An Infinite Fugue #1 - 12 Tone

This composition could be regarded as a concept piece. It is one measure long, but can be played indefinitely without ever repeating. This instance is a one measure recording. A longer instance is available on YouTube; the video displays the full score and plays a one measure instance followed by a 10 minute instance:

https://youtu.be/-jK45dRXqEU

Scoring of notes for each instrument family represents the chromatic range for that family, with a few exceptions. The highest notes for some instruments are not always pleasant sounding – these notes have been excluded.

Some instruments are scored with whole notes and some with pairs of half notes. Use of half notes gives more opportunity for rhythmic complexity.

The driver at the center of the composition is the Probability Gate Value function (as it is called in the Logic DAW). This function is applied to each note in each chord for each instrument on every pass through the music. The result of this process determines whether or not any specific note sounds for an instrument. It is a randomizing process. Each application of the function is independent of all others. PGV values can be from 0% (no note will sound) to 100% (each note/all notes will sound). With large numbers of notes in a chord, smaller percentages assure that the sound does not become too dense and make single notes or silence highly improbable.

And now some math.

As we are dealing with independent events, the Probability of hearing a specific note, say Eb, is given by:
P(NS) = (PGV)^NS
where NS = the # of notes in the chord (Note Stack) and PGV = a value between 0% and 100%.
Consider an instrument with a range of the 12 note chromatic scale.  NS = 12. With PGV = 11% , the calculation is:
(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11) = 3.14E-012 approx., or 1 in 3,140,000,000,000.

The calculation for Probability of an exact repeat of a series of events is:
P(C) = ((PGV)^NS)^C
where C = # of chords (Note Stacks) in the Instrument Track. If the piece of music is 20 measures long and the instrument is playing whole notes, the previous calculated value (3.14E-12) is raised to the power of 20, approx. 8.60E-231, a very small number.

For a composition using several instruments, the calculation for Probability of an exact repeat of a series of events is:
P(M) = (((PGV)^NS )^C )^M
where M = # of Instrument Tracks (for simplicity, assume the number of chords is the same in each track).
For a composition with 4 Instrument Tracks, say a string quartet, my spreadsheet program offered 0 [zero] as the P(M); so a number too small for the spreadsheet to calculate.

Circling back to the single event scenario, if each note sounds for 1 second (or does not sound, but the second is taken up by silence), then if you wanted to hear Eb - or you have heard it and want to hear it again - the calculation using the P(NS) = (PGV)^NS result for how long you might wait looks like this:

3,140,000,000,000 events / 86,400 seconds per day = 36,342,593 days, or 99,569 years.

While this result is astonishing, the math appears to be correct.

For Infinite Fugue #1, probability values for most instruments are set to 1%. Higher values cause too many notes to sound and single notes or silence become increasingly unlikely. The probability calculations result in very small numbers. The Bassoon, e.g., has a NS = 25 notes and a single chord in the one measure; with PGV = 1%, there is a 1 in 1.00E+50 probability of a specific, chosen note playing or repeating once heard. When the composition is taken as a whole, the probability numbers are vanishingly small.

This composition could actually be played by an orchestra provided with a cueing mechanism to tell each player what notes to play and when as the piece progresses - perhaps a score that appears on a tablet display one measure ahead of where the performer is in the piece.

An Infinite Fugue #1 - 12 Tone, Measure 1

An Infinite Fugue #2 - A Major, no accidentals

Another concept piece using the method of Infinite Fugue #1. It is one measure long, but can be played indefinitely without ever repeating. This instance is a 75 measure recording, 5 minutes long. Why no accidentals? Random inclusion of accidentals usually ends up sounding bad, so no accidentals.

The scoring of notes for each instrument family represents the range for that family in the chosen Key, with a few exceptions. The highest notes for some instruments are not particularly pleasant sounding; these notes have been excluded.

Some instruments are scored with whole notes and some with pairs of half notes. The use of half notes gives more opportunity for rhythmic complexity.

The driver at the center of this composition is the Probability Gate Value function (as it is called in the Logic DAW). This function is applied to each note in each chord for each instrument on every pass through the music. The result of this process determines whether or not any specific note sounds for an instrument. It is a randomizing process. Each application of this function is independent of all others. PGV values can be from 0% (no note will sound) to 100% (each note/all notes will sound). With large numbers of notes in a chord, smaller percentages assure that the sound does not become too dense and make single notes or silence highly improbable.

