Yerevan Komitas State Conservatory Entrance Exam Set

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a concept fundamental to analyzing melodic and harmonic structures in music theory, particularly relevant for advanced students at Yerevan Komitas State Conservatory. A harmonic progression is a sequence of numbers where the reciprocals form an arithmetic progression. If \(a, b, c\) are in harmonic progression, then \(1/a, 1/b, 1/c\) are in arithmetic progression. This means the difference between consecutive terms is constant: \(1/b – 1/a = 1/c – 1/b\). To find the third term of a harmonic progression given the first two, we can rearrange this equation. \(2/b = 1/a + 1/c\) \(2/b – 1/a = 1/c\) To find \(c\), we first find \(1/c\): \(1/c = \frac{2a – b}{ab}\) Therefore, \(c = \frac{ab}{2a – b}\). Given the first two terms of a harmonic progression are \(1/4\) and \(1/7\). Let \(a = 1/4\) and \(b = 1/7\). We need to find \(c\). Using the formula: \(c = \frac{(1/4)(1/7)}{2(1/4) – (1/7)}\) \(c = \frac{1/28}{1/2 – 1/7}\) To subtract the fractions in the denominator, find a common denominator, which is 14: \(1/2 – 1/7 = 7/14 – 2/14 = 5/14\) So, \(c = \frac{1/28}{5/14}\) Dividing by a fraction is the same as multiplying by its reciprocal: \(c = \frac{1}{28} \times \frac{14}{5}\) \(c = \frac{14}{28 \times 5}\) \(c = \frac{1}{2 \times 5}\) \(c = \frac{1}{10}\) The third term of the harmonic progression is \(1/10\). This demonstrates how understanding the underlying arithmetic progression of the reciprocals allows for the prediction of subsequent elements in a harmonic sequence, a skill crucial for analyzing and composing music with specific intervallic relationships. The ability to manipulate these relationships conceptually, without necessarily performing complex numerical calculations, reflects a deeper grasp of musical structure and theoretical principles taught at Yerevan Komitas State Conservatory.

The question probes the understanding of harmonic progression and its application in musical intervals, specifically focusing on the relationship between the overtone series and consonant intervals. A perfect fifth is represented by a frequency ratio of 3:2. In a harmonic series, the frequencies are integer multiples of the fundamental frequency. If the fundamental is \(f\), the harmonic series is \(f, 2f, 3f, 4f, 5f, 6f, \dots\). The interval between the third harmonic (\(3f\)) and the second harmonic (\(2f\)) has a frequency ratio of \(\frac{3f}{2f} = \frac{3}{2}\), which corresponds to a perfect fifth. Similarly, the interval between the sixth harmonic (\(6f\)) and the fourth harmonic (\(4f\)) also yields a ratio of \(\frac{6f}{4f} = \frac{3}{2}\), another perfect fifth. The interval between the fifth harmonic (\(5f\)) and the third harmonic (\(3f\)) is \(\frac{5f}{3f} = \frac{5}{3}\), which is a major third. The interval between the fourth harmonic (\(4f\)) and the third harmonic (\(3f\)) is \(\frac{4f}{3f} = \frac{4}{3}\), a perfect fourth. The interval between the sixth harmonic (\(6f\)) and the fifth harmonic (\(5f\)) is \(\frac{6f}{5f} = \frac{6}{5}\), a minor third. Therefore, the most fundamental and acoustically pure representation of a perfect fifth within the initial segments of the harmonic series arises from the relationship between the third and second harmonics, or any subsequent pair of harmonics whose ratio simplifies to 3:2, such as the sixth and fourth. The question asks for the interval that is *most directly* and *purely* represented by the simplest integer ratio within the harmonic series that defines a perfect fifth. This is the ratio of 3:2, which occurs between the third and second harmonics. This foundational relationship is crucial for understanding consonance in Western music theory, a core concept at the Yerevan Komitas State Conservatory.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a concept fundamental in music theory and composition, particularly in analyzing voice leading and chord structures. A harmonic progression is a sequence of chords where the roots of the chords form a harmonic series. The question asks to identify a sequence of roots that, when their frequencies are considered, would exhibit a harmonic progression. A harmonic progression is characterized by the roots of the chords being related by perfect fifths or perfect fourths. In terms of frequencies, this means that if the first root has a frequency \(f\), the subsequent roots will have frequencies that are multiples of \(f\) by factors related to the harmonic series, typically \(f\), \(2f\), \(3f/2\), \(4f/3\), \(5f/4\), \(3f/2\), etc. However, in the context of root movement in Western harmony, a harmonic progression is often simplified to a sequence of roots that move by perfect fifths (downward) or perfect fourths (upward). Let’s analyze the options in terms of root movement: Option a) C – G – D – A. This sequence represents movements of a perfect fifth down (C to G), a perfect fifth down (G to D), and a perfect fifth down (D to A). This is a classic example of a harmonic progression based on the circle of fifths. Option b) C – E – G – B. This sequence represents movements of a major third (C to E), a minor third (E to G), and a major third (G to B). These are diatonic relationships within a scale but not a harmonic progression in the sense of root movement by fifths or fourths. Option c) C – F – Bb – Eb. This sequence represents movements of a perfect fourth up (C to F), a perfect fourth up (F to Bb), and a perfect fourth up (Bb to Eb). This is also a harmonic progression, moving in the opposite direction of the circle of fifths. Option d) C – D – E – F. This sequence represents movements of a major second (C to D), a major second (D to E), and a minor second (E to F). These are stepwise movements within a scale and do not constitute a harmonic progression. The question asks for a sequence that *demonstrates* a harmonic progression. Both option a) and option c) represent harmonic progressions. However, the Yerevan Komitas State Conservatory Entrance Exam often emphasizes foundational concepts in Western music theory, where the progression by descending fifths (or ascending fourths) is a cornerstone. Without further context or a specific definition provided in the exam syllabus that might favor one direction over the other, both are technically correct harmonic progressions. Given the need to select one correct answer, and assuming the question is designed to test the most commonly recognized form of harmonic progression in classical analysis, the sequence of descending fifths is often the primary example. Let’s re-evaluate the options with the understanding that the question is likely testing the most fundamental and widely taught harmonic progression. The progression C-G-D-A represents a series of roots moving by perfect fifths downwards. This is a direct application of the circle of fifths, which is a fundamental concept in tonal harmony and is extensively studied in music theory programs like those at Yerevan Komitas State Conservatory. This type of progression is crucial for understanding cadences, modulation, and the underlying structure of many musical compositions. The other options represent diatonic scalar movement or other intervallic relationships that, while musically significant, do not define a harmonic progression in the same way that root movement by fifths or fourths does. The Conservatory’s curriculum would certainly prioritize the understanding of the circle of fifths and its implications for harmonic movement. Therefore, the sequence C-G-D-A is the most representative example of a harmonic progression as typically understood in classical music theory and taught at institutions like the Yerevan Komitas State Conservatory.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a fundamental concept in music theory and composition, particularly relevant to the harmonic analysis taught at Yerevan Komitas State Conservatory. A sequence \(a_1, a_2, a_3, \ldots\) is in harmonic progression if the reciprocals of its terms, \(1/a_1, 1/a_2, 1/a_3, \ldots\), form an arithmetic progression. Let the harmonic progression be \(h_1, h_2, h_3, h_4\). The reciprocals form an arithmetic progression: \(1/h_1, 1/h_2, 1/h_3, 1/h_4\). Let the common difference of this arithmetic progression be \(d\). So, \(1/h_2 = 1/h_1 + d\), \(1/h_3 = 1/h_1 + 2d\), \(1/h_4 = 1/h_1 + 3d\). We are given \(h_1 = 1/2\) and \(h_4 = 1/11\). Therefore, \(1/h_1 = 2\) and \(1/h_4 = 11\). Using the formula for the \(n\)-th term of an arithmetic progression, \(a_n = a_1 + (n-1)d\): For the fourth term: \(1/h_4 = 1/h_1 + (4-1)d\) \(11 = 2 + 3d\) \(9 = 3d\) \(d = 3\) Now we can find the terms of the arithmetic progression: \(1/h_1 = 2\) \(1/h_2 = 1/h_1 + d = 2 + 3 = 5\) \(1/h_3 = 1/h_1 + 2d = 2 + 2(3) = 2 + 6 = 8\) \(1/h_4 = 1/h_1 + 3d = 2 + 3(3) = 2 + 9 = 11\) The terms of the harmonic progression are the reciprocals of these values: \(h_1 = 1/2\) \(h_2 = 1/5\) \(h_3 = 1/8\) \(h_4 = 1/11\) The question asks for the third term of the harmonic progression, which is \(h_3\). \(h_3 = 1/8\). This question tests the understanding of the definition of a harmonic progression and the ability to apply the properties of arithmetic progressions to solve for unknown terms. In musical contexts, harmonic progressions are not just mathematical sequences but can relate to the spacing of pitches or the structure of chords, requiring a deep conceptual grasp beyond rote memorization. Understanding how to derive terms in such sequences is crucial for analyzing musical structures and potentially for compositional techniques explored at Yerevan Komitas State Conservatory. The ability to manipulate these relationships demonstrates a foundational analytical skill applicable to various musical disciplines.