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\centerline{{\Large {\bf {\em From Music to Mathematics: Exploring the Connections}}}}

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\centerline{{\large {\bf Gareth E. Roberts}}}


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\centerline{{\Large {\bf Recommended Music to Accompany Chapter 2}}}

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\centerline{{\Large {\bf Introduction to Music Theory}}}

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The selections below illustrate the tremendous changes that have occurred in classical music over
the last 1500 years, ranging from Gregorian chant to the atonal compositions
of Arnold Schoenberg.  At first, we hear music with few voices and simple
harmonies.  Later, the idea of tonality begins to take shape and the use of a specific
major or minor key becomes central to the underlying harmonic structure of a piece.
A radical shift occurs at the start of the twentieth century, when many composers discard
tonality altogether seeking more freedom and flexibility in utilizing all the
notes of the chromatic scale, rather than favoring one over the others.
Arnold Schoenberg championed this philosophy, stating that ``[t]onality is not an eternal law of music, but simply a
means toward the achievement of musical form.''\footnote{J. Machlis, {\em The Enjoyment of Music: An Introduction
to Perceptive Listening}, 5th ed., W. W. Norton \& Company, p. 471.}

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\noindent {\bf What to listen for:}  Obviously this is but a small survey of the 
evolution of Western classical music.  The best way to become familiar 
with the genre is to listen frequently and to vary the era of composition.
Try to listen for contrasts in the
different styles with an eye (or ear) to the key musical concepts discussed in Chapter~2:
pitch, tonality or atonality, intervals, chords, and harmony.  
Can you recognize how polyphony (multiple voices playing or singing together)
has evolved?  Can you hear the difference between a piece in a major versus a minor key?  
Can you hear the tonic-dominant relationship in Mozart's {\it Eine Kleine Nachtmusik}?
Can you hear how Barber's {\it Agnus Dei} is more consonant (harmonically
pleasing) than Schoenberg's work?  Can you hear why Schoenberg's piece
is described as atonal?  


\begin{enumerate}

\item  {\it Christus factus est pro nobis}, Gradual (Modo V), Gregorian chant (anonymous).
Track 3 from an Angel Records CD titled \underline{Chant} performed by the
Benedictine Monks of Santo Domingo De Silos.  {\em Gregorian chant} is some of the oldest music
of the human race.  The name refers to Pope Gregory the Great, who reigned from 590 to
604 and commanded that the singing of the monks of the Catholic church be transcribed
and preserved for the future.  The melodic texture is monophonic, consisting of a
single melody with voices in unison.  There is no rhythmic structure but rather a 
free-verse rhythm to accompany the Latin text.  


\item  {\it Salve cleri speculum/Salve iubar presulum}, English motet, Anonymous Four.
Track 15 from a Harmonia Mundi (USA) CD titled \underline{Legends of St. Nicholas:
Medieval Chant and Polyphony} by the quartet Anonymous Four.  Moving forward to
the 13th and 14th centuries, the {\em motet} emerged as one of the earliest forms of
polyphonic music (multiple voices playing or singing together).  Often a fragment of
Gregorian chant was taken and arranged into a more precise rhythmic structure (usually
with longer notes) that formed the structural blueprint for the piece.  This became
known as the {\em cantus firmus} (fixed melody).  Other faster lines of countermelodies
accompanied and contrasted with the cantus firmus creating polyphonic music.  In this
piece the cantus firmus is sung in perfect fourths and fifths, the start of a basic
harmonic vocabulary.  Notice the sound is fuller than Gregorian chant but not as rich
as the following pieces by Palestrina and Bach.

\pagebreak


\item  Giovanni Pierluigi da Palestrina (c. 1525/6 - 1594), 
{\it Lamentation I, Music for Maundy Thursday}.
Track 1 on a Chaconne CD titled \underline{Music For Maundy Thursday: Palestrina}.
Palestrina was the most famous composer of the Roman school of musical composition in
the 16th century.  His style blends the melodic lines of different voices into a 
smooth polyphonic texture.  Tonality, the idea of having a specific key,
is not quite established here, but the cadences (the music leading up to a final
chord) are suggestive of a dominant to tonic relationship.  Palestrina was a 
deeply religious composer.  This Lamentation of the Prophet Jeremiah 
would be sung at the beginning of the Thursday service of Holy Week.