And now some math.

What is the probability of hearing a specific note in a chord for a particular instrument?

As we are dealing with independent events, the Probability of hearing a specific note, say A, is given by:
P(NS) = (PGV)^NS
where NS = the # of notes in the chord (Note Stack) and PGV = a value between 0% and 100%.
Consider an instrument that has a range of the 12 note chromatic scale. NS = 12. With PGV = 11% (an arbitrary choice), the calculation is:
(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11) = 3.14E-012 approx., or 1 in 3,140,000,000,000.

Moving from a single event to a series of events, an Instrument Track, the calculation for Probability of an exact repeat of a specified series of events is:
P(C) = ((PGV)^NS)^C
where C = # of chords (Note Stacks, chromatic octaves in this example) in the Instrument Track. If the piece of music is 20 measures long and the instrument is playing whole notes, the previous calculated value (3.14E-12) is raised to the power of 20, and is approx. 8.60E-231, a very small number.

Finally, moving from one series of events to multiple series, a composition using several instruments, the calculation for Probability of an exact repeat of a specified series of events for the composition is:  
P(M) = (((PGV)^NS )^C )M
where M = # of Instrument Tracks (for simplicity, assume the number of chords is the same in each track; if not then the P(C) value for each track would be multiplied together to arrive at P(M) — e.g., P(C-violin) x P(C-cello) x P(C-viola) x P(C-bass)).
For a composition with 4 Instrument Tracks, say a string quartet, my spreadsheet program offered 0 [zero] as the P(M); so a number too small for the spreadsheet to calculate.

Circling back to the single event scenario, if each note sounds for 1 second (or does not sound, but the second is taken up by silence), then if you wanted to hear Eb - or you have heard it and want to hear it again - the calculation using the P(NS) = (PGV)^NS result for how long you might wait looks like this:

3,140,000,000,000 events / 86,400 seconds per day = 36,342,593 days, or 99,569 years.

While this result is astonishing, the math appears to be correct.

This composition could actually be played by an orchestra provided with a cueing mechanism to tell each player what notes to play and when as the piece progresses — perhaps a score that appears on a tablet display one measure ahead of where the performer is in the piece.

An Infinite Fugue #2, A Major, 75 Bars, 5 minutes

An Infinite Fugue #3 - Eb Minor, no accidentals

Another concept piece using the method of Infinite Fugue #1. It is one measure long, but can be played indefinitely without ever repeating. This instance is a 75 measure recording, 5 minutes long. Why no accidentals? Random inclusion of accidentals usually ends up sounding bad, so no accidentals.

The scoring of notes for each instrument family represents the range for that family in the chosen Key, with a few exceptions. The highest notes for some instruments are not particularly pleasant sounding; these notes have been excluded.

Some instruments are scored with whole notes and some with pairs of half notes. The use of half notes gives more opportunity for rhythmic complexity.

The driver at the center of this composition is the Probability Gate Value function (as it is called in the Logic DAW). This function is applied to each note in each chord for each instrument on every pass through the music. The result of this process determines whether or not any specific note sounds for an instrument. It is a randomizing process. Each application of this function is independent of all others. PGV values can be from 0% (no note will sound) to 100% (each note/all notes will sound). With large numbers of notes in a chord, smaller percentages assure that the sound does not become too dense and make single notes or silence highly improbable.

And now some math.

What is the probability of hearing a specific note in a chord for a particular instrument?

As we are dealing with independent events, the Probability of hearing a specific note, say Eb, is given by:
P(NS) = (PGV)^NS
where NS = the # of notes in the chord (Note Stack) and PGV = a value between 0% and 100%.
Consider an instrument that has a range of the 12 note chromatic scale. NS = 12. With PGV = 11% (an arbitrary choice), the calculation is:
(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11)(0.11) = 3.14E-012 approx., or 1 in 3,140,000,000,000.

Moving from a single event to a series of events, an Instrument Track, the calculation for Probability of an exact repeat of a specified series of events is:
P(C) = ((PGV)^NS)^C
where C = # of chords (Note Stacks, chromatic octaves in this example) in the Instrument Track. If the piece of music is 20 measures long and the instrument is playing whole notes, the previous calculated value (3.14E-12) is raised to the power of 20, and is approx. 8.60E-231, a very small number.

