Lesson 47 Avant-Garde and Electronic music

Lesson 47 - Avant-Garde and electronic music

Part I - The Avant-Garde Period

Avant-Garde composers sought to redefine the parameters of music in bold new ways.  The Avant-Garde period pieces contained a large element of chance.  At about this time, electronic means of producing sound became possible.  Composers sought to use these new means of creating sound in bold new ways.

Part II - Aleatoric Music

Pieces in this style would almost guarantee never having a repeat performance.  The most famous Aleatoric piece would be John Cage's "4:33"  Where all of the windows in the concert hall were opened and the piece consisted of the atmospheric sounds of every day life that happened during those four minutes and thirty-three seconds of the piece.  

here is the score for 4'33"

"Tacet" is an instruction for performers to remain silent for a section of music.

As can be seen, there are actually movements.  

Another famous Aleatoric piece is Terry Riley's "In C."  This piece consists of fifty-three snippets of music.  Each member of an ensemble would pitch which snippet they wanted to play, and would repeat that figure for however long they desire.  Then the performer would choose another figure and repeat that for however long they choose.  The piece is over once the conductor chooses.   The best part of the piece is that there is a very slim chance of a repeat performance given all of the liberties the performers have with the performance of the piece.

Part III - A Brief History of Electronic instruments.

The use of electronics as a means of composition has been around since the early 1900s, however the first music producing computer, caled the Telharmonium was the first synthesizer.  It was however, not exactly portable.  It weighed two hundred tons, and had to be transported by thirty railroad cars! Fortunately, as technology improved, the electronics got smaller and smaller.  The Moog modular synthesizer gave composers a chance to create their own sounds for pieces.

The Moog modular synthesizer.

Minimoog synthesizer.

Part IV - Electroacoustic Music

Electronic and electroacoustic music is defined as music that uses electronics to create sounds and music.  Once sounds are created, they are often mixed together using programs such as ProTools.  

Often, instruments play an excerpt and a computer records that sound then plays it back distorted in some way (time, register, etc.)

Lesson 46 Minimalism and Rhythmic techniques

Lesson 46 - Minimalism and Rhythmic Techniques

Part I - Minimalism

Minimalism is a technique inspired by the techniques of Indian music.  Composers if the latter half of the twentieth century developed a style of music that uses a very limited rhythmic source material. Minimalist music takes a brief melodic motif.  Often only three or four measures in length as its starting point.  Over time, the composition takes this brief rhythmic motif and gradually evolves and modulates it over time.

Part II - Twentieth Century Rhythmic Techniques

Composers of the Twentieth Century developed a wide variety of techniques dealing with the element of rhythm.  They did so by using irregular meters, multiple simultaneous meters, frequent changing meters, or even use of no meter at all.

Part III - Metric Modulation

This process creates the illusion of tempo changes through the use of meter change, or which subdivision of the beat gets the  downbeat.

Lesson 45 Serialism

Lesson 45 - Serialism

Part I Twelve-Tone Technique

The goal of Twelve-tone compositions is to give each pitch Equal importance.  In other words to destroy the Dominant-Tonic relationship.
In order to compose a twelve-tone piece, one must first make a series of compositional choices.  Prior to writing a piece, a composer creates a specific order in which the twelve pitch classes will be used (C-B chromatic scale).

Part II - The Tone Row

In order to compose a Serial piece, the composer must first construct a tone row.  The tone row must use all pitches in a chromatic scale, but they may occur in any order.  Here is a row from Schoenberg's Op.25 "Intermezzo." 

 The pitches in the preceding figure are numbered in the order that they appear in the piece.

