81.
Another important point which Aristoxenus stresses frequently is
that the genera exhibit characteristic sounds which are capable of
accomodating an *infinite* variety of intervals within their
defined ranges. Thus:
- all _lichanoi_ from the highest diatonic to the lowest diatonic
are to be considered 'diatonic',
- all _lichanoi_ from the lowest diatonic to the lowest chromatic are
to be considered 'chromatic', and
- all _lichanoi_ below the lowest chromatic are to be considered
'enharmonic';
- all _parhypatai_ below the lowest chromatic are to be considered
'enharmonic', and
- all other _parhypatai_ are to be considered common to diatonic and
chromatic.
He further explains on this last point that mixed genera sometimes occur,
which make use of a diatonic _lichanos_ with a chromatic _parhypate_.
|
[2.52.5 ...] tettaron d' ouson parypaton,
he [6] men enarmonios idia esti tes harmo-[7]nias;
hai de treis koinai tou te diatonou,
[8]kai tou chromatos.
|
Of the four _parhypatai_, the enharmonic one is peculiar
to the enharmonic genus, while the other three are common
to the diatonic and chromatic.
|
Mathiesen illustrates distinctions in Aristoxenus's terminology at
this point, which have been missed by most editors and translators:
|
[1.26.30-33, text from Mathiesen 1976, eliminating emendations]
parypates de duo eisi topoi - ho men koinos tou te diatonou kai
tou chromatos, ho d' heteros idios tes harmonias - ; koinonei
gar duo gene ton parypaton. enarmonios men oun esti parypate
pasa he barytera tes barytates chromatikos, ...
|
[Mathiesen 1976, p 14, translating Aristoxenus 1.26.30-33]
There are two positions for the parhypate, the one common to
the diatonic and _color_, the other unique to the _Harmonia_,
for the two genera share the parhypate. For every parhypate
is _enharmonic_ which is lower than the lowest _chromatic_
[parhypate], ...
|
According to the emphasis Mathiesen gives these two pairs of words,
Aristoxenus makes the important point [1.26] that there are two
different *types* of _parhypate_: one that can be moved within
the locus for all the shades of diatonic and chromatic (regardless
of which of these two genera is in use), and the other one unique
to the _harmonia_, 'the divine ordering of the Universe', which
would be used only in the enharmonic genus.
We will thus assume that the largest interval which may appear at
the bottom of the tetrachord, in either the diatonic or chromatic
genera, will be the 'limma' 256/243.
|
[2.51.32] Lichanoi men oun eisin hex.
mia enarmo-[33]nios, treis chromatikai, kai duo diato-[34]noi.
{parypatai de tettares,} hosai per ai
[2.52.1] ton tetrachordon diaipeseis.
parypatai [2] de duo{in} elattous. te gar hemitoniaia
[3] chrometha pros te tas diatonous, kai
[4] pros ten tou toniaiou chromatos diaipe-
[5]sin.
|
There are thus as many _lichanoi_ as there are divisions of the
tetrachord [i.e., six], and two fewer _parhypatai_, since we use
that which stands at a semitone both for the diatonic divisions
and for that of the tonic chromatic.
|
In order to ascertain as precisely as possible the boundaries
of the generic ranges, it is necessary to determine first the exact
locations of the 'highest' and 'lowest' _lichanoi_ and _parhypatai_
acceptable into each of Aristoxenus's genera.
LOCATING THE MOVEABLE NOTES
Aristoxenus presents [1.24.16-32] six different _systemata_,
each one bounded by an interval which is successively larger
than the last. The four smallest are referred to as _pykna_,
but the last two are not; as Barker says, they are:
[Barker 1989, p 143, note 100]
Called by the general name _systema_ because [they exceed]
the magnitude properly called _pyknon_.
===========
[1.24.15-25.11]
[1.24.15-25.11]
> [Meibom:]
15> Touton houtos horismenon, pros toi
16> baryteroi ton menonton phthongon eile-
17> phtho to elachiston. pyknon d' estai to
18> ek duo dieseon enarmonion kai chro-
19> matikon elachiston. esontai duo de li-
20> chanoi eilemmenai duo genon bary-
21> tatai. he men harmonias; he de, chro-
22> matos. katholou gar barytatai
23> men hai enarmonioi lichanoi esan.
24> echomenai de, hai chromatikai. syn-
25> tonotatai de, hai diatonoi. Meta tau-
26> ta triton eilephtho pyknon pros toi
27> autoi. tetarton eilephtho pyknon to-
28> niaion. pempton de pros toi autoi,
29> to ex hemitoniou kai hemioliou diastem-
30> matos synestekos systema eilephtho.