Finally, moving from one series of events to multiple series, a composition using several instruments, the calculation for Probability of an exact repeat of a specified series of events for the composition is:  
P(M) = (((PGV)^NS )^C )M
where M = # of Instrument Tracks (for simplicity, assume the number of chords is the same in each track; if not then the P(C) value for each track would be multiplied together to arrive at P(M) — e.g., P(C-violin) x P(C-cello) x P(C-viola) x P(C-bass)).
For a composition with 4 Instrument Tracks, say a string quartet, my spreadsheet program offered 0 [zero] as the P(M); so a number too small for the spreadsheet to calculate.

Circling back to the single event scenario, if each note sounds for 1 second (or does not sound, but the second is taken up by silence), then if you wanted to hear Eb - or you have heard it and want to hear it again - the calculation using the P(NS) = (PGV)^NS result for how long you might wait looks like this:

3,140,000,000,000 events / 86,400 seconds per day = 36,342,593 days, or 99,569 years.

While this result is astonishing, the math appears to be correct.

This composition could actually be played by an orchestra provided with a cueing mechanism to tell each player what notes to play and when as the piece progresses — perhaps a score that appears on a tablet display one measure ahead of where the performer is in the piece.

An Infinite Fugue #3, E-Flat Minor, 75 Bars, 5 minutes

A meditation in structured composition.

Soundtrack for the movie "Buddha" which can be viewed here:

https://youtu.be/lHSxXuZU5M0

Music Copyright © 2021, Joseph Fraizer

Buddha

Study, 12 Tone, #1

My first 12 tone serial composition.  This piece uses a tone row and matrix derived therefrom for pitch sequences.

Rules are:

Select any row/column in the matrix:  original, inverted, retrograde, or inverted retrograde.

Once a row is begun for a given instrument, follow it to completion; play all the pitches in order, do not skip any pitches, and do not repeat any pitches.  
Exception: Octave doubling for any single Note.

Notes may occur in any octave and may last any duration.  Two or more notes may sound simultaneously, as long as they occur sequentially in the row.  

Any number of rows may be played concurrently.  They may begin and end at any time; rows do not have to start or end together.  Any type of row may be played concurrently.

I called this piece a Study as it is more of an exploration of serial composition possibilities than anything else.  The term étude is generally reserved for a single instrument and is regarded as a means for improving technique; as this piece is not technically difficult and uses four instruments, étude did not seem appropriate.

Joseph Fraizer
December, 2022

Music Copyright © 2022, Joseph Fraizer

Study, 12 Tone, #1

Study, 12 Tone, #2

For Cellos, Clarinet, English Horn, French Horn, Trombone, Tuba, Harp, and Piano. 

The piece uses a tone row and matrix derived therefrom for pitch sequences.

Rules are:

Select any row/column in the matrix:  original, inverted, retrograde, or inverted retrograde.

Once a row is begun for a given instrument, follow it to completion; play all the pitches in order, do not skip any pitches, and do not repeat any pitches.  
Exception: Octave doubling for any single Note.

Notes may occur in any octave and may last any duration.  Two or more notes may sound simultaneously, as long as they occur sequentially in the row.  

Any number of rows may be played concurrently.  They may begin and end at any time; rows do not have to start or end together.  Any type of row may be played concurrently.

Joseph Fraizer
January, 2023

Music Copyright © 2023, Joseph Fraizer

Study, 12 Tone, #2

Study, 12 Tone, #3 for Piano

The piece uses a tone row and matrix derived therefrom for pitch sequences.

Rules are:

Select any row/column in the matrix:  original, inverted, retrograde, or inverted retrograde.

Once a row is begun, follow it to completion; play all the pitches in order, do not skip any pitches, and do not repeat any pitches.  
Octave doubling is allowed for any single Note.
A given pitch may be repeated any number of times.

Notes may occur in any octave and may last any duration.  Two or more notes may sound simultaneously, as long as they occur sequentially in the row.  

Any number of rows may be played concurrently.  They may begin and end at any time; rows do not have to start or end together.  Any type of row may be played concurrently.

Joseph Fraizer
January, 2023

Music Copyright © 2023, Joseph Fraizer

Study, 12 Tone, #3 for Piano

Study, 12 Tone, #4 for 9 Instruments

Vibraphone, Xylophone, Flute, Bassoon, French Horn, Tuba, Violin, Cello, Timpani

The piece uses a tone row and matrix derived therefrom for pitch sequences.