Part III - the Matrix

THe original version of the matrix is known as the prime row, Shown here is an example of a prime row:

 Of course, the pitches can occur in any order
To build the Inverted row (The vertical rows), there are two methods.
1.  Invert the intervals from pitch to pitch (E to G is a m3 up: a m3rd down from E is C#)
2.  Invert the intervals as such: F to G is a M2nd up: a M2nd down from E flat is D flat)
Here is the completed prime inverted row:

                                         RI⁰
1. The Rows from left to right are called the Prime rows
2. The Rows from right to left are called the Retrograde (backwards) rows
3. The Columns from top to bottom are called the Inversion rows
4. The Columns from bottom to top are called the retrograde inversion rows.
The Vertical rows (from bottom to top are called the retrograde inversion rows (meaning the Backwards inverted prime rows)


The next step is to assign names to all of the rows.

Before analyzing Twelve-tone music, you must first complete a matrix as I have done above.  This is the matrix for Schoenberg's Op.25.

Part IV - Analyzing Twelve-tone Music.

After determining the tone row and completing the matrix for a composition, you can analyze the composition.Simply determine what row is being used and write the name of that row.  For example, R⁷, or P³. Once all rows are represented, a piece is complete.

Part V - Total Serialization

This technique involves creating a second matrix for the duration and dynamics of the notes as well as a matrix for the order of the notes.  

A total Serialization matrix would look similar to this:

The idea of serializing the pitch component of a piece led to later composers serializing other elements of their music, such as duration of notes and manner of attack.  

Lesson 44 Non-Tertian Harmonic techniques

Lesson 44 - Non-Tertian Harmonic Techniques

Part I - Quartal Harmony

Tonal music is said to be based on the principles of Tertian Harmony, or chords built in thirds.  It is also posible to create harmonic systems that use other intervals as their basis.  Quartal harmony is built on the interval of a fourth.  That is the trichords are stacked in fourths.

Quartal trichords are considered to be in root position when all of the consecutive intervals are fourths.  Notice that, in each chord, the outer interval is a 7^th.  When the trichords are inverted, the outer interval becomes a 5^th.  

Paul Hindemith is one of the most important composers to work in the realm of quartal harmony.  He wrote many pieces for four part harmony (SATB) using chords stacked in fourths.  Hindemith showed that it is possible to convey the sense of a tonic using quartal harmony.  In his piece "A Swan," in measure five, the final resting chord, although it is a quartal trichord, can be considered 'tonic.'  

Part II - clusters and Secundal Harmony

Sucundal harmony can be defined as that which the chords are built on and stacked in seconds.  Bartók used this technique extensively in many of his works.  Most famously Mikrokosmos.  In Book VI, #144 in particular is called "Minor Seconds, Major Sevenths."  The seconds and sevenths dominate the entire texture of this particular composition.  

A parallel technique to secundal harmony is clusters:

The example in the treble clef shows a cluster where every note is notated.  The example in the bass clef show an alternate notation to the cluster on the right would be the example on the left. 

A good example of secundal harmony is in Charles Ives', "Hawthorne."  Also, Henry Cowell's, "Tiger"

Part III - Microtones and Sound Mass

Inspired by the clusters of secundal harmony, some composers took it one step further and notated for every note within the range of an instrument.

This included notes between the normal half step.  These notes are called microtones. A microtone is defined as any interval smaller than a second.  

The string family is used most often to achieve microtones.  One way to notate microtones is with the use of quarter sharps and flats.

THese are used to notate an interval smaller than a half step.  

Lesson 43 Pitch class set theory

Lesson 43 - Pitch Class Set Theory

Unfortunately, there is no quick way to explain this, so A WALL OF TEXT!  The good news is, we're getting to the end of the theory lessons!

Part I - Introduction

Pitch class set theory is a way of analyzing atonal music using the principles of mathematical set theory.

Analyzing tonal music in terms of chordal structure makes sense, but analyzing atonal music in terms of chordal structure doesn't make sense.  Pitch class set theory provides a means of studying similarities and differences of pitch content as well as organizational aspects of atonal music within groups of notes called "sets."
This theoretical system was developed by Allen Forte in his book, The Structure of Atonal Music(New Haven:Yale University Press, 1973) Pitch class sets are collections of three or more pitches.  