\item  Johann Sebastian Bach, Opening Chorale, {\it Wachet auf, ruft uns die Stimme}
from the Cantata {\it Wachet auf, ruft uns die Stimme} BWV 140, 1731.  Track 1
from a Deutsche Grammophon CD titled \underline{Bach Cantatas}, performed
by the Monteverdi Choir and the English Baroque Soloists under the direction
of John Eliot Gardiner.  Bach wrote over two hundred {\em cantatas} to be performed
as part of Sunday religious services in Leigzig, Germany.  Note the mixture of
orchestra and voices in contrast to the earlier pieces.  Notice also how the
melody is carried in long notes (sopranos) while the lower voices react with
quick, staggered, imitative lines.  


\item  Johann Sebastian Bach, Final Chorale, {\it Gloria sei dir gesungen} 
from the Cantata {\it Wachet auf, ruft uns die Stimme} BWV 140, 1731.  Track 7
from the previous CD recording.  The final chorale in the cantata, the main melody
announced in the first chorale is now gloriously sung by all in hymn-like style
to close the service.  Note that the main melody is now sung twice as fast as
the opening chorale.  This is vintage four-part choral writing by one of the
great masters.


\item  Wolfgang Amadeus Mozart, Allegro from {\it Eine Kleine Nachtmusik} 
(A Little Night Music), K. 525, 1787.  Track 1 from \underline{The Best of
Mozart}, Madacy Entertainment Group, Inc, performed by the ``Zazerkalye'' Chamber
Orchestra conducted by Pavel Bubelnikov.   This famous piece is in $\timesig{4}{4}$
time in the key of G major.  The opening theme (shown in Figure~2.46 of the text),
alternates between the tonic G (first two measures)
and the dominant D (next two measures).  The dominant 7th chord built on D contains the notes D, F\#, A
and C, and the tritone between F\# and C strongly wants to resolve to the major
third between G and B.  Listen for the tonic-dominant relationship throughout the piece.


\item  Ludwig Van Beethoven, Adagio Sostenuto from the {\it Moonlight Sonata},
Sonata No. 14 in C sharp minor, Op. 27, No. 2, 1801.  Track 4 from a Philips
CD titled \underline{Beethoven: Favourite Piano Sonatas} performed by
pianist Alfred Brendel.  In contrast with the jovial night music of Mozart in
G major, Beethoven's eerie yet sublimely beautiful opening movement to this sonata
is in a minor key.   In general, minor keys evoke sorrow and sound a bit more
haunting than the major keys, although this is not a definitive rule.  This is
clearly evident when comparing this piece to Mozart's.  Beethoven composed the
sonata in honor of the 17-year-old Countess Giulietta Guicciardi, his pupil, whom
he evidently was in love with.


\item  Richard Wagner, {\em Tristan und Isolde}, Prelude to Act 1, 1865.  Track 2 from 
\underline{The Wagner Collection}, the Decca Record Company Limited (London, 1992),
performed by the Vienna Philharmonic under the direction of Sir Georg Solti.  
Wagner's groundbreaking opera featured a new and unconventional approach to tonality
and harmonic structure.   Noteworthy was the manner in which the composer experimented
with musical tension, often delaying or failing to resolve dissonant chords.  This is immediately apparent 
the start with the famous ``Tristan'' chord: F, B, D\#, and G\#, a tense combination built over
the unstable tritone between F and B (see Figure~2.47 of the text).  This chord and the lack of resolution
are repeated throughout the prelude.  Wagner's seminal work is widely viewed to have laid the
foundation for later composers to challenge and eventually discard the commonly held tenants of
major-minor tonality.  