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a fundamental concept in music theory and composition, particularly relevant to advanced harmonic analysis taught at Yerevan Komitas State Conservatory. A sequence \(a_1, a_2, a_3, \dots\) is in harmonic progression if the reciprocals of its terms, \(1/a_1, 1/a_2, 1/a_3, \dots\), form an arithmetic progression. Let the harmonic progression be \(H_1, H_2, H_3, H_4\). The reciprocals form an arithmetic progression: \(1/H_1, 1/H_2, 1/H_3, 1/H_4\). Let the first term of the arithmetic progression be \(A = 1/H_1\) and the common difference be \(d\). Then, \(1/H_2 = A + d\), \(1/H_3 = A + 2d\), and \(1/H_4 = A + 3d\). We are given that the first term of the harmonic progression is \(H_1 = 1/2\). So, \(A = 1/(1/2) = 2\). We are also given that the fourth term of the harmonic progression is \(H_4 = 1/11\). Therefore, \(1/H_4 = 1/(1/11) = 11\). Using the formula for the nth term of an arithmetic progression, \(a_n = a_1 + (n-1)d\), we have for the fourth term: \(1/H_4 = 1/H_1 + (4-1)d\) \(11 = 2 + 3d\) \(11 – 2 = 3d\) \(9 = 3d\) \(d = 9/3 = 3\) Now we can find the second and third terms of the arithmetic progression: \(1/H_2 = A + d = 2 + 3 = 5\) \(1/H_3 = A + 2d = 2 + 2(3) = 2 + 6 = 8\) The terms of the harmonic progression are the reciprocals of these values: \(H_1 = 1/2\) \(H_2 = 1/5\) \(H_3 = 1/8\) \(H_4 = 1/11\) The question asks for the sum of the second and third terms of the harmonic progression. Sum = \(H_2 + H_3 = 1/5 + 1/8\) To add these fractions, we find a common denominator, which is 40: Sum = \((1 \times 8)/(5 \times 8) + (1 \times 5)/(8 \times 5)\) Sum = \(8/40 + 5/40\) Sum = \((8+5)/40\) Sum = \(13/40\) This calculation demonstrates the process of identifying the underlying arithmetic progression, determining its common difference, and then using that to find the missing terms of the harmonic progression before summing them. Understanding such relationships is crucial for analyzing complex harmonic structures and developing compositional techniques, aligning with the rigorous curriculum at Yerevan Komitas State Conservatory. The ability to manipulate sequences and understand their theoretical underpinnings is a hallmark of a well-prepared music scholar.

The core of this question lies in understanding the principles of counterpoint and harmonic progression as applied in Baroque-era composition, a cornerstone of study at Yerevan Komitas State Conservatory. Specifically, it probes the candidate’s ability to identify a harmonic resolution that maintains voice-leading integrity and adheres to stylistic conventions. Consider a two-part invention in C major. If the first voice (soprano) has just completed a melodic phrase ending on a G, and the second voice (alto) is harmonically supporting this by holding a C, the implied harmony is C major. The subsequent melodic movement in the soprano is to an F. To resolve this F in the alto voice, we must consider the harmonic implications and the rules of counterpoint. A common and stylistically appropriate resolution from a C major chord (implied by G in soprano and C in alto) to a chord that accommodates the F in the soprano would be to move to a G major chord. In a G major chord, the F is the leading tone. The alto voice, currently on C, must move to a note that creates a G major chord. The most logical and stylistically sound movement for the alto voice, considering the F in the soprano, would be to move to the dominant note of the implied G major chord, which is D. This creates a G major chord (G-B-D) with the F in the soprano acting as a non-chord tone (specifically, a suspension or passing tone resolving to E, or part of a secondary dominant leading to C). However, if we are looking for a direct resolution of the implied C major to a chord that accommodates the F, and assuming the F is a chord tone, then a chord where F is a chord tone and that follows logically from C major is F major. In F major, the F is the tonic. If the alto moves from C to F, and the soprano moves to F, this creates an F major chord. However, the question implies a progression from a C major context. A more sophisticated understanding of Baroque harmony would consider secondary dominants or modulations. If the F in the soprano is intended to resolve to E (a common melodic movement), then the preceding harmony could be a dominant seventh chord of A minor (E7), which contains G#, B, D, and F. However, the initial implied harmony is C major. Let’s re-evaluate the progression from C major with a G in the soprano and C in the alto. The implied harmony is C major. The soprano moves to F. If this F is to be a chord tone in the next harmony, and we are still in the orbit of C major, a common progression would be to a G chord (dominant). In a G major chord, the notes are G, B, D. The F in the soprano is not a chord tone of G major. However, if the F is a passing tone or suspension resolving to E, then the underlying harmony could be G major. In that case, the alto, currently on C, could move to D, the fifth of the G major chord. This would create a G major chord (G-B-D) with the F in the soprano as a non-chord tone. Alternatively, consider the F as the tonic of a new implied harmony. If the alto moves from C to F, and the soprano is on F, this creates an F major chord. This is a common progression (IV chord in C major). The question asks for the resolution of the alto’s C when the soprano moves to F. If the implied harmony is C major (G in soprano, C in alto), and the soprano moves to F, a stylistically appropriate next harmony could be F major. In F major, the alto’s C is the fifth. Therefore, the alto holding the C and the soprano moving to F would imply an F major chord. The question asks for the resolution of the alto’s C. If the alto moves from C to F, this is a valid resolution within an F major chord. Let’s consider the context of Yerevan Komitas State Conservatory’s emphasis on rigorous harmonic analysis. If the implied harmony is C major (G in soprano, C in alto), and the soprano moves to F, a common progression is to the subdominant (F major). In F major, the alto’s C is the fifth. Therefore, the alto moving from C to F is a valid harmonic and melodic resolution within the F major chord. This maintains smooth voice leading and adheres to common practice period harmonic progressions. The soprano’s F is the tonic of this new F major harmony. Final Answer Calculation: Initial implied harmony: C major (G in soprano, C in alto). Soprano moves to F. Consider the progression to the subdominant chord, F major. In F major, the notes are F, A, C. The alto is currently on C. If the alto moves to F, it becomes the tonic of the F major chord. This is a valid resolution and progression. Therefore, the alto moving from C to F is the correct resolution.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a fundamental concept in music theory and composition, particularly relevant to advanced harmonic analysis taught at Yerevan Komitas State Conservatory. A harmonic progression is a sequence of musical chords or tones that follows a specific pattern of root movement. When the *reciprocals* of the frequencies of tones in a harmonic progression form an arithmetic progression, the original frequencies themselves form a harmonic progression. Let the frequencies of the tones be \(f_1, f_2, f_3, \dots\). If \( \frac{1}{f_1}, \frac{1}{f_2}, \frac{1}{f_3}, \dots \) form an arithmetic progression, then there exists a common difference \(d\) such that: \( \frac{1}{f_2} = \frac{1}{f_1} + d \) \( \frac{1}{f_3} = \frac{1}{f_1} + 2d \) and so on. This implies: \( f_2 = \frac{1}{\frac{1}{f_1} + d} \) \( f_3 = \frac{1}{\frac{1}{f_1} + 2d} \) The question asks for the characteristic of the sequence of frequencies \(f_1, f_2, f_3, \dots\). Consider the relationship between consecutive terms. \( \frac{1}{f_n} = \frac{1}{f_1} + (n-1)d \) \( f_n = \frac{1}{\frac{1}{f_1} + (n-1)d} \) Let’s examine the relationship between \(f_n\) and \(f_{n+1}\). \( \frac{1}{f_{n+1}} = \frac{1}{f_1} + nd \) \( \frac{1}{f_{n+1}} – \frac{1}{f_n} = (\frac{1}{f_1} + nd) – (\frac{1}{f_1} + (n-1)d) = d \) This means that the difference between the reciprocals of consecutive frequencies is constant. This is the definition of a harmonic progression. The question is essentially asking to identify the sequence whose reciprocals form an arithmetic progression. This is precisely the definition of a harmonic progression. The scenario describes a composer at Yerevan Komitas State Conservatory exploring novel harmonic structures. The composer has identified a sequence of fundamental frequencies that, when their reciprocals are taken, yield a constant difference between successive terms. This mathematical property directly defines a harmonic progression. Therefore, the sequence of frequencies itself constitutes a harmonic progression. This concept is crucial for understanding complex overtone series, spectral music, and advanced tuning systems, all areas of study at the conservatory. Understanding these relationships allows for the creation of unique sonic textures and the analysis of contemporary compositional techniques. The ability to recognize and manipulate such mathematical underpinnings of sound is a hallmark of advanced musical study.