There are a number of founding premises that must be understood in order to begin working with Pitch class set theory.  They are:
1. Octave equivalence.  No distinction is made to represent octave/register.
2.  Enharmonic equivalence.  All spellings of a given pitch are considered equal.
3.  Integer notation.  pitch classes are represented by numbers instead of letters.
4.  Sets are not represented on a staff but instead grouped within brackets and separated with comas as such: [0,1,2]
5. Elements are always listed in ascending numerical order regardless of the order in which they appear as notated.

These five premises are crucial to the beginning of the study of set theory.  Pitch class is conveyed in the following system: C=0, C#(or any of it's enharmonic equivalents)=1, D=2, and so on.

Part II - Naming sets

In order to work with sets, it is necessary to have a naming system to distinguish unique sets.  Naming sets was developed by Allen Forte and sometimes referred to as "Forte Numbers."  A set's name consists of two elements.  The first number is the number of elements in a set.  The second number describes the ordering of the set.  

Here is a chart of some sets as named by Forte:

Part III - Inversion of sets

To start the process of inverting a set, we must understand the principle of inversional equivalence.  Just as pitch classes can be represented with integers, intervals can be represented in the same way:

prime (unison) = 0

minor 2^nd = 1

major 2^nd = 2

minor 3^rd = 3

major 3^rd = 4

Perfect 4^th = 5

tritone = 6

perfect 5^th = 7

minor 6^th = 8

major 6^th = 9

minor 7^th = 10

major 7^th = 11

octave = 12
Of course, interval inversion is also acceptable.
0 = 12
1 = 11
2 = 10
3 = 9
4 = 8
5 = 7
6 = 6
When a set has been placed in normal order and transposed to start with 0 but still does not appear in the list of Forte Numbers, it is because the set has not been packed so that the smallest intervals appear on the left side of the set and thus must be inverted.

Part IV - Interval Vectors

Apart from identifying the names of the set, many other types of relationships can be demonstrated through the use of pitch class set theory.  A representation of all of the intervals contained within a set is called its interval vector.  

The Cardinal 5 set, [0,1,4,5,8] will be used to demonstrate interval vectors. Each interval class is determines through subtraction.  

0          1          4          5          8

|     ic1  |

|             ic4      |

|                     ic5           |

|                               ic4              |

             |     ic3  |

             |            ic4        |

             |                   ic5                | 

                          |     ic1   |

                          |             ic4         |

                                        |     ic3   |

Using a six digit string of integers, each of which represents one of the interval classes, it is possible to map the entire interval content of a pc set.  The first digit of the string will represent how many times that interval appears in a set.  The second number will represent the interval class 2, and so on.  Interval class 0 is disregarded.
The interval vector for set [0,1,4,5,8] contains two occurrences of ic 2, two occurrences of ic3, four of ic4, one of ic5, and none of ic6.  The interval vector would be represented in this manner: 202420.
Pairs of unique Forte sets that have unique Forte names, but identical interval vectors are described as being Z-related pairs.

Lesson 42 Techniques of non-tonal music

Techniques of Non-Tonal Music

Part I - Atonality

In some aspects, the impressionists paved the way for the next generation of composers in the twentieth century.In their music, they cast aside many of the conventions of tonal music.  The terms Atonal, non-tonal, and post-tonal can all be used interchangeably as they all refer to the same style of music.
Atonality its self has many forms and techniques. Some of these techniques include, Pandiatonicism, Pointillism, and Bichordality/Polychordality.

Part II - The Octatonic Scale

In addition to the scales discussed in the previous lesson, Several other new scales.  One such scale is the Octatonic scale.  The octatonic scale is an alteration of the Whole-tone scale.  Put simply, the octatonic scale is al alternation of whole and half steps.  Here are two iterations of the octatonic scale.  First, the Half step-Whole step pattern.  Secondly the Whole step-Half Step pattern.  Using the Octatonic scale, a triad built on any pitch will yield a diminished triad, and a seventh chord will yield a fully diminished seventh.