\item  Samuel Barber, {Agnus Dei} (Based on {\it Adagio for Strings}), 1967.  Track 8 on the
Telarc CD \underline{Evocation of the Spirit} conducted by Robert Shaw and performed by the
Robert Shaw Festival Singers.  Barber, one of the most well-known 20th century American composers,
took the slow movement from his String Quartet in B minor, Op. 11, 1936, and reworked it into
the glorious {\it Adagio for Strings} for string orchestra in 1938.  This was the music 
used for Oliver Stone's great Vietnam War film {\it Platoon}.  Much later in his career, Barber
set the music to the traditional Latin ``Agnus Dei'' text.  The piece is in a minor key and drips with luscious
harmonies and a tantalizingly slow, somber melody that appears to be continuously climbing
up the B$^{\displaystyle{\flat}}$ minor scale (see Figure~5.11 in the text).  
Barber's work has often been described as neo-romantic, harkening back to
the great Romantic composers (e.g., Beethoven and Brahms.)  Although the harmonic structure
is more complicated, it is quite distant from the atonal works of his peers 
(note the contrast with Schoenberg's {\it Friede auf Erden}.)


\item  Arnold Schoenberg, {\it Friede auf Erden} (Peace on Earth), Op. 13, 1907.  Track 9 on the
Telarc CD \underline{Evocation of the Spirit} conducted by Robert Shaw and performed by the
Robert Shaw Festival Singers.  This is a challenging vocal work set to a poem
by Conrad Ferdinand Meyer.  Schoenberg was the first composer to truly break away
from the notion of using a principle key.  This piece has no key or tonal center, 
but experiments harmonically between consonance and dissonance.  After listening to the
other examples described above, it should be apparent how radically different this work sounds.
This piece belongs to Schoenberg's atonal-expressionist period, preceding 
his famous 12-tone method discussed in Chapter~7 of the text.


\item  Bobby McFerrin, {\it The 23rd Psalm}, 1990.  Track 12 from the CD \underline{Medicine Music},
Prob Noblem Music BMI.  Here we have an elegant and moving vocal arrangement of Psalm 23.  
McFerrin dedicated the piece to his mother and adjusts
the gender specific pronouns accordingly.  The work is clearly tonal although certain 
voicings come from jazz-style chords, creating a sustained dissonance before resolving to the tonic
at the end of each verse, a modern version of ``tension and release.''


\item  Meade ``Lux'' Lewis, {\em Medium Blues}, (recorded in 1944).  Track 6 from the CD \underline{Classic Piano Blues},
Smithsonian Folkways, \copyright 2008.   This
is a standard blues tune by one of the most popular boogie-woogie pianists, Meade ``Lux'' Lewis (1905--64).
After a brief intro, the left-hand lays down a bass line that follows the standard 12-bar blues pattern:
four bars on I (tonic), two bars on IV (subdominant), two bars on I, one bar on V (dominant), one bar on IV, and finally
two bars back on I (see page~63 of the text).   
The second time through the sequence (as well as in later passes), the second measure shifts briefly
to IV, so that the initial four bars are I-IV-I-I.  This is a common variant to the opening of a blues.
Try and count along with each measure in order to follow the chord changes underneath the melody.  


\item  Roosevelt ``The Honeydripper'' Sykes, {\em Sweet Old Chicago}, (recorded in 1961).  Track 18 from the same
CD as the previous tune.   Sykes (1906--84) was a significant blues pianist originally from St. Louis, who later lived and played
in Chicago.  He recorded under a number of different pseudonyms such as Dobby Bragg, Willie Kelly, and The Honeydripper.
As with {\em Medium Blues}, this famous song (originally written by Woody Payne) also follows the standard
12-bar blues pattern, using the opening I-IV-I-I variation.  Notice that the lyrics for each verse follow an A-A-B pattern, where
the opening line sung over the first four bars is repeated over the next four (bars 5--8).


\end{enumerate}


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