The question probes the understanding of harmonic progression and its relationship to arithmetic progression, a fundamental concept in music theory and composition, particularly relevant to the harmonic analysis taught at Yerevan Komitas State Conservatory. A harmonic progression is a sequence of musical chords. A sequence of numbers \(a_1, a_2, a_3, \dots\) is in harmonic progression if their reciprocals \(1/a_1, 1/a_2, 1/a_3, \dots\) are in arithmetic progression. Let the first three terms of the harmonic progression be \(h_1, h_2, h_3\). Their reciprocals are \(1/h_1, 1/h_2, 1/h_3\). These reciprocals form an arithmetic progression. Let the common difference be \(d\). So, \(1/h_2 = 1/h_1 + d\) and \(1/h_3 = 1/h_2 + d = 1/h_1 + 2d\). We are given that the first term of the harmonic progression is \(1/2\), so \(h_1 = 1/2\). The third term is \(1/6\), so \(h_3 = 1/6\). Substituting these values into the reciprocal relationships: \(1/h_1 = 1/(1/2) = 2\) \(1/h_3 = 1/(1/6) = 6\) Now, using the arithmetic progression property: \(1/h_3 = 1/h_1 + 2d\) \(6 = 2 + 2d\) \(4 = 2d\) \(d = 2\) The second term of the arithmetic progression is \(1/h_2 = 1/h_1 + d\). \(1/h_2 = 2 + 2 = 4\) To find the second term of the harmonic progression, \(h_2\), we take the reciprocal of \(1/h_2\): \(h_2 = 1/(1/h_2) = 1/4\) Therefore, the second term of the harmonic progression is \(1/4\). This concept is crucial for understanding voice leading, chord progressions, and the construction of musical phrases, all core elements of a rigorous music education at Yerevan Komitas State Conservatory. Understanding how harmonic relationships can be described through underlying arithmetic structures allows students to analyze and create music with greater depth and precision. The ability to recognize and manipulate these progressions is essential for composers, theorists, and performers alike, fostering a sophisticated musical intuition.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a concept fundamental to analyzing musical intervals and their theoretical underpinnings, particularly relevant for advanced music theory studies at Yerevan Komitas State Conservatory. A harmonic progression is a sequence of numbers where the reciprocals form an arithmetic progression. If \(a_1, a_2, a_3, \dots\) is a harmonic progression, then \( \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \dots \) is an arithmetic progression. Let the harmonic progression be \(h_1, h_2, h_3, \dots\). The reciprocals form an arithmetic progression: \( \frac{1}{h_1}, \frac{1}{h_2}, \frac{1}{h_3}, \dots \). Let the common difference of this arithmetic progression be \(d\). So, \( \frac{1}{h_2} = \frac{1}{h_1} + d \) and \( \frac{1}{h_3} = \frac{1}{h_1} + 2d \). Given the first three terms of a harmonic progression are \( \frac{1}{2}, \frac{1}{5}, \frac{1}{8} \). The reciprocals are \( 2, 5, 8 \). This is an arithmetic progression with the first term \(a_1 = 2\) and a common difference \(d = 5 – 2 = 3\). The next term in the arithmetic progression would be \( 8 + 3 = 11 \). Therefore, the next term in the harmonic progression is the reciprocal of 11, which is \( \frac{1}{11} \). To verify the relationship between harmonic and arithmetic progressions in a musical context, consider the concept of perfect fifths. If we have a sequence of pitches whose frequencies are in a harmonic progression, their relationships can be analyzed through the underlying arithmetic progression of their reciprocals. For instance, in tuning systems, the relationships between intervals are often expressed as ratios. A harmonic progression can represent a series of intervals where the underlying mathematical structure, when inverted, reveals a constant additive relationship, mirroring how musical intervals can be understood through their frequency ratios or, in this case, the reciprocals of those ratios. Understanding this allows for deeper analysis of consonance, dissonance, and the mathematical basis of musical scales and temperaments, which are core to advanced music theory at Yerevan Komitas State Conservatory. The ability to identify and extend such progressions is crucial for theoretical musicianship and compositional analysis.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a concept fundamental to analyzing musical intervals and their theoretical underpinnings, particularly relevant for advanced music theory studies at Yerevan Komitas State Conservatory. A harmonic progression is a sequence of numbers where the reciprocals form an arithmetic progression. If \(a_1, a_2, a_3, \dots\) is a harmonic progression, then \( \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \dots \) is an arithmetic progression. Let the harmonic progression be \(h_1, h_2, h_3, \dots\). The reciprocals form an arithmetic progression: \( \frac{1}{h_1}, \frac{1}{h_2}, \frac{1}{h_3}, \dots \). Let this arithmetic progression be \(a_1, a_2, a_3, \dots\). We are given the first three terms of the harmonic progression: \(h_1 = \frac{1}{3}, h_2 = \frac{1}{5}, h_3 = \frac{1}{7}\). The corresponding arithmetic progression is: \(a_1 = \frac{1}{h_1} = \frac{1}{1/3} = 3\) \(a_2 = \frac{1}{h_2} = \frac{1}{1/5} = 5\) \(a_3 = \frac{1}{h_3} = \frac{1}{1/7} = 7\) This arithmetic progression has a common difference \(d = a_2 – a_1 = 5 – 3 = 2\). We can verify this with \(a_3 – a_2 = 7 – 5 = 2\). The \(n\)-th term of an arithmetic progression is given by \(a_n = a_1 + (n-1)d\). So, the \(n\)-th term of this arithmetic progression is \(a_n = 3 + (n-1)2\). The \(n\)-th term of the harmonic progression \(h_n\) is the reciprocal of the \(n\)-th term of the arithmetic progression: \(h_n = \frac{1}{a_n} = \frac{1}{3 + (n-1)2}\). We need to find the 5th term of the harmonic progression, \(h_5\). First, find the 5th term of the arithmetic progression, \(a_5\): \(a_5 = a_1 + (5-1)d = 3 + (4)2 = 3 + 8 = 11\). Now, find the 5th term of the harmonic progression: \(h_5 = \frac{1}{a_5} = \frac{1}{11}\). The question tests the understanding of the reciprocal relationship between harmonic and arithmetic progressions and the ability to apply the formula for the \(n\)-th term of an arithmetic progression. This is crucial for analyzing tonal relationships, voice leading, and structural elements in music theory, where proportional relationships are paramount. Understanding how these sequences manifest in musical contexts, such as the overtone series or specific compositional techniques, requires a firm grasp of these foundational mathematical concepts as applied to musical structures. The ability to derive and manipulate these sequences demonstrates a candidate’s capacity for abstract reasoning and its application within a musical framework, aligning with the rigorous analytical demands at Yerevan Komitas State Conservatory.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a common area of study in music theory and composition, particularly relevant to the foundational theoretical knowledge expected at Yerevan Komitas State Conservatory. A harmonic progression is a sequence of numbers where the reciprocals form an arithmetic progression. If \(a, b, c\) are in harmonic progression, then \(1/a, 1/b, 1/c\) are in arithmetic progression. This means \(1/b – 1/a = 1/c – 1/b\). Rearranging this, we get \(2/b = 1/a + 1/c\), or \(b = \frac{2ac}{a+c}\). Let the three consecutive terms of the harmonic progression be \(h_1, h_2, h_3\). Their reciprocals, \(1/h_1, 1/h_2, 1/h_3\), form an arithmetic progression. Let the common difference of this arithmetic progression be \(d\). Then, \(1/h_2 = 1/h_1 + d\) and \(1/h_3 = 1/h_2 + d = 1/h_1 + 2d\). The problem states that the first term of the harmonic progression is 12, so \(h_1 = 12\). The third term is 18, so \(h_3 = 18\). We need to find the second term, \(h_2\). Using the property of harmonic progressions: \(1/h_1, 1/h_2, 1/h_3\) are in arithmetic progression. So, \(1/h_2\) is the arithmetic mean of \(1/h_1\) and \(1/h_3\). \(1/h_2 = \frac{(1/h_1) + (1/h_3)}{2}\) Substitute the given values: \(1/h_2 = \frac{(1/12) + (1/18)}{2}\) To add the fractions in the numerator, find a common denominator, which is 36: \(1/12 = 3/36\) \(1/18 = 2/36\) So, \(1/h_2 = \frac{(3/36) + (2/36)}{2}\) \(1/h_2 = \frac{5/36}{2}\) \(1/h_2 = 5/72\) Now, to find \(h_2\), we take the reciprocal: \(h_2 = 72/5\) \(h_2 = 14.4\) This calculation demonstrates that if the first and third terms of a harmonic progression are 12 and 18 respectively, the second term must be 14.4. This concept is fundamental in understanding tonal relationships and intervallic structures in music, where proportional relationships between frequencies or pitches are crucial. Understanding how harmonic and arithmetic progressions relate allows composers and theorists to analyze and construct musical sequences with specific sonic qualities and structural integrity, a core skill for students at Yerevan Komitas State Conservatory. The ability to manipulate and understand these numerical relationships underpins advanced harmonic analysis and compositional techniques.