Octatonic On C

 Part III - Other scales

Of course, compositions can be written on scales other than those previously mentioned.  One such scale is the Gypsy scale.  This is a scale where not only Major and minor seconds are used, but also minor thirds. Here is the Gypsy scale on C:

Note the intervals are:Hole step, Half step, minor third, Half step, Half step, minor third, half step.

Finally, composers were free to simply make up a scale if they wanted!  This is called a synthetic scale.  A synthetic scale could have as many pitches as the composer wanted.  Here is an example of a synthetic scale with nine separate distinct pitches.  Of course, one could have more!

Part IV - Pandiatonicism 

This is a technique of using the pitch resources of a major scale without regard to the typical harmonic relationships that are found in tonal music. This is often called non-functional harmony.  A very famous example of pandiatonicism is Stravinsky's "Petrushka."  All of the chords can be named in the C Major scale, however, they are not used in a way that can be associated with the C Major scale.
Another example of pandiatonicism comes from Samuel Barber in the piece, "Excursions." In the third movement, He uses the Key signature and pitch resources of the G flat major scale, But the chord relationships are not tonal therefore it is incorrect to analyze these pieces in their respective key signatures.

Part V - Pointillism

In the visual arts, pointillism os a technique where individual dots of color are placed next to each other in order to create a full image.  Pointillism in music uses the same technique of "points" in this case notes spaced out very widely often separated by time and register.  Instead of thick textures and full lush chords, a pointillistic composition is often austere, since a minimum of material is used in the composition process.  An example of a pointillistic piece is Webern's "Piano variations Op.27 #3"  

Part VI - Bichordality and Polychordality

Another harmonic technique of the early twentieth century is Bichordality, or Polychordality.  This is the sounding of two or more distinct triads.  Thus intentionally avoiding a key center.  Each horizontal chordal sequence is non-functional.  Here is the begining of William Schumann's "A three score set" Measures 1-4.  Notice the use of apparently unrelated chords in each of the two registers.  

Lesson 41 Jazz theory

Lesson 41 - Jazz Theory

Part I - Introduction to Jazz Concepts

Jazz theory shares a lot of the same building blocks as traditional theory.  It is the application of these concepts where Jazz differs from traditional theory.

There are two primary types of jazz.  The first is improvisational jazz, The vast majority of western art music is notated to convey the exact emotion the composer is trying to convey.  However in jazz, a majority of the score may be left up to the performer to improvise, giving every performance a different feel.

The second style associated with jazz is swing.  In the simplest definition, the performer gives the subdivisions of the beat unequal length

A steady series of notes would be interpreted as as pairs of long-short durations 

Part II - Scales and Modes

Jazz makes use of the modes more so than normal major/minor scales 

These modes in particular are used most:
Mode 1 Ionian - melodic minor

Mode 2 Dorian - no common designation(might be called Phrygian #6)

Mode 3 Phrygian - Lydian Augmented

Mode 4 Lydian - Lydian flat7 or Lydian dominant 

Mode 5 Mixolydian - Mixolydian (No common designation in Jazz)

Mode 6 Aeolian - Half-diminished scale or Locrian #2 or Locrian #9

Mode 7 Locrian - Altered dominant or diminished whole-tone scale

Part III - Additional Scales

Apart from the Major/minor scales Several other scales 

The first to know is the Octatonic scale

Here are two different versions of the Octatonic scale:

Half step-whole step pattern:

 whole step-half step pattern:

One primary feature of both of these scales is that any chord built with every other note of the scale will always be a fully-diminished seventh chord.