31> hekton de, ex hemitoniou kai tonou. Hai
32> men oun ta duo ta prota lephthen-
33> ta pykna horizousai lichanoi eiren-
34> tai; he de to triton pyknon horizousa
[25]
1> lichanos, chromatike men estin; kalei-
2> tai de to chroma, en hoi estin, hemiolion.
3> he de to tetarton pyknon horizousa li-
4> chanos, chromatike men estin; kaleitai
5> de to chroma, en hoi esti, toniaion. he de
6> to pempton lephthen systema horizousa
7> lichanos, ho meizon ede pyknou en.
8> epeideper isa esti ta duo toi heni, ba-
9> rytate diatonos estin. he de to hekton
10> lephthen systema horizousa lichanos,
11> syntonotate diatonos estin.
> [Macran:]
> Touton houtos horismenon pros toi baryteroi ton menonton
> phthongon eilephtho to elachiston pyknon; touto d' estai to ek
> duo dieseon autoi; touto de estai to ek duo dieseon> chromatikon elachiston.
> esontai de duo li|chanoi eilemmenai duo genon barytatai,
> he men harmonias he de chromatos. katholou gar barytatai men
> hai enarmonioi lichanoi esan, echomenai d' hai chromatikai,
> syn|tonotatai d' hai diatonoi. Meta tauta triton eilephtho
> pyknon pros toi autoi; tetarton eilephtho pyknon toniaion;
> pempton de pros toi autoi, to ex hemitoniou kai hemioliou
> diastem|matos synestekos systema eilephtho; hekton de to ex
> hemitoniou kai tonou. Hai men oun ta duo [ta] prota lephthenta
> pykna horizousai lichanoi eirentai; he de to triton pyknon
> horizousa || lichanos chromatike men estin, kaleitai de to
> chroma en hoi estin hemiolion. He de to tetarton pyknon horizousa
> lichanos chromatike men estin, kaleitai | de to chroma en hoi
> esti toniaion. he de to pempton lephthen systema horizousa
> lichanos, ho meizon ede pyknou en, epeideper isa esti ta duo
> toi heni, barytate diatonos estin. he de to hekton lephthen |
> systema horizousa lichanos syntonotate diatonos estin.
>
>> Given these definitions, let us take the smallest pyknon, placed
>> next to the lower of the fixed notes. This will be the one
>> composed of two enharmonic or two of the smallest chromatic
>> diesis. The two lichanoi thus specified will be the lowest in
>> each of the two genera, one in the enharmonic, the other in the
>> chromatic: for we have explained that, taken overall, the
>> enharmonic lichanoi are the lowest, the chromatic next, and the
>> diatonic the highest. After these, consider a third pyknon
>> placed next to the same note, and then a fourth one, which
>> is a tone: fifthly, from the same note take the systema composed
>> of a semitone and an interval one and a half times as great, and
>> sixthly, that composed of a semitone and a tone.
>> We have already spoken of the lichanoi bounding the first two
>> pykna listed. The one that bounds the third is chromatic, and
>> the chroma in which it is is called hemiolic. That bounding the
>> fourth pyknon is chromatic, and the chroma in which it is is
>> called tonic. The lichanos bounding the fifth systema mentioned,
>> which was specified as greater than a pyknon, since the two
>> intervals are equal to the one, is the lowest diatonic: that
>> which bounds the sixth systema mentioned is the highest diatonic.
[1.25.11-26.7]
[Meibom:]
[25]
11> .................. He men oun
12> barytate chromatike lichanos tes
13> enarmoniou barytates hektoi merei to-
14> nou oxytera estin. epeideper he chro-
15> matike diesis tes enarmoniou dieseos
16> dodekatemorioi tonou meizon esti. Dei
17> gar to tou autou tritemorion tou tetartou
18> merous dodekatemorioi hyperechein.
19> hai de duo chromatikai ton duo
20> enarmonion delon hos toi diplasioi.
21> touto de estin hektemorion elatton dia-
22> stema tou elachistou ton meloidoume-
23> non. Ta de toiauta ameloideta
24> estin. ameloideton gar legomen, ho me
25> tattetai kath' heauto en systemati. He de
26> barytate diatonos tes barytates
27> chromatikos hemitonioi kai dodekate-
28> morioi tonou oxytera estin. epi men gar
29> ten tou hemioliou chromatos lichanon
30> hemitonion en ep' autes. apo de tes hemio-
31> liou epi ten enarmonion, diesis. apo
32> de tes enarmoniou epi ten baryta-
33> ten chromatiken, hektemorion. apo de
34> tes barytates chromatikes epi ten hemiolion, dodekatemorion tonou. to
[26]
1> de tetartemorion ek trion dodeka-
2> temorion synkeitai. host' einai phane-
3> ron, hoti to eiremenon diastema estin
4> apo tes barytates diatonou, epi ten
5> barytaten chromatiken. he de syn-
6> tonotate diatonOS tes barytates
7> diatonou, diesei esti syntonotera.