Rules are:

Select any row/column in the matrix:  original, inverted, retrograde, or inverted retrograde.

Once a row is begun, follow it to completion; play all the pitches in order, do not skip any pitches, and do not repeat any pitches.  
Octave doubling is allowed for any single Note.
A given pitch may be repeated any number of times.
The current pitch and the pitch immediately previous in the row may be alternated any number of times (trill).

Notes may occur in any octave and may last any duration.  Two or more notes may sound simultaneously, as long as they occur sequentially in the row.  

Any number of rows may be played concurrently.  They may begin and end at any time; rows do not have to start or end together.  Any type of row may be played concurrently.

Joseph Fraizer
January, 2023

Music Copyright © 2023, Joseph Fraizer

Study, 12 Tone, #4 for 9 Instruments

Study #6, 12 Tone, for Flute, Clarinet, Oboe, and Bassoon

The piece uses a tone row and matrix derived therefrom for pitch sequences.

Rules are:

Select any row/column in the matrix:  original, inverted, retrograde, or inverted retrograde.

Once a row is begun, follow it to completion; play all the pitches in order, do not skip any pitches, and do not repeat any pitches.  
Octave doubling is allowed for any single Note.
A given pitch may be repeated any number of times.
The current pitch and the pitch immediately previous in the row may be alternated any number of times (trill).

Notes may occur in any octave and may last any duration.  Two or more notes may sound simultaneously, as long as they occur sequentially in the row.  

Any number of rows may be played concurrently.  They may begin and end at any time; rows do not have to start or end together.  Any type of row may be played concurrently.

Joseph Fraizer
January, 2023

Music Copyright © 2023, Joseph Fraizer

Study, 12 Tone, #6 for Woodwinds

Study, 12 Tone, #7 for Piano, Invention 9 by J.S.Bach Re-imagined

The Bach piece supplies the underlying rhythmic structure for this study.

Listen to Invention 9 in F Minor, BWV 780, by Bach:
https://www.youtube.com/watch?v=1ytvth9cXeo

And here is an interesting analysis of Invention 9 by Samuel Andreyev:
https://www.youtube.com/watch?v=8Vx3IqkOaJw&t=260s

The study uses a tone row and matrix derived therefrom for pitch sequences.

Rules are:

Select any row/column in the matrix:  original, inverted, retrograde, or inverted retrograde.

Once a row is begun, follow it to completion; play all the pitches in order, do not skip any pitches, and do not repeat any pitches.  
Octave doubling is allowed for any single Note.
A given pitch may be repeated any number of times.
The current pitch and the pitch immediately previous in the row may be alternated any number of times (trill).

Notes may occur in any octave and may last any duration.  Two or more notes may sound simultaneously, as long as they occur sequentially in the row.  

Any number of rows may be played concurrently.  They may begin and end at any time; rows do not have to start or end together.  Any type of row may be played concurrently.

For this study, rows were selected separately from the note matrix for right hand and left hand.

Joseph Fraizer
March, 2023

Music Copyright © 2023, Joseph Fraizer

Study, 12 Tone, #7 for Piano, Invention 9 by J.S.Bach Re-imagined

Study, 12 Tone, #8 for Piano, Violin, and Cello

The piece uses a tone row and matrix derived therefrom for pitch sequences.

Each row used is expressed in a 3 measure phrase.  Each phrase is made up of combinations of eighth notes and quarter notes.  The phrases overlap one another in various ways.

Rules are:

Select any row/column in the matrix:  original, inverted, retrograde, or inverted retrograde.

Once a row is begun, follow it to completion; play all the pitches in order, do not skip any pitches, and do not repeat any pitches.  
Octave doubling is allowed for any single Note.
A given pitch may be repeated any number of times.
The current pitch and the pitch immediately previous in the row may be alternated any number of times (trill).

Notes may occur in any octave and may last any duration.  Two or more notes may sound simultaneously, as long as they occur sequentially in the row.  

Any number of rows may be played concurrently.  They may begin and end at any time; rows do not have to start or end together.  Any type of row may be played concurrently.

Joseph Fraizer
March, 2023

Music Copyright © 2023, Joseph Fraizer

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