The next scale is the blues scale.  The feature of this scale is the lowered third, fifth, and seventh.  Traditionally considered 'blues notes' because of their use in blues.  A twelve-bar progression iteration of jazz.

the blues scale: 

The final category of scales used in Jazz is the Bebop scale, of which there are four:

Bebop dominant - Mixolydian scale with a chromatic passing tone between the seventh and eighth scale degrees.

Bebop Dorian - Dorian scale with a chromatic passing tone between the third and fourth scale degrees.

Bebop Major - Major scale with a chromatic passing tone between the fifth and sixth scale degrees.

Bebop melodic minor - Melodic minor scale with chromatic passing tone between the fifth and sixth scale degrees.

Part IV - Harmony

In tonal music, melody is closely related to harmony.  Over any given triad in a chord progression, The melody will primarily be comprised of chord tones from that triad, with non-harmonic tones added for color and ornamentation. The same principle is true in jazz.  The chords used in jazz harmony are associated with specific modes and melodies are improvised over each chord drawn from the pitch resources of that mode.

Many of the chord progressions are the same in jazz as they are in tonal harmony.  For example, the ii-V-I and vi-ii-V-I have counterparts in jazz.  

The V⁷ Chord is a major-minor seventh is always referred to as the dominant seventh chord.  The seventh chord built on the tonic C-E-G-B is a Major-major seventh.  This is the biggest break from traditional harmony this far.  A tonic seventh chord is virtually unknown in music of the common practice period.
The symbol used for the major seventh chord is unique to jazz: C∆7.  The triangle symbol indicates that the interval of the seventh is of a major quality.
Here is the standard ii-V-I progression in Jazz harmony:

 Part V - Tritone substitution

One of the most interesting aspects of jazz harmony is the ability to substitute the tritone in the V⁷ chord.  Illustrated here is the typical way a tritone substitution would manifest in a ii-V-I progression.

Notice that the substituted V⁷ chord is just the same as the ii⁷ chord except for the three lowered notes.  This makes part writing easy.

Part VI - Other considerations

In order to allow performers the freedom to improvise using the modes and techniques described, a style of jazz developed that involves repetition of very few chords.  This style of jazz based on very few chord changes is known as modal jazz.  An example of this type of jazz would be "I got Rhythm" by Gershwin.  

A wide variety of symbols are used for commonly used chords.  Here is a list of commonly used chord symbols and other symbols that might be used for that chord.

Chord Quality                        Common symbols         Other symbols

Major triad with added 6th           C6, or C13                  CMaj6; CM6

minor triad with added 6th            C-6                             Cmin6; Cm6

Major 7th chord                             C∆                            CMaj7; CM7

Dominant 7th chord                       C7

Minor 7th chord                             C-7                            Cmin7; Cm7

Minor-Major 7th chord                   C∆7 or C-(∆7)           Cmin/Maj7

Augmented-Major 7th                    C+∆                            Caug/Maj7

Fully diminished sevnth                  C°7                             Cdim7

Half diminished seventh Chord       C^ø7                      C-7(b5); Cm7(b5) 

Lesson 40 Impressionist techniques

Lesson 40 - Impressionist Techniques

Part I - Impressionism.

The Impressionist period was a short movement sandwiched between two larger style periods: The Romantic, and the the Twentieth Century.  Despite its length, it still held importance as a period of dramatic changes.

There are seven style elements that identify the Impressionist period.

1. Emphasis on tone color rather than melody.

2. Absence of most brass and percussion from the orchestra--in most cases only the French horn and small delicate percussion such as finger cymbals remained.

3. Dreamy and atmospheric textures and moods.

4. Lack of strong rhythmic impulses to propel the music.

5. Programatic writing (meaning that the music usually contains some sort of extramusical meaning, even if it is not an overt story (such as in "The Moldau"))

6.  Interest in writing music about water

7.  Use of non-functional progressions instead of functional harmony.

Part II - Parallelism

In all of functional harmony, parallelism is avoided and seen as bad part writing (Because the parallel intervals are difficult to sing and hear). However, in Impressionist music, Parallelism is actually encouraged.  In this example, there are two different examples of parallelism.    