[Macran:]
> He men oun barytate chromatike lichanos tes
> enarmoniou barytates hektoi merei tonou oxytera estin, epeideper
> he chro|matike diesis tes enarmoniou dieseos dodekatemorioi
> tonou meizon esti. Dei gar to tou autou tritemorion tou tetartou
> merous dodekatemorioi hyperechein, hai de duo chromatikai ton
> duo | enarmonion delon hos toi diplasioi. touto de estin
> hektemorion, elatton diastema tou elachistou ton meloidoumenon.
> Ta de toiauta ameloideta estin, ameloideton gar legomen ho me |
> tattetai kath' heauto en systemati. He de barytate diatonos
> tes barytates chromatikos hemitonioi kai dodekatemorioi tonou
> oxytera estin. epi men gar ten tou hemioliou chromatos lichanon |
> hemitonion en ap' autes, apo de tes hemioliou epi ten
> enarmonion diesis, apo de tes enarmoniou epi ten barytaten
> chromatiken hektemorion, apo de tes barytates chromatikes epi
> ten hemiolion dodekatemorion tonou. to || de tetartemorion
> ek trion dodekatemorion synkeitai, host' einai phaneron, hoti to
> eiremenon diastema estin apo tes barytates diatonou epi ten |
> barytaten chromatiken. He de syntonotate diatonos tes barytates
> diatonou diesei esti syntonotera.
>
>> Thus the lowest chromatic lichanos is higher than the lowest
>> enharmonic by a sixth part of a tone, since the chromatic diesis
>> is greater by a twelfth part of a tone than the enharmonic diesis.
>> A third part of anything must exceed a quarter of the same thing
>> by a twelfth part, and the two chromatic dieses must evidently
>> exceed the two enharmonic ones by twice that amount. This is
>> a sixth part, an interval smaller than the least of the melodic
>> intervals. Such intervals are unmelodic, since we call 'unmelodic'
>> any interval that is not placed in a systema in its own right.
>> The lowest diatonic lichanos is higher than the lowest chromatic
>> by a semitone and a twelfth part of a tone. We said that it is
>> a semitone from the lichanos of the hemiolic chromatic, and from
>> there to the enharmonic lichanos is a diesis: from the enharmonic
>> to the lowest chromatic lichanos is a sixth part of a tone, and
>> from the lowest chromatic to the hemiolic lichanos is a twelfth
>> part of a tone. Now a quarter is composed of three twelfth parts,
>> so that it is clear that from the lowest diatonic to the lowest
>> chromatic lichanos is the interval stated. The highest diatonic
>> lichanos is higher than the lowest diatonic by a diesis.
[Monzo edit - it's a poem!]
>>
>> Thus the lowest chromatic lichanos
>> is higher than the lowest enharmonic
>> by a sixth part of a tone,
>>
>> since the chromatic diesis
>> is greater by a twelfth part of a tone
>> than the enharmonic diesis.
>>
>>
>> A third part of anything
>> must exceed a quarter of the same thing
>> by a twelfth part,
>>
>> and the two chromatic dieses
>> must evidently exceed the two enharmonic ones
>> by twice that amount.
>>
>> This is a sixth part,
>> an interval smaller
>> than the least of the melodic intervals.
>>
>> Such intervals are unmelodic,
>> since we call 'unmelodic'
>> any interval that is not placed in a systema in its own right.
>>
>> The lowest diatonic lichanos
>> is higher than the lowest chromatic
>> by a semitone and a twelfth part of a tone.
>>
>> We said that it is a semitone
>> from the lichanos of the hemiolic chromatic,
>>
>> and from there
>> to the enharmonic lichanos
>> is a diesis:
>>
>> from the enharmonic
>> to the lowest chromatic lichanos
>> is a sixth part of a tone,
>>
>> and from the lowest chromatic
>> to the hemiolic lichanos
>> is a twelfth part of a tone.
>>
>> Now a quarter
>> is composed of three twelfth parts,
>>
>> so that it is clear that
>>
>> from the lowest diatonic
>> to the lowest chromatic lichanos
>> is the interval stated.
>>
>> The highest diatonic lichanos
>> is higher than the lowest diatonic
>> by a diesis.
******************
Put them all together into a song that would help Aristoxenus's
students remember this important part of his lecture:
