In this section I will discuss harmonics, and how harmonics play a key role in composition and why certain notes are grouped together to form chords or motifs in a piece of music. I will then briefly discuss chromaticism and the twelve-tone technique. At the end of this page is a list of references.
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Introduction to Harmonics
It was Pathagoras who discovered the fact that two similar strings under the same tension, differing only in length where the lengths of the strings are in a ratio of two integers, and when sounded together gave a pleasing effect to the listener. If one string length is twice the length of the of the other string length, then they would be considered an octave apart from one another. If the lengths of these strings are 2 to 3, then they would correspond to the interval between C and G, this is called a fifth. It turns out that these intervals are very pleasing to hear, especially when sounded together to create a chord.
Out of all the sounds you hear there is one called noise, and this type of sound corresponds to an irregular vibration of the eardrum that is produced by some type of irregular vibration. The sound of music has a different character. Music is characterized by the production of sustained tones or musical notes. A musical note differs from a noise in that there is a periodic character to its frequency spectrum. There are three aspects of musical tone: loudness, pitch, and quality of sound. The loudness corresponds to the magnitude of pressure change. The pitch corresponds to the period of time for one repetition of the basic pressure function, low notes have longer periods and high notes have relatively shorter periods. The quality of a tone has to do with the structure of the repeating pattern.
The sound produced by a vibrating string generates a wave motion. These wave will travel in both directions of the string, and will be reflected at the ends. The period if repetition is the time required for the wave to travel two full lengths of the string. The time is the same for waves that start out in either direction. The sound wave produced must also have the same repetition. This is how the string produced a musical tone.
A vibrating string has various natural modes of oscillation, and any vibration that is set up by the starting conditions can be considered a combination of several of the natural modes oscillating together. For a string the normal modes of oscillation has the frequencies ω0, 2ω0, 3ω0, …
The most general motion of a vibrating string is composed of the sum of sinusoidal oscillations at the fundamental frequency ω0, another at the second harmonic frequency 2ω0, another at the third harmonic frequency 3ω0, etc… The fundamental mode repeats itsef every period T1 = 2π/ω0. The second harmonic mode repeats itself every T = 2π/ω0. It also repeats itself every T1 = 2T2, after two of its periods. The third harmonic mode repeats itself after a time T1 which is 3 of its periods. This is how a vibrating string repeats its whole pattern with a periodicity of T1, and how a musical tone is produced.
If f(t) represents the air pressure as a function of time for a musical tone, then expect that f(t) can be written as the sum of a number of simple harmonic functions of time, like cos(ωt) for each of the harmonics. If the period of the vibration is T, the fundamental angular frequency is ω= 2π/T, and the harmonics will be 2ω, 3ω, etc…
There is one slight issue. For each frequency the starting phases will not likely be the same for all frequencies. It is probably a good idea to use functions like cos(ωt + φ), where φ represents the phase. However, it is more traditional to use both the sine and cosine functions for each frequency. Therefore:
and since φ is a constant, any sinusoidal oscillation at the frequency ω can be represented as the sum of a term with cosωt and another term with sinωt.
It turns out that any function f(t) that is periodic with time period T can be written as:
where ω = 2π/T and the a’s and b’s are numerical constants which tell you how much of each component oscillation is present in the oscillation f(t). This series is called the Fourier series for f(t).
Any sound wave can be made up of such a sum.
Lets get back to the term “quality” of a musical tone. It is the relative amounts of the various harmonics, the values of the a’s and b’s. A tone with only the first harmonic is a pure tone. A tone with many strong harmonics is a rich tone. A violin produces a different portion of harmonics than does a trumpet.
Getting back to to the similar strings, it is true that strings with lengths in the ratio of 2 to 3 will have fundamental frequencies in the ratio of 2 to 3. So, why should they sound pleasant together? Remember,look at the properties of the frequencies of the harmonics. The second harmonic of the lower shorter string will have the same frequency as the third harmonic of the longer string.
Consider making the following rules. Notes sound consonant when they have harmonics with the same frequency. Notes sound dissonant if their upper harmonics have frequencies near to each other but far enough apart that there are rapid beats between the two. You may ask why is it that beats do not sound pleasant, and why unison of upper harmonics do sound pleasant, it is something that we do not know how to define or describe.
Here is an exercise for you to check the harmonic relationships we have been discussing. On a piano, label the 3 successive C’s near the middle of the keyboard by C, C’, and C”, and the G’s just above by G, G’, and G”. Then the fundamentals will have the following relative frequencies:
These harmonic relationships can be demonstrated in the following way: Suppose you press C’ slowly so that it does not sound but you can cause the damper to be lifted. If you then sound C, it will produce its own fundamental and some second harmonic. The second harmonic will set the string of C’ into vibration. If you now release C (keeping C’ pressed) the damper will stop the vibration of the C strings, and you will hear the note of C’ as it dies away. In a similar way, the third harmonic of C can cause a vibration of G’. Or the sixth of C can set up a vibration in the fundamental of G”.
The major scale can be defined by the following three chords: (F, A, C), (C, E, G), and (G, B, D). Each of these chords represent tone sequences with the frequency ratio of 4:5:6, these ratios plus the fact that an octave, C to C’, has the ratio of 1:2 determine the whole scale for the ideal case for what is called “just intonation”. It is interesting to note that keyboard instruments like the piano are not usually tuned in this manner, but a little bit of tweaking is done so that frequencies are approximately correct for all possible starting tones. For this tuning, which is called “tempered” the octave (with ratio 1:2) is divided into 12 equal intervals for which the frequency ratio is 21/12. In this case, a fifth no longer has the frequency ratio 3/2, but instead has a ratio of 27/12 = 1.499, which is close enough for most ears. It is still difficult to determine whether the ear is matching harmonics or performing mathematical steps when you decide a sound is pleasing to hear.
Arnold Schoenberg (1874 – 1951) created a twelve-tone row technique that changed keys every few notes, this creates a disturbing feel to the music. Schoenberg’s technique is often used in soundtracks of horror movies. If you play C and C’ on the white keys of a piano, this is a major scale which sounds pleasant. If you play A and A’ on the white keys, this is a minor scale that sounds tragic. Then, play B to B’, this generates a blues scale for the melancholy jazz sound. Not enough for you, then go ahead and play D to D’ on the white keys, this generates a bright sound used in Spanish guitar music.
Why do different combinations of notes “feel” different, it is because of the math. Why does a flute and violin sound different when they play the same note? The answer is harmonics! Harmonics is why scales have different feels to them. Play an A (440 Hz) with full harmonics, you will not only hear the 440 Hz tone, but also the 880 Hz tone at half the volume (first harmonic), a 1320 Hz tone at a third the volume (second harmonic), a 1760 Hz tone at one-quarter the volume (third harmonic), etc….until the frequency gets too high of the volume gets too low.
Play an A at 110 Hz, 220 Hz is the first harmonic, 330 Hz is the second harmonic corresponding to an E on the second octave. The third harmonic is also an A two octaves up from 440 Hz, 550 Hz is the fourth harmonic which is a C# on the third octave, the fifth harmonic is another E on the same octave. The significance of this is that A, C# and E are the notes that make up the A major chord. These notes sound natural and pleasant together, this is because they emphasize one another’s harmonic pattern.
The following represent some chords of C major and c minor, and displays how harmonics emphasize one another:
|Note||Fundamental||1st overtone||2nd overtone||3rd overtone||4th overtone||5th overtone|
|Note||Fundamental||1st overtone||2nd overtone||3rd overtone||4th overtone||5th overtone|
Note, the fundamental frequency represents the 1st harmonic, the 2nd harmonic represents the 1st overtone, etc… The missing harmonic correspondence creates a darker feel to the chord. Chords in different scales create different harmonic patterns. The more divergent they are the more disturbing they seem.
When putting chords together it is important to know how the harmonics are associated with the scales and octaves. The following represents this association:
|3rd Harmonic||5th over octave|
|4th Harmonic||2nd octave|
|5th Harmonic||3rd over 2nd octave|
|6th Harmonic||5th over 2nd octave|
|7th Harmonic||minor 7th over 2nd octave|
|8th Harmonic||3rd octave|
|9th Harmonic||whole tone over 3rd octave|
|10th Harmonic||3rd over 3rd octave|
For completeness, the following describe emotional properties of the Major and minor scales.
|Bb minor||A “dark” key, a feeling of being all alone. Tchaikovsky’s Symphony No. 4.|
|F minor||Passion, Beethoven’s Appassionata Sonata. A highly tinted and grey feel.|
|C minor||Heroic struggle. Beethoven, Brahms’ Symphony No. 1.|
|G minor||Sadness and tragedy.|
|D minor||Bach’s The Art of Fugue. Albert Einstein associated this key with counterpoint and chromaticism.|
|A minor||Expressing the sad effect.|
|E minor||Lots of classical guitar, heavy metal. Mendelssohn used this key.|
|B minor||The key of passive suffering, a quiet acceptance of fate. Bach used this key a lot. Very melancholic and grieving feel. Eagle’s Hotel California, Pink Floyd’s Comfortably Numb. This is a common key for rock, folk, and country music.|
|F# minor||Has been described as “light red”. Can lead to great distress and is more love-sick then lethal. Gloomy, and can be dark and evil.|
|C# minor||Very popular with piano. Beethoven, Scarlatti, Brahms|
|G# minor||Lady Gaga’s Poker Face. Rarely used in orchestral music.|
|D# minor||not used that much.|
|Db Major||Majestic, great for the flute or other instruments that stress the fundamental.|
|Ab Major||Peaceful, serene feel, used by Schubert and Chopin.|
|Eb Major||Bold, heroic,, majestic and serious. Superior to C Major|
|Bb Major||Most useful for wind instruments.|
|F Major||Fantastic for the English horn.|
|C Major||Very common, simple. This is the key of strength or happiness.|
|G Major||Lot of guitar music.|
|D Major||Violin music, open string resonate sympathetically with the D string producing a brilliant sound. Lots of trumpet music.|
|A Major||Innocence, love, the hope of seeing one’s beloved again when parting.|
|E Major||Music of contemplation. Bach’s third partita for violin. Bells of London’s Clock Tower.|
|B Major||Very easy to play. Chopin used this key quite a bit.|
|F# Major||Common for piano.|
The idea that any note that has a periodic sound can be represented by a suitable combination of harmonics. It would be interesting to find out what amount of each harmonic is required. It is straight forward to compute f(t) if you are given all the coefficients “a” and “b”. The more interesting question is, if you are given f(t) how do you know what the coefficients of the various harmonic terms should be? In other words, it is east to make a cake from a recipe, but can you write down the recipe if you are given a cake?
Fourier discovered that is is not very difficult to do this. The term a0 is easy to calculate, it is just the average value of f(t) over one period (from t = 0 to t = T). It is east to see how this is so, the average value of a sine or cosine function over one period is zero, over two, or three, or any whole number of periods, it is also zero. So, the average value of all the terms on the right hand side of the equation of f(t) is zero, except for a0.
The average of a sum is the sum of the averages, so the average of f(t) is just the average of a0. however, a0 is a constant, so its average is just the same as its value. It can get rather involved to explain the details of how to analyze a periodic wave into its harmonic components. So, for now, I will leave it at that. Just to touch on this topic a bit more, this procedure is called Fourier analysis, and the separate terms are called Fourier components. Once you find all the Fourier components and add them together, you get f(t).
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The increased use of chromaticism in the early to mid 1800’s is often the main cause or sign of the breakdown of tonality in the form of increased importance or use of: mode mixing, leading tones, and tonicization. The use of chromaticism was the start of the post-tonal era. The chromatic scale became the basis of modern writing using the twelve-tone technique, where the tone row is defined as being a specific ordering of series of the chromatic scale. A chromatic note is a note that does not belong to the scale of the key that a particular piece is written in. The chromatic scale proceeds entirely by semitones, that is, 12 equal steps. In fact, a lot of jazz music uses chromaticism, all improvised lines usually use non-harmonic chromatic notes. This is similar to the C-major bebop scale:
|C D E F G G# A B C|
Chromatic chords use at least one note not belonging to the diatonic scale associated with the key that the piece of music is written in. Here is a list of some chromatic chords:
|1. Dominant seventh chords
2. Augmented sixth chords
3. Diminished seventh chords
4. altered chords
5. Expanded chords
Mode mixing: A borrowed chord, also called mode mixture, is a chord borrowed from a parallel key. These chords are used to “color” the sound. There are six chords borrowed from the parallel minor key commonly found in Baroque, Classical, and Romantic periods. Here are the six chords:
|D F Ab
D F Ab C
Eb G Bb
F Ab C
Ab C Eb
B D F Ab
Leading tones: A pitch or notch that resolves or “leads” to a note one semitone higher or lower. The leading tone is usually the seventh scale degree of the diatonic scale with a string attachment to the tonic. To refresh your memory, the scale degrees for the major scale are as follows:
|1st – Tonic
2nd – Supertonic
3rd – Mediant
5th – Dominant
6th – Submediant
7th – Leading Tone
8th – Tonic (or Octave)
For example, the leading tone of a C-major scale is the B note. The leading chord for this scale would be: B D F, a diminished triad.
Tonicization: is the treatment of a pitch other than the overall tonic as a temporary key. A tonicized chord is a chord where the secondary dominant progresses.
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The twelve-tone technique orders the 12 notes of the chromatic scale, forming a row or series and provides for a basis of melody and harmony. It is difficult not to speak of serialism when explaining the twelve-tone technique. The term Serialism is used here because it represents a recurring series of ordered elements which are using in a way to give a piece of music unity.
The twelve-tone technique was developed by Arnold Schoenberg in 1921. Traditionally, harmony is based on resolution of dissonance to consonance. The tonic chord is the strongest consonance, followed by a dominant chord, etc… However, in atonal music no such hierarchy exists. Schoenberg’s solution was to extend Brahms’ idea (who developed an entire piece, “Developing Variation” from a single motif using retrograde or inversion), so that one motif contains both melodic and harmonic material, and each motif contains twelve tones. Any row can be modified in serial ways that maintain the original order of pitches. Consecutive pitches can be played simultaneously to create a chord, but the order of intervals must always be the same.
The following explain the basic aspects of the twelve-tone technique.
To understand octave equivalence, you need to be aware of two different main aspects. The first aspect is existence of a pitch-interval template that enables a listener to match successive tones of any kind to the particular pitch interval of an octave. The second aspect is the listeners affinity of tones being in an octave interval that is based on commonality of pitches. Therefore, octave equivalence is a property of the octave interval that is determined by the multiple pitches elicited by harmonic tones, and that the pitch template corresponding to the octave interval develops from the existence of harmonic complex tones with variable oscillation frequencies. Octave equivalence give unity and coherence the the musical piece.
Pitch Class: A pitch is a tone with a certain frequency associated with it, a pitch class represents a group of pitches with a common name. Pitch class A, contains all the pitches named A, for example different octaves of A.
Enharmonic Equivalence:Usually a Db is not the same as a C#. First of all, they represent different scale degrees and play different musical roles on melody and harmony. These distinctions do not exist in atonal music, notes that are enharmonically equivalent (Db and C#) are also functionally equivalent.
Integer Notation: Composers in the 20th Century used staff notation where Db and C# are noted differently from one another. However, for analytical purposes, the use of integers from 0 to 11 are going to be used to refer to different pitches. The following table describe this relationship.
|Integer Name||Pitch class|
|0||B#, C, Dbb|
Numbers and mathematics are used to model interesting properties of the music, but music itself is not mathematical and more than time is mathematical just because we keep track of what time it is.
Mod 12: Every pitch belongs to one of the twelve pitch classes. Going up and down the octaves, adding or subtracting 12 semitones) will just produce another member of the same pitch class. For example, if you start on E above middle C (pitch class 4) and go up 12 semitones, you will have ended back at pitch class 4. In the world of pitch classes, 4 + 12 = 16 = 4. Any number larger than 12 or smaller than 0 is equivalent to some integer from 0 to 11. To figure out which integer, just add or subtract 12 or any multiple of 12. For this example, and scales in general, twelve is calleed the modulus, and follows the rules of modulo 12 or mod 12.
To create a circular pitch class space, just attach the pitches p and p + 12 together. The result is a circular pitch class space that is called Z/12Z. The following diagram represents a circular pitch class space, where C = 0, D# = 3, F3 = 6, A = 9, etc…
Equal temperament is represented here by having each pair of adjacent notes be required to have an identical frequency. Other models of pitch class space, such as the circle of fifths, attempt to describe the special relationship between pitch classes related by perfect fifths. In equal temperament, twelve successive fifths equate to seven octaves, and in terms of pitch class, closes onto itself, forming a circle. A cyclic group of order 12, with residue class modulo 12.
Pitch space represents the modal relationships between pitches. There is also chord space, that is, relationships between chords. We will now discuss what is termed circular pitch class space. The simplest pitch space model is a linear line defined by real numbers as follows:
Intervals: Because of enharmonic equivalence, it is no longer required that you need different names for intervals with the same absolute size, for example, the minor 3rd or augmented 4th. In atonal music it is easier to just name intervals according to the number of semitones they contain. The interval between C and E and between C and Fb both contain four semitones and are both assigned the number 4. The following table explain these relationships.
|major 2nd, diminished 3rd||2|
|major 3rd, augmented 2nd||3|
|major 3rd, diminished 4th||4|
|augmented 3rd, perfect 4th||5|
|augmented 4th, diminished 5th||6|
|perfect 5th, diminished 6th||7|
|augmented 5th, minor 6th||8|
|major 6th, diminished 7th||9|
|augmented 6th, minor 7th||10|
Pitch Intervals: A pitch interval is the distance between two pitches, with units of semitones. Pitch intervals are created when you move from pitch to pitch in pitch space. You can consider ordered or unordered intervals. The ordered intervals focus on the contour of the line of music, its balance of rising and falling. The unordered pitch intervals concentrate entirely on the spaces between the pitches.
Ordered Pitch Class Intervals: A pitch class interval is the distance between tow pitch classes. It can never be larger than 11 semitones. To calculate pitch class intervals, it is best to think about the circular clack above. The ordered interval from pitch class “x” to pitch class “y” is y – x (mod)12. You will find it much faster to think about the circular clock, to find the ordered pitch class interval between C# and A, just count the number of steps (semitones) you need to go upward to the nearest A.
Unordered Pitch Class Intervals: For unordered intervals, it no longer matters if you count upward or downward, all you care about is the space between two pitch classes.
Interval Class: An unordered pitch class interval is also called an interval class. Each pitch class contains many individual pitches, each interval class contains many individual pitch intervals. Due to octave equivalence, compound intervals are considered equivalent to their counterparts within the octave. Pitch class intervals larger than six are considered equivalent to their complements in mod 12. (0 = 12, 1 = 11, 2 = 10, 3 = 9, 4 = 8, 5 = 7, 6 = 6). Therefore, intervals 23, 13, 11 and 1 are all members of interval class 1. The following table shows the seven different interval classes and some of their content.
To summarize, there are four ways of talking about intervals: ordered pitch intervals, unordered pitch intervals, ordered pitch class intervals, and unordered pitch class intervals.
Interval Class Content: The quality of a sound (or chord) can be estimated by listing all the intervals it contains. To keep everything under control, only unordered interval classes are considered. The number of interval classes a sound contains depends on the number of distinct pitch classes in the sound. The more pitch classes the greater number of interval classes. The following table describes the number of interval class content in each pitch class.
|No. of Pitch Classes||No. of Interval Classes|
Note that now you are counting all the intervals in the sound, not just those formed from from notes that are right next to each other.
The difference in sound can be correlated with the sounds interval class content. The interval class content is represented as a string of six numbers, this is called an interval vector. The first number in the interval vector gives the number of occurrences of interval class 1; the second number gives the number of occurrences of interval class 2, etc… The interval vector is not very useful when speaking about traditional tonal music, because here only a few basic sounds; four kinds of triads and five kinds of seventh chords, are regularly in use. In atonal music, you are confronted with a huge variety of musical ideas. The interval vector give a very simple and elegant way of representing the sound of music. There are 200 total interval vectors, 208 total prime forms, 352 total unique qualities, and 4096 total pitch collections.
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Basic Twelve-Tone Operations
Twelve-Tone Series: The previous discussion mostly talked about unordered sets, in this section you will be dealing with ordered sets of pitch classes called series. A series is a line of pitch classes. In a series, the pitches occur in a particular order, the identity of the series changes if the order changes. A series of twelve different pitch classes is called a set or row. Music that uses such a series is its basic structure is known as twelve-tone music. In some ways the twelve-tone row is like a theme that recurs in various ways throughout a piece. In other ways, its like a scale, the basic reference collection from which harmony and melody are derived. Thee series is a source of structural relations in a twelve-tone piece, from the surface to the deepest structural level, the series shapes the music.
Basic Operations:The twelve-tone series can be subjected to various operations like transposition and inversion for developing contrast and continuity. There is one important caveat, when a set of fewer than twelve elements is transposed or inverted, the content of the set usually changes.
The series is usually used for four different orderings: prime, retrograde, inversion, and retrograde inversion. For any series you will have a family of 48 series forms: 12 primes, 12 retrogrades, 12 inversions, and 12 retrograde inversions. In studying or composing a twelve-tone piece of music, it is useful to be able to calculate all 48 forms. The simplest way to do this is to create a 12×12 matrix. To construct this matrix, begin by writing P0 horizontally across the top and I0 vertically down the left hand side. Then write the remaining elements of the matrix by writing the remaining prime orderings in the rows from left to right. The second row will contain P6, the third row will contain P8. The following represents a maxtrix for P0 = 0, 11, 7, 8, 3, 1, 2, 10, 6, 5, 4, 9.
The rows of the matrix, reading left to right, contain all of the prime forms. Reading the rows right to left, contain all the retrograde forms. The columns of the matrix reading top to bottom contain all the inverted forms. Reading from bottom to top, contain all the retrograde inversion forms. It is important to note that most twelve-tone pieces use far less than 48 different forms. You as a composer can build into the original series, or for that matter the entire family of 48 forms, all kinds of structure and relations. A composition based on that series can express those structures and relationships in many different ways. This is the heart of the twelve-tone technique!
Subset Structure: Rows may be derived from a subset of any pitch class that is a divisor of 12, the most common is the trichord. The trichord may then undergo transposition, inversion, retrograde, or any combination to produce other parts of the row.
Invariance is a side effect of a derived row. Lets work through an example. Here is a row derived from a trichord:
|B Bb D Eb G F# G# E F C C# A|
Setting B = 0, the row can be displayed as:
|0, 11, 3, 4, 8, 7, 9, 5, 6, 1, 2, 10|
Notice that the third trichord 9, 5, 6, is the same as the first trichord 0, 11, 3 backwards 3, 11, 0 and transposed by 6. Here, you just add 6 to each value: #+6, 11+6, and 0+6 ==> 9, 5, 6 (the third trichord). You may have noticed that 11+6 = 17, but since we are working in mod12, then this really equates to 17-12 = 5. Remember from the above discussion that modular math numbers wrap around after they reach a certain value, in this case the value is 12.
Combinatoriality: A side effect of a derived row where combining different segments, or sets, such that the pitch class fulfills certain criteria, that is, a twelve-tone row and one of its transformations combine to form a pair of the total chromatic scale.. Usually the combination of hexachords completes the full chromatic.
Invariance: When listening to twelve-tone music, you do not need to be able to identify the forms of the series. Instead, you only need to hear the musical consequences of the series, the musical results of its ongoing transformations. Any musical quality or relationship preserved when the series is transformed is called invariant, and as we hear our way through a piece, our ear is often led via a chain of invariants. If all the discrete trichords of P0 are members of 3-3 (014), for example, then that will also be true for the other 48 series forms. No matter how the series os transposed or inverted, or retrograded, you will be able to hear the constant presence of those subsets.
The important message to convey is that in all twelve-tone music, it is not the mere presence of the series, but the audible musical content and the chain of associations created by its transformations. The series is not a static object that is mechanically repeated again and again, but a rich network of musical relationships and correlations that are expressed and developed in a multitude of ways.
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Piston, W. (1987) Harmony.
Piston, W. (1947) Counterpoint.
Piston, W. (1960) Principals of Harmonic Analysis.
Schoenberg, A. (1969) Structural Functions of Harmony.
Schoenberg, A. (1978) Theory of Harmony.
Strauss, J. N. (2000) Introduction to Post-Tonal Theory.
Perle, G. (1991) Serial Composition and Atonality.
Fux, J.J.; Mann, A. (1971). The Study of Counterpoint from Johann Joseph Fux’s Gradus ad parnassum.
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