 This example is called that because the parallelism (The fifth) is not a perfect fifth                               This is named as such, because the parallelism IS perfect fifths.

Part III - The Modes

The Modes that we learned in the previous chapter are used constantly in the music of the impressionist period.  The modes that one typically sees are: Aeolian, Dorian, Phrygian, Lydian, and Mixolydian.  Typically melodies in the Impressionist period are based on scales other than Major and minor.

Part IV - Other Scales

Of course, the modes aren't the only scales that can be used to make music.  Examples of other scales include: The Whole-tone scale, Pentatonic scale, 

 Whole-tone scales:
In addition to the modes, a composer could also use the whole-tone scale or the pentatonic scale

• The Whole-tone scale is constructed of six tones, each equally distant from each other
• Any pitch in the scale can serve as the "tonic"
• The whole-tone scale can be started on any pitch class
• Unlike traditional major and minor scales, the whole-tone scale is only made up of six distinct pitches.

Four different iterations of a Whole-tone scale:

The next type if scale that can be used is the Pentatonic scale.  This is a scale where only five distinct pitches are present.  The pattern of intervals is: M2, m3, M2, m3.  Any note of the scale can serve as it's starting point (Without altering the pitches).  

Pentatonic scale on F#:

Other patterns for the pentatonic scale:

                                         M2          m3        M2      m3       M2                        m3         M2       m3       M2        M2

                                           M2       m3        M2      M2        m3                            m3       M2      M2       m3       M2

Lesson 39 Modes

Lesson 39 Modes

Part I - Modes

The earliest examples of music notation date back to 900 AD.  It was back during the reign of Charlemagne that the modes were named.  

Each mode had two iterations, The first is where the final note is the lowest note in the scale, the "tonic" if you will.

The second id where the final note of the scale is the fourth scale degree, called the Plagal iteration of the mode.

Dorian Mode:

Authentic:                        :                                          Plagal

Phrygian Mode:
 Authentic:                                                                 Plagal:

Lydian Mode:

Authentic:                                                                   Plagal:

Myxolydian Mode:

Authentic:                                                                   Plagal:

An example of the Hypomyxolydian mode used on gregorian chant.

Part II - Forming the modes

Mode                                White key scale                                    Similar to:

Aeolian                                             A to A                                                         natural minor scale

Locrian                                             B to B                              natural minor with lowered 2nd and 5th scale degree

Ionian                                               C to C                                                           Major scale

Dorian                                              D to D                                      Natural minor with raised 6th scale degree

Phrygian                                           E to E                                       Natural minor with lowered 2nd scale degree

Lydian                                               F to F                                          Major with raised 4th scale degree

Mixolydian                                       G to G                                           Major with lowered 2nd scale degree

Here are a few examples of the relationships between modes and their related keys.

Listen:

Listen:

Listen:

Listen:

Listen:

Part III - Key signatures

Mode                                          Key signature

Aeolian                                       Parallel minor

Locrian                                       Add two sharps or subtract two flats from the parallel minor

Ionian                                         parallel Major

Dorian                                        Subtract one flat or add one sharp from the parallel minor

Phrygian                                     Add one sharp or subtract one flat from the parallel minor

Lydian                                         Subtract one flat or add one sharp from the parallel Major

Mixolydian                                 Add one flat or subtract one sharp from the parallel major

Part IV - Modal harmony

A variety of chord qualities and progressions arise when using modes.  For example, in the dorian mode the chord qualities are:
i  ii  III  IV  v  vi°  VII    

in the Phrygian mode they are:

i   II   III   iv   v°   VI   vii

in the Lydian mode:

I  ii  iii   iv°   V   vi   vii   

and in the Mixolydian mode:

I   ii   iii°   IV   v   vi   VII

Some standard chord progressions in the modes are

Dorian: i   v   i   IV   VII   i

Phrygian:  i   VI   iv   II   i

Lydian:  I   vi   II   V   I

Mixolydian:  I   ii   IV   I   vi   VII   I

Here is a standard chord progression in the key of C Dorian:

Listen:

Lesson 38 Enharmonic modulation with fully diminished seventh chords

Lesson 38 - Enharmonic modulation using fully diminished seventh chords

Part I - The fully diminished seventh chord's symmetry 

Ready to have your mind blown?

There are only THREE fully diminished seventh chords, and using them, one can modulate from any key, to any key.

No matter what inversion the chord is in, it is always three minor thirds stacked on top of each other.

Here's an example of the symmetry of a fully diminished seventh chord on B flat:

The only three fully diminished seventh chord in existence are:

1. Those that contain C (or its enharmonic equivalent)

2. Those that contain C# (or its enharmonic equivalent)

3. Those that contain D (or its enharmonic equivalent)

Go ahead, try it! with those spellings, one can create all fully diminished seventh chords on every note.  And since no matter what inversion the chord is in, it is always made up of minor thirds, one can create every fully diminished seventh chord on every pitch center!

Part II - Enharmonic modulation with the fully diminished 7th Chord

With the symmetrical properties of the fully diminished seventh chord, that chord can be resolved to a different pitch that would be indicated by the spelling.

Here is an example of respelling a single note in order to resolve to a chord that would not be immediately apparent. 

Listen:

Here's the same example, but it modulates to G Major instead of A Major:

G#BDAF

BDFAb

Listen:

In this example, the fully diminished seventh Chord has three possible spellings:

                         ACEbGb--given spelling

D#F#AC

                               F#ACEb--Preferred spelling

Listen:

Lesson 37 Enharmonic Modulation with German Sixths

Lesson 37: Enharmonic Modulation using German Sixths.

Part I

Enharmonic modulation takes advantage of a common chord spelled differently in two separate key signatures. For example, in C Major, the V⁷ Chord is spelled: G, B, D, F. At the same time, the Gr^{6 chord in B Major is spelled E#, G, B, D,} 

^Listen:

Part II - Using the Gr⁶ and V⁷ to modulate 

In this example, The point of modulation is the V⁷ chord in C Major (Which is also the Gr⁶ chord in F# minor). 

Listen:

Part III - Modulation using secondary dominants

The Secondary Dominant of any chord in any key can also function as the Gr⁶ in another key. Notice how the V⁷/IV in C Major is the same as the Gr⁶ in E minor.

Listen:

The point of modulation is a V⁷/V in C Major and an enharmonic Gr⁶ in the ending key of E minor.

Lesson 36 review

Beethoven  piano sonata op. 49 no.1

this piece is full of secondary dominants

Schubert Impromptu op. 142 No.3 B flat major

In measures 1-8, there are secondary dominants of the dominant

Mendelssohn Songs Without Words Op.19 No.1

Here, there are many secondary dominants of the dominant

Felix Mendelssohn - Songs without Words - Op.53, No.1

In this piece, there are many secondary dominants of the subdominant

Franz Schubert - Andante in A Major (D. 604)

This piece is full of secondary dominants of the submediant

Schumann Piano Concerto in a minor, Op.54, 2nd & 3rd movements 

This piece modulates using secondary dominants

SCHUMANN, Intermezzo op 4 No 4 

This piece is full of borrowed chords 

Chopin Nocturne in C# minor, op. posth.

Near the end of the piece, there is a borrowed ii°

At the bebining of the piece, there is a German sixth as well.

Chopin prelude, Op.28 No. 20

this piece contains the Neapolitan triad 

Beethoven, Piano Sonata Op. 31, No. 1, 2nd movement

This piece modulates using secondary dominants.

Beethoven Piano sonata Op. 81a 1st movement

This piece contains the Augmented 6th chord