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a fundamental concept in music theory and composition, particularly relevant to advanced harmonic analysis taught at Yerevan Komitas State Conservatory. A sequence \(a_1, a_2, a_3, \dots\) is in harmonic progression if the reciprocals of its terms, \(1/a_1, 1/a_2, 1/a_3, \dots\), are in arithmetic progression. Let the harmonic progression be \(h_1, h_2, h_3, h_4\). Their reciprocals form an arithmetic progression: \(1/h_1, 1/h_2, 1/h_3, 1/h_4\). Let the common difference of this arithmetic progression be \(d\). So, \(1/h_2 = 1/h_1 + d\), \(1/h_3 = 1/h_1 + 2d\), \(1/h_4 = 1/h_1 + 3d\). We are given \(h_1 = 1/2\) and \(h_4 = 1/11\). Therefore, \(1/h_1 = 2\) and \(1/h_4 = 11\). Using the formula for the \(n\)-th term of an arithmetic progression, \(a_n = a_1 + (n-1)d\): For the fourth term of the reciprocal sequence: \(1/h_4 = 1/h_1 + (4-1)d\) \(11 = 2 + 3d\) \(9 = 3d\) \(d = 3\) Now we can find the terms of the reciprocal sequence: \(1/h_1 = 2\) \(1/h_2 = 1/h_1 + d = 2 + 3 = 5\) \(1/h_3 = 1/h_1 + 2d = 2 + 2(3) = 2 + 6 = 8\) \(1/h_4 = 1/h_1 + 3d = 2 + 3(3) = 2 + 9 = 11\) The terms of the harmonic progression are the reciprocals of these values: \(h_1 = 1/2\) \(h_2 = 1/5\) \(h_3 = 1/8\) \(h_4 = 1/11\) The question asks for the third term of the harmonic progression, which is \(h_3\). \(h_3 = 1/8\). This question tests the understanding of the definition of a harmonic progression and its relationship with arithmetic progressions, requiring the candidate to apply the properties of arithmetic sequences to solve for terms in a related harmonic sequence. Mastery of such concepts is crucial for analyzing complex harmonic structures and understanding historical compositional techniques, areas of significant focus within the curriculum at Yerevan Komitas State Conservatory. The ability to translate between different types of progressions demonstrates a deep grasp of mathematical underpinnings in music theory.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a fundamental concept in music theory and composition, particularly relevant to advanced studies at Yerevan Komitas State Conservatory. A harmonic progression is a sequence of numbers where the reciprocals form an arithmetic progression. If a sequence \(a_1, a_2, a_3, \dots\) is a harmonic progression, then the sequence \(\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \dots\) is an arithmetic progression. Let the harmonic progression be \(h_1, h_2, h_3, \dots\). We are given that the first three terms are \(h_1 = \frac{1}{2}\), \(h_2 = \frac{1}{5}\), and \(h_3 = \frac{1}{8}\). Consider the reciprocals of these terms: \(\frac{1}{h_1} = \frac{1}{1/2} = 2\) \(\frac{1}{h_2} = \frac{1}{1/5} = 5\) \(\frac{1}{h_3} = \frac{1}{1/8} = 8\) This sequence of reciprocals, \(2, 5, 8, \dots\), is an arithmetic progression. The common difference (\(d\)) of this arithmetic progression is \(5 – 2 = 3\) or \(8 – 5 = 3\). The \(n\)-th term of an arithmetic progression is given by \(a_n = a_1 + (n-1)d\). In this case, the \(n\)-th term of the reciprocal sequence is \(a_n = 2 + (n-1)3\). The \(n\)-th term of the harmonic progression (\(h_n\)) is the reciprocal of the \(n\)-th term of the arithmetic progression. So, \(h_n = \frac{1}{a_n} = \frac{1}{2 + (n-1)3}\). We need to find the 7th term of the harmonic progression, \(h_7\). First, find the 7th term of the arithmetic progression: \(a_7 = 2 + (7-1)3\) \(a_7 = 2 + (6)3\) \(a_7 = 2 + 18\) \(a_7 = 20\) Now, find the 7th term of the harmonic progression by taking the reciprocal of \(a_7\): \(h_7 = \frac{1}{a_7} = \frac{1}{20}\) Therefore, the 7th term of the harmonic progression is \(\frac{1}{20}\). This understanding of how harmonic progressions are derived from arithmetic progressions is crucial for analyzing melodic and harmonic structures, understanding voice leading, and composing in various styles, all of which are central to the curriculum at Yerevan Komitas State Conservatory. The ability to identify and extend such sequences is a foundational skill for any aspiring composer or music theorist.

The question probes the understanding of harmonic progression and its relationship to arithmetic progression, a common concept in music theory and composition, particularly relevant to the foundational studies at Yerevan Komitas State Conservatory. A harmonic progression is a sequence of musical chords that are related by harmonic function. The progression of roots of these chords often follows patterns. If the roots of three successive chords in a harmonic progression form an arithmetic progression, their frequencies would also form an arithmetic progression. However, the question asks about the *intervals* between the roots, not the frequencies themselves. Let the roots of three successive chords be represented by their fundamental frequencies \(f_1, f_2, f_3\). If these frequencies form an arithmetic progression, then \(f_2 – f_1 = f_3 – f_2\), or \(2f_2 = f_1 + f_3\). The intervals between these roots are typically measured in semitones. Let the interval between \(f_1\) and \(f_2\) be \(I_1\) semitones, and the interval between \(f_2\) and \(f_3\) be \(I_2\) semitones. The relationship between frequency and semitones is logarithmic: \(f = f_0 \cdot 2^{n/12}\), where \(f_0\) is a reference frequency and \(n\) is the number of semitones. So, \(f_2 = f_1 \cdot 2^{I_1/12}\) and \(f_3 = f_2 \cdot 2^{I_2/12} = f_1 \cdot 2^{I_1/12} \cdot 2^{I_2/12} = f_1 \cdot 2^{(I_1+I_2)/12}\). If \(f_1, f_2, f_3\) form an arithmetic progression, then \(2f_2 = f_1 + f_3\). Substituting the frequency relationships: \(2 \cdot (f_1 \cdot 2^{I_1/12}) = f_1 + (f_1 \cdot 2^{(I_1+I_2)/12})\) Divide by \(f_1\): \(2 \cdot 2^{I_1/12} = 1 + 2^{(I_1+I_2)/12}\) \(2^{1 + I_1/12} = 1 + 2^{(I_1+I_2)/12}\) \(2^{(12+I_1)/12} = 1 + 2^{(I_1+I_2)/12}\) This equation relates the intervals \(I_1\) and \(I_2\). The question asks about the nature of the intervals themselves. If the *roots* form an arithmetic progression, it implies a constant difference in frequency. However, musical intervals are typically perceived and notated based on ratios or logarithmic scales (semitones). Consider the case where the roots themselves are not frequencies but scale degrees or tonal centers. In a harmonic progression, the root movement often follows specific patterns. If the root movement is an arithmetic progression in terms of scale degrees (e.g., C, D, E), the intervals in semitones would be 2 semitones (C to D) and 2 semitones (D to E). This is a constant interval. However, the question is more abstract, referring to the *intervals* between the roots. If the roots themselves are considered as abstract entities whose relationships are defined by intervals, and these relationships form an arithmetic progression, it means the difference between successive intervals is constant. Let the intervals be \(I_1\) and \(I_2\). If the sequence of intervals forms an arithmetic progression, then \(I_2 – I_1 = d\), where \(d\) is a constant difference. This means the intervals themselves are not necessarily equal, but their difference is constant. For example, if the intervals were 2 semitones and 4 semitones, the difference is 2 semitones. This is an arithmetic progression of intervals. The core concept here is that a harmonic progression implies a functional relationship between chords. If the *roots* of these chords are considered in a way that their progression forms an arithmetic sequence, it implies a consistent step-wise movement. In musical contexts, this often translates to equal steps in pitch space, which are measured by intervals. Therefore, if the underlying progression of roots is arithmetic, the intervals between them must also exhibit a consistent, arithmetic relationship. This means the difference between consecutive intervals is constant. The correct answer is that the intervals themselves form an arithmetic progression. This means the difference between the first interval and the second interval is the same as the difference between the second interval and the third interval, and so on. This is the definition of an arithmetic progression applied to the intervals.

The question probes the understanding of harmonic progression and its relationship to arithmetic progression in the context of musical intervals, a core concept in music theory relevant to the Yerevan Komitas State Conservatory. A harmonic series is based on integer multiples of a fundamental frequency. The intervals formed by successive partials in a harmonic series, when expressed as frequency ratios, are not constant in terms of semitones. However, the *reciprocals* of these frequencies, or equivalently, the *periods* of these partials, form an arithmetic progression. Let the fundamental frequency be \(f_0\). The frequencies of the harmonic series are \(f_0, 2f_0, 3f_0, 4f_0, \dots, nf_0, \dots\). The periods of these frequencies are \(T_0 = \frac{1}{f_0}, T_1 = \frac{1}{2f_0}, T_2 = \frac{1}{3f_0}, T_3 = \frac{1}{4f_0}, \dots, T_{n-1} = \frac{1}{nf_0}, \dots\). Let’s examine the differences between successive periods: \(T_1 – T_0 = \frac{1}{2f_0} – \frac{1}{f_0} = \frac{1-2}{2f_0} = -\frac{1}{2f_0}\) \(T_2 – T_1 = \frac{1}{3f_0} – \frac{1}{2f_0} = \frac{2-3}{6f_0} = -\frac{1}{6f_0}\) \(T_3 – T_2 = \frac{1}{4f_0} – \frac{1}{3f_0} = \frac{3-4}{12f_0} = -\frac{1}{12f_0}\) This shows that the periods themselves do not form an arithmetic progression. Now consider the reciprocals of the frequencies, which are the periods: \(T_n = \frac{1}{(n+1)f_0}\) for \(n=0, 1, 2, \dots\). The sequence of reciprocals of frequencies is \(\frac{1}{f_0}, \frac{1}{2f_0}, \frac{1}{3f_0}, \frac{1}{4f_0}, \dots\). Let \(a_n = \frac{1}{nf_0}\) for \(n=1, 2, 3, \dots\). The differences between successive terms are: \(a_2 – a_1 = \frac{1}{2f_0} – \frac{1}{f_0} = -\frac{1}{2f_0}\) \(a_3 – a_2 = \frac{1}{3f_0} – \frac{1}{2f_0} = -\frac{1}{6f_0}\) \(a_4 – a_3 = \frac{1}{4f_0} – \frac{1}{3f_0} = -\frac{1}{12f_0}\) This confirms that the reciprocals of frequencies do not form an arithmetic progression. The question is about the *intervals* formed by successive partials. The frequency ratios of successive partials are: \(\frac{2f_0}{f_0} = 2\) (Octave) \(\frac{3f_0}{2f_0} = \frac{3}{2}\) (Perfect Fifth) \(\frac{4f_0}{3f_0} = \frac{4}{3}\) (Perfect Fourth) \(\frac{5f_0}{4f_0} = \frac{5}{4}\) (Major Third) \(\frac{6f_0}{5f_0} = \frac{6}{5}\) (Minor Third) The question states that the *reciprocals of the frequencies* form an arithmetic progression. This is a misunderstanding of the harmonic series’ properties. The harmonic series itself is defined by frequencies that are integer multiples of a fundamental. The *intervals* between successive partials are what are musically significant. The statement that the reciprocals of frequencies form an AP is incorrect. However, the question asks what property *would* hold if this were true. If the reciprocals of frequencies formed an arithmetic progression, let the frequencies be \(f_1, f_2, f_3, \dots\). Then \(\frac{1}{f_1}, \frac{1}{f_2}, \frac{1}{f_3}, \dots\) is an AP. This means \(\frac{1}{f_2} – \frac{1}{f_1} = \frac{1}{f_3} – \frac{1}{f_2} = d\) (a constant difference). Rearranging, \(\frac{f_1 – f_2}{f_1 f_2} = \frac{f_2 – f_3}{f_2 f_3} = d\). This implies \(\frac{f_2 – f_1}{f_1 f_2} = -\frac{f_3 – f_2}{f_2 f_3}\). Or, \(\frac{1}{f_1} – \frac{1}{f_2} = \frac{1}{f_2} – \frac{1}{f_3}\). This means the frequencies themselves would be in harmonic progression. A harmonic progression is a sequence where the reciprocals form an arithmetic progression. Therefore, if the reciprocals of the frequencies of a series of musical tones formed an arithmetic progression, the frequencies themselves would be in a harmonic progression. This is a fundamental concept in understanding the mathematical underpinnings of musical intervals and scales, a key area of study at the Yerevan Komitas State Conservatory. The relationship between arithmetic and harmonic progressions is crucial for analyzing tonal systems and the construction of scales. Understanding this inverse relationship is vital for advanced music theory and composition. The correct answer is that the frequencies themselves would form a harmonic progression.

The question probes the understanding of harmonic progression and its relationship to arithmetic progression, specifically in the context of musical intervals. A harmonic series is based on integer multiples of a fundamental frequency. If we consider the frequencies of notes in a harmonic series, they form a sequence where the reciprocals of the frequencies are in an arithmetic progression. For example, if the fundamental frequency is \(f_0\), the frequencies in the harmonic series are \(f_0, 2f_0, 3f_0, 4f_0, \dots\). The reciprocals are \(\frac{1}{f_0}, \frac{1}{2f_0}, \frac{1}{3f_0}, \frac{1}{4f_0}, \dots\). This sequence of reciprocals is a harmonic progression. The question asks about a scenario where the *intervals* between successive notes in a harmonic series are perceived to be equal in a specific tuning system. This implies that the *ratios* of frequencies are not necessarily simple integers as in a pure harmonic series, but are adjusted to create equal perceived steps. In equal temperament, the octave is divided into 12 equal semitones, where the frequency ratio between adjacent semitones is the twelfth root of 2, or \(2^{1/12}\). If we consider a sequence of notes \(n_1, n_2, n_3, \dots\) where the interval between \(n_i\) and \(n_{i+1}\) is a constant semitone, then the frequencies \(f_1, f_2, f_3, \dots\) would be \(f_1, f_1 \cdot 2^{1/12}, f_1 \cdot (2^{1/12})^2, f_1 \cdot (2^{1/12})^3, \dots\). The question then asks about the relationship between the *number of vibrations* (frequency) of these notes and their position in the sequence. If the frequencies form a geometric progression with a common ratio \(r = 2^{1/12}\), then the sequence of frequencies is \(f_1, f_1 r, f_1 r^2, f_1 r^3, \dots\). The reciprocals of these frequencies would be \(\frac{1}{f_1}, \frac{1}{f_1 r}, \frac{1}{f_1 r^2}, \frac{1}{f_1 r^3}, \dots\). This sequence of reciprocals is a geometric progression with a common ratio of \(\frac{1}{r} = \frac{1}{2^{1/12}}\). A sequence where the reciprocals form a geometric progression is a harmonic progression. Therefore, the frequencies themselves form a geometric progression, and their reciprocals form a harmonic progression. The question is subtly asking about the relationship between the *number of vibrations* (frequency) and the *position* in a sequence of equally tempered intervals. If the intervals are equal, the frequencies form a geometric progression. The question is framed around the concept of a harmonic series, which is fundamentally related to integer ratios and thus harmonic progressions of frequencies. However, the scenario describes equal temperament, which implies geometric progression of frequencies. The core of the question is to identify the progression of the *reciprocals* of these frequencies. Let the frequencies be \(f_1, f_2, f_3, \dots\). In equal temperament, \(f_{n+1} = f_n \cdot 2^{1/12}\). This means the frequencies form a geometric progression with common ratio \(r = 2^{1/12}\). The reciprocals are \(\frac{1}{f_1}, \frac{1}{f_2}, \frac{1}{f_3}, \dots\). The ratio of successive terms in the reciprocal sequence is \(\frac{1/f_{n+1}}{1/f_n} = \frac{f_n}{f_{n+1}} = \frac{f_n}{f_n \cdot r} = \frac{1}{r} = \frac{1}{2^{1/12}}\). Since the ratio of successive terms is constant, the reciprocals form a geometric progression. A sequence whose reciprocals form a geometric progression is, by definition, a harmonic progression. The question is designed to test the understanding of how different tuning systems relate to mathematical progressions, particularly the distinction between the harmonic series (integer ratios, harmonic progression of frequencies) and equal temperament (geometric progression of frequencies). The scenario describes a situation that, while using the term “harmonic series” loosely to refer to a sequence of notes, actually defines equal temperament. The key is to recognize that equal temperament implies a geometric progression of frequencies, and therefore a harmonic progression of their reciprocals. The correct answer is that the reciprocals of the frequencies form a harmonic progression.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a fundamental concept in music theory and composition, particularly relevant to the analytical skills fostered at Yerevan Komitas State Conservatory. A harmonic progression is a sequence of numbers where the reciprocals form an arithmetic progression. If \(a, b, c\) are in harmonic progression, then \(1/a, 1/b, 1/c\) are in arithmetic progression. This means \(1/b – 1/a = 1/c – 1/b\). Rearranging this, we get \(2/b = 1/a + 1/c\), or \(b = \frac{2ac}{a+c}\). In this scenario, the fundamental frequencies of three instruments are in harmonic progression. Let these frequencies be \(f_1, f_2, f_3\). We are given \(f_1 = 440\) Hz (A4) and \(f_3 = 880\) Hz (A5). We need to find \(f_2\). Using the harmonic progression formula: \[ f_2 = \frac{2 \times f_1 \times f_3}{f_1 + f_3} \] \[ f_2 = \frac{2 \times 440 \text{ Hz} \times 880 \text{ Hz}}{440 \text{ Hz} + 880 \text{ Hz}} \] \[ f_2 = \frac{2 \times 440 \times 880}{1320} \text{ Hz} \] \[ f_2 = \frac{880 \times 880}{1320} \text{ Hz} \] \[ f_2 = \frac{774400}{1320} \text{ Hz} \] \[ f_2 = 586.67 \text{ Hz} \] (approximately) This calculation demonstrates the precise mathematical relationship that defines a harmonic progression. Understanding this relationship is crucial for advanced harmonic analysis, understanding overtone series, and the construction of musical intervals. At Yerevan Komitas State Conservatory, students are expected to not only recognize these theoretical underpinnings but also to apply them in compositional and analytical contexts. The ability to derive the middle term of a harmonic sequence is a foundational skill that informs a deeper appreciation of acoustic principles and their manifestation in musical structures. This question tests the candidate’s grasp of abstract mathematical relationships as they apply to concrete musical phenomena, a hallmark of rigorous conservatory training.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a foundational concept in music theory and composition, particularly relevant to advanced harmonic analysis taught at Yerevan Komitas State Conservatory. A harmonic progression is a sequence where the reciprocals of the terms form an arithmetic progression. If a sequence \(a, b, c\) is in harmonic progression, then \(1/a, 1/b, 1/c\) is in arithmetic progression. This means the difference between consecutive terms in the reciprocal sequence is constant: \(1/b – 1/a = 1/c – 1/b\). To solve this, we are given that \(1/3, 1/x, 1/y\) are in arithmetic progression. This implies \(1/x – 1/3 = 1/y – 1/x\). Rearranging this, we get \(2/x = 1/3 + 1/y\). We are also given that \(x, 12, y\) are in harmonic progression. This means \(1/x, 1/12, 1/y\) are in arithmetic progression. Therefore, \(1/12 – 1/x = 1/y – 1/12\). Rearranging this, we get \(2/12 = 1/x + 1/y\), which simplifies to \(1/6 = 1/x + 1/y\). Now we have a system of two equations: 1) \(2/x = 1/3 + 1/y\) 2) \(1/6 = 1/x + 1/y\) Substitute equation (2) into equation (1): \(2/x = 1/3 + (1/6 – 1/x)\) \(2/x = 1/3 + 1/6 – 1/x\) \(2/x + 1/x = 1/3 + 1/6\) \(3/x = 2/6 + 1/6\) \(3/x = 3/6\) \(3/x = 1/2\) Cross-multiplying gives \(3 \times 2 = 1 \times x\), so \(x = 6\). Now substitute the value of \(x\) back into equation (2) to find \(y\): \(1/6 = 1/6 + 1/y\) \(1/6 – 1/6 = 1/y\) \(0 = 1/y\) This implies that \(y\) would have to be infinitely large, which is not a standard interpretation for a finite harmonic progression in this context. Let’s re-examine the problem statement and the properties of harmonic progressions. A more direct approach using the definition of harmonic progression: if \(a, b, c\) are in harmonic progression, then \(b = \frac{2ac}{a+c}\). Given \(x, 12, y\) are in harmonic progression, we have \(12 = \frac{2xy}{x+y}\). This simplifies to \(12(x+y) = 2xy\), or \(6(x+y) = xy\). Given \(1/3, 1/x, 1/y\) are in arithmetic progression, the common difference is \(d = 1/x – 1/3\). Also, \(1/y = 1/x + d = 1/x + (1/x – 1/3) = 2/x – 1/3\). So, \(1/y = \frac{2}{x} – \frac{1}{3} = \frac{6-x}{3x}\). This means \(y = \frac{3x}{6-x}\). Now substitute this expression for \(y\) into the equation \(6(x+y) = xy\): \(6(x + \frac{3x}{6-x}) = x(\frac{3x}{6-x})\) Divide both sides by \(x\) (assuming \(x \neq 0\)): \(6(1 + \frac{3}{6-x}) = \frac{3x}{6-x}\) \(6(\frac{6-x+3}{6-x}) = \frac{3x}{6-x}\) \(6(\frac{9-x}{6-x}) = \frac{3x}{6-x}\) Multiply both sides by \((6-x)\) (assuming \(x \neq 6\)): \(6(9-x) = 3x\) \(54 – 6x = 3x\) \(54 = 9x\) \(x = 54 / 9\) \(x = 6\) Now substitute \(x=6\) back into the equation for \(y\): \(y = \frac{3x}{6-x} = \frac{3(6)}{6-6} = \frac{18}{0}\). This again leads to an undefined value for \(y\). This indicates that the initial assumption of a standard finite harmonic progression might be too restrictive or that there’s a subtlety in the problem setup. Let’s re-evaluate the arithmetic progression: \(1/3, 1/x, 1/y\). The condition for arithmetic progression is \(2 \times (\text{middle term}) = (\text{first term}) + (\text{third term})\). So, \(2/x = 1/3 + 1/y\). For the harmonic progression \(x, 12, y\): The condition for harmonic progression is \(2 \times (\text{middle term}) = \frac{1}{\frac{1}{\text{first term}} + \frac{1}{\text{third term}}}\) is incorrect. The correct definition is that the reciprocals are in arithmetic progression. So, \(1/x, 1/12, 1/y\) are in arithmetic progression. Thus, \(2/12 = 1/x + 1/y\), which simplifies to \(1/6 = 1/x + 1/y\). We have the system: 1) \(2/x = 1/3 + 1/y\) 2) \(1/6 = 1/x + 1/y\) From (2), \(1/y = 1/6 – 1/x\). Substitute this into (1): \(2/x = 1/3 + (1/6 – 1/x)\) \(2/x = 1/3 + 1/6 – 1/x\) \(2/x + 1/x = 2/6 + 1/6\) \(3/x = 3/6\) \(3/x = 1/2\) \(x = 6\) Now substitute \(x=6\) into \(1/6 = 1/x + 1/y\): \(1/6 = 1/6 + 1/y\) \(0 = 1/y\) This implies \(y\) approaches infinity. However, in the context of typical conservatory entrance exams, such scenarios usually imply a misunderstanding or a specific interpretation. Let’s consider the possibility that the question implies a specific relationship that avoids infinities. Let’s re-read the question carefully. The question asks for the value of \(x\). We have definitively found \(x=6\) through consistent application of the definitions. The issue with \(y\) might be a distractor or indicate a degenerate case that doesn’t affect the value of \(x\). The core of the problem lies in the relationship between harmonic and arithmetic progressions. The calculation for \(x\) is robust. Final check: If \(x=6\), then \(1/3, 1/6, 1/y\) are in AP. The common difference is \(1/6 – 1/3 = 1/6 – 2/6 = -1/6\). So, \(1/y = 1/6 + (-1/6) = 0\), meaning \(y\) is infinite. If \(x=6\), then \(6, 12, y\) are in HP. This means \(1/6, 1/12, 1/y\) are in AP. The common difference is \(1/12 – 1/6 = 1/12 – 2/12 = -1/12\). So, \(1/y = 1/12 + (-1/12) = 0\), meaning \(y\) is infinite. Both conditions are met with \(x=6\), even though \(y\) becomes infinite. The question specifically asks for \(x\). The calculation for \(x\) is: Given \(1/3, 1/x, 1/y\) are in arithmetic progression (AP), then \(2/x = 1/3 + 1/y\). Given \(x, 12, y\) are in harmonic progression (HP), then \(1/x, 1/12, 1/y\) are in AP. Thus, \(2/12 = 1/x + 1/y\), which simplifies to \(1/6 = 1/x + 1/y\). From the second equation, \(1/y = 1/6 – 1/x\). Substitute this into the first equation: \(2/x = 1/3 + (1/6 – 1/x)\). Combine terms involving \(x\): \(2/x + 1/x = 1/3 + 1/6\). Simplify: \(3/x = 2/6 + 1/6\). \(3/x = 3/6\). \(3/x = 1/2\). Cross-multiply: \(3 \times 2 = 1 \times x\). \(x = 6\). The value of \(x\) is 6. This question tests the fundamental understanding of the relationship between arithmetic and harmonic progressions, which is crucial for analyzing harmonic structures and voice leading in Western classical music, a core component of the curriculum at Yerevan Komitas State Conservatory. Understanding these progressions allows students to deconstruct complex harmonic sequences, identify underlying patterns, and apply theoretical knowledge to compositional and analytical tasks. The scenario, while seemingly abstract, reflects the kind of analytical rigor expected when studying advanced harmony and counterpoint, where recognizing such relationships is key to interpreting musical scores and developing one’s own compositional voice. The potential for an infinite term in the sequence is a mathematical consequence that highlights the properties of these progressions and requires students to focus on the specific variable requested, demonstrating precision in their analytical approach.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a fundamental concept in music theory and composition, particularly relevant to advanced harmonic analysis taught at Yerevan Komitas State Conservatory. A harmonic progression is a sequence of numbers where the reciprocals form an arithmetic progression. If \(a, b, c\) are in harmonic progression, then \(1/a, 1/b, 1/c\) are in arithmetic progression. This means the difference between consecutive terms is constant: \(1/b – 1/a = 1/c – 1/b\). To find the third term of a harmonic progression given the first two, we can rearrange this equation: \(2/b = 1/a + 1/c\) \(2/b – 1/a = 1/c\) \( (2a – b) / ab = 1/c \) \( c = ab / (2a – b) \) Let the first term be \(a = 1/2\) and the second term be \(b = 1/3\). Substituting these values into the formula for the third term \(c\): \( c = (1/2) * (1/3) / (2 * (1/2) – 1/3) \) \( c = (1/6) / (1 – 1/3) \) \( c = (1/6) / (2/3) \) \( c = (1/6) * (3/2) \) \( c = 3/12 \) \( c = 1/4 \) Therefore, the third term of the harmonic progression is \(1/4\). The sequence is \(1/2, 1/3, 1/4, \dots\). The reciprocals are \(2, 3, 4, \dots\), which form an arithmetic progression with a common difference of 1. This demonstrates a core principle of harmonic relationships that underpins tonal music and its theoretical frameworks, crucial for students at Yerevan Komitas State Conservatory to grasp for advanced compositional and analytical studies. Understanding these underlying mathematical structures in musical relationships is vital for developing sophisticated musical intuition and technical proficiency.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a concept fundamental to analyzing melodic and harmonic structures in music theory, particularly relevant to advanced studies at Yerevan Komitas State Conservatory. A harmonic progression is a sequence of numbers where the reciprocals form an arithmetic progression. If \(a_1, a_2, a_3, \dots\) is a harmonic progression, then \(\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \dots\) is an arithmetic progression. Let the harmonic progression be \(h_1, h_2, h_3, h_4\). The reciprocals form an arithmetic progression: \(\frac{1}{h_1}, \frac{1}{h_2}, \frac{1}{h_3}, \frac{1}{h_4}\). Let the common difference of this arithmetic progression be \(d\). We are given that the first term of the harmonic progression is \(h_1 = \frac{1}{2}\) and the fourth term is \(h_4 = \frac{1}{11}\). For the corresponding arithmetic progression, the first term is \(a_1 = \frac{1}{h_1} = \frac{1}{1/2} = 2\). The fourth term is \(a_4 = \frac{1}{h_4} = \frac{1}{1/11} = 11\). In an arithmetic progression, the \(n\)-th term is given by \(a_n = a_1 + (n-1)d\). For the fourth term, we have \(a_4 = a_1 + (4-1)d\). Substituting the known values: \(11 = 2 + 3d\). Solving for \(d\): \(11 – 2 = 3d\) \(9 = 3d\) \(d = \frac{9}{3} = 3\). Now we need to find the third term of the harmonic progression, \(h_3\). This corresponds to the third term of the arithmetic progression, \(a_3\). Using the formula for the \(n\)-th term of an arithmetic progression: \(a_3 = a_1 + (3-1)d\). \(a_3 = 2 + (2)(3)\) \(a_3 = 2 + 6\) \(a_3 = 8\). Since \(a_3 = \frac{1}{h_3}\), we can find \(h_3\) by taking the reciprocal of \(a_3\): \(h_3 = \frac{1}{a_3} = \frac{1}{8}\). The third term of the harmonic progression is \(\frac{1}{8}\). This understanding of reciprocal relationships between harmonic and arithmetic progressions is crucial for analyzing intervallic relationships and constructing harmonic sequences in composition and performance, aligning with the rigorous theoretical training at Yerevan Komitas State Conservatory. The ability to manipulate these sequences demonstrates a deep grasp of foundational music theory principles.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a fundamental concept in music theory and composition, particularly relevant for advanced students at Yerevan Komitas State Conservatory. A harmonic progression is a sequence of numbers where the reciprocals form an arithmetic progression. If the first three terms of a harmonic progression are \(h_1, h_2, h_3\), then their reciprocals \(1/h_1, 1/h_2, 1/h_3\) form an arithmetic progression. This means that the difference between consecutive terms is constant: \(1/h_2 – 1/h_1 = 1/h_3 – 1/h_2\). Let the harmonic progression be \(h_1, h_2, h_3\). We are given \(h_1 = 1/4\) and \(h_3 = 1/10\). The reciprocals form an arithmetic progression: \(1/h_1, 1/h_2, 1/h_3\). So, \(1/(1/4), 1/h_2, 1/(1/10)\) form an arithmetic progression. This simplifies to \(4, 1/h_2, 10\). In an arithmetic progression, the middle term is the average of the first and third terms. Therefore, \(1/h_2 = \frac{4 + 10}{2}\). \(1/h_2 = \frac{14}{2}\). \(1/h_2 = 7\). To find \(h_2\), we take the reciprocal of \(1/h_2\): \(h_2 = 1/7\). This calculation demonstrates that if the first term of a harmonic progression is \(1/4\) and the third term is \(1/10\), the second term must be \(1/7\). This concept is crucial for understanding intervallic relationships, voice leading, and the construction of musical sequences that adhere to specific mathematical relationships, reflecting the rigorous theoretical training expected at Yerevan Komitas State Conservatory. Understanding how harmonic progressions relate to arithmetic progressions allows students to analyze and create music with sophisticated underlying structures, moving beyond simple melodic or rhythmic patterns to explore deeper mathematical and aesthetic connections within musical composition.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a concept fundamental to certain theoretical aspects of music theory and composition, particularly in analyzing melodic or harmonic structures. A harmonic progression is a sequence of numbers where the reciprocals of the terms form an arithmetic progression. If \(a, b, c\) are in harmonic progression, then \(1/a, 1/b, 1/c\) are in arithmetic progression. This means \(1/b – 1/a = 1/c – 1/b\). Rearranging this, we get \(2/b = 1/a + 1/c\), or \(b = \frac{2ac}{a+c}\). Consider a scenario where the first three terms of a harmonic progression are \(x, y, z\). If these terms represent, for instance, the durations of successive rhythmic units or the intervals between specific pitches in a compositional framework, their reciprocals would form an arithmetic progression. Let the common difference of this arithmetic progression be \(d\). Then, \(1/y = 1/x + d\) and \(1/z = 1/y + d = 1/x + 2d\). The question asks for the relationship between \(x, y, z\) if they form a harmonic progression. From the definition, we know that \(1/x, 1/y, 1/z\) are in arithmetic progression. Therefore, the middle term \(1/y\) is the arithmetic mean of the other two: \(1/y = \frac{1/x + 1/z}{2}\). To find the relationship between \(x, y, z\), we can manipulate this equation: \(2/y = 1/x + 1/z\) To combine the terms on the right side, we find a common denominator: \(2/y = \frac{z + x}{xz}\) Now, we can take the reciprocal of both sides: \(y/2 = \frac{xz}{x+z}\) Finally, multiply both sides by 2 to isolate \(y\): \(y = \frac{2xz}{x+z}\) This relationship is crucial in understanding how harmonic sequences function, potentially influencing melodic contour, rhythmic organization, or even the spacing of sonic events in a composition studied at Yerevan Komitas State Conservatory Entrance Exam University. Understanding this mathematical underpinning allows for a deeper appreciation of compositional techniques that might employ such progressions, reflecting a rigorous analytical approach to musical structure. The ability to derive this relationship demonstrates a grasp of fundamental mathematical sequences as applied to musical concepts, a skill valued in advanced music studies.

The question probes the understanding of harmonic progression and its relationship to arithmetic progression, a common concept in music theory and composition, particularly relevant for advanced students at Yerevan Komitas State Conservatory. A harmonic progression is a sequence of musical chords or intervals that are perceived as a coherent progression. In a harmonic progression, the reciprocals of the frequencies of the notes form an arithmetic progression. If the frequencies of three notes in a harmonic progression are \(f_1, f_2, f_3\), then \(1/f_1, 1/f_2, 1/f_3\) form an arithmetic progression. This means \(1/f_2 – 1/f_1 = 1/f_3 – 1/f_2\), or \(2/f_2 = 1/f_1 + 1/f_3\). Let the frequencies be \(f_1, f_2, f_3\). We are given that these frequencies form a harmonic progression. This implies that their reciprocals form an arithmetic progression. So, \(1/f_1, 1/f_2, 1/f_3\) are in arithmetic progression. This means \(1/f_2 = (1/f_1 + 1/f_3) / 2\). We are given \(f_1 = 220\) Hz (A3) and \(f_3 = 440\) Hz (A4). We need to find \(f_2\). Substituting the given values into the arithmetic progression formula: \(1/f_2 = (1/220 + 1/440) / 2\) To add the fractions, find a common denominator, which is 440: \(1/f_2 = (2/440 + 1/440) / 2\) \(1/f_2 = (3/440) / 2\) \(1/f_2 = 3 / (440 \times 2)\) \(1/f_2 = 3 / 880\) Now, to find \(f_2\), we take the reciprocal of both sides: \(f_2 = 880 / 3\) \(f_2 \approx 293.33\) Hz. This frequency corresponds to a note that is conceptually between A3 and A4, and its position within the harmonic progression is crucial for understanding tonal relationships and voice leading in composition. The concept of harmonic progression is fundamental to Western music theory, influencing chord construction, melodic movement, and the overall structure of musical pieces. Understanding that harmonic progressions are based on arithmetic progressions of reciprocals allows composers and theorists to predict and analyze musical relationships with greater precision. For students at Yerevan Komitas State Conservatory, grasping this principle is essential for advanced studies in harmony, counterpoint, and composition, enabling them to analyze complex musical textures and create their own sophisticated works. The specific frequencies chosen (A3 and A4) are common reference points in music, making the application of the concept tangible. The calculation demonstrates how a seemingly simple musical relationship is rooted in mathematical principles, a core tenet of musical education at institutions like Yerevan Komitas State Conservatory.

The core of this question lies in understanding the principles of harmonic progression and its application in musical composition, specifically concerning voice leading and the resolution of dissonances. A perfect fifth, when considered in terms of its frequency ratio, is approximately 3:2. In a harmonic progression, the movement between chords is governed by established voice-leading rules and harmonic function. The diminished seventh chord, often built on the leading tone of a key, contains intervals that, when resolved, create specific harmonic movements. A common resolution of a diminished seventh chord involves moving to a tonic chord. For instance, in C major, a B diminished seventh chord (B-D-F-Ab) can resolve to C major (C-E-G). The diminished seventh interval (B to Ab) is enharmonically equivalent to a major sixth. When considering the progression from a dominant seventh chord to a tonic chord, the leading tone (the seventh degree of the dominant chord) resolves upwards to the tonic. The seventh of the dominant chord resolves downwards by step to the third of the tonic chord. The diminished seventh chord, however, presents a more complex scenario due to its inherent instability and multiple possible resolutions. The question asks about the *most* harmonically stable resolution, implying a movement that adheres most closely to traditional tonal principles and minimizes awkward leaps or dissonances in the inner voices. A diminished seventh chord, when functioning as a secondary dominant or a leading-tone chord, typically resolves to a chord a semitone or whole tone above or below. The diminished seventh chord itself contains two tritones, which are inherently dissonant and require careful resolution. The most common and stable resolution of a diminished seventh chord is to a tonic chord where the diminished seventh interval resolves outwards by semitone to form a major third in the tonic chord, and the other notes move by step or common tone. For example, in the key of G major, the F# diminished seventh chord (F#-A-C-Eb) resolves to G major (G-B-D). The F# (leading tone) resolves to G. The A resolves to B. The C resolves to B. The Eb resolves to D. This creates a smooth progression. The question, however, is framed around the *harmonic progression* itself, implying a sequence of chords. The diminished seventh chord is often used to modulate or to create chromatic tension. Its resolution to a chord a semitone higher is a very common and stable practice, particularly when the diminished seventh chord is acting as a leading-tone chord to the dominant of the target key, which then resolves to the tonic. For example, in C major, the E diminished seventh chord (E-G-Bb-Db) can resolve to F major (F-A-C). The E (leading tone to F) resolves to F. The G moves to A. The Bb moves to C. The Db moves to C. This is a stable resolution. The question asks about the *harmonic progression* of a diminished seventh chord resolving to a chord a semitone higher. This implies a movement like vii°7/V to V, or vii°7 to I in a different key. The stability comes from the resolution of the tritones within the diminished seventh chord. The diminished seventh interval (e.g., B-Ab in C major) resolves outwards to a major third (C-E). The other tritone (D-Ab) resolves to a perfect fifth (D-G). This creates a smooth and tonally sound resolution. Therefore, the most harmonically stable progression involving a diminished seventh chord is its resolution to a chord a semitone higher, as this often facilitates smooth voice leading and resolves the inherent dissonances effectively within the framework of tonal harmony.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a concept fundamental to analyzing melodic and harmonic structures in music theory, particularly relevant for advanced students at Yerevan Komitas State Conservatory. A harmonic progression is a sequence where the reciprocals of the terms form an arithmetic progression. If we have a harmonic progression \(h_1, h_2, h_3, \dots\), then the sequence \(\frac{1}{h_1}, \frac{1}{h_2}, \frac{1}{h_3}, \dots\) is an arithmetic progression. Let the harmonic progression be \(h_1, h_2, h_3\). The reciprocals form an arithmetic progression: \(\frac{1}{h_1}, \frac{1}{h_2}, \frac{1}{h_3}\). This means the difference between consecutive terms is constant: \(\frac{1}{h_2} – \frac{1}{h_1} = \frac{1}{h_3} – \frac{1}{h_2}\) We are given the harmonic progression \(1/2, 1/5, 1/8\). Let’s verify if this is indeed a harmonic progression by checking if its reciprocals form an arithmetic progression. The reciprocals are \(2, 5, 8\). The difference between the second and first term is \(5 – 2 = 3\). The difference between the third and second term is \(8 – 5 = 3\). Since the differences are equal, the reciprocals form an arithmetic progression with a common difference of 3. Therefore, \(1/2, 1/5, 1/8\) is a harmonic progression. The question asks for the next term in this harmonic progression. This means we need to find the next term in the arithmetic progression of reciprocals and then take its reciprocal. The arithmetic progression of reciprocals is \(2, 5, 8, \dots\). The common difference is \(d = 3\). The next term in the arithmetic progression is \(a_4 = a_3 + d = 8 + 3 = 11\). The next term in the harmonic progression is the reciprocal of this next term in the arithmetic progression. So, the next harmonic term is \(\frac{1}{11}\). This concept is crucial for understanding voice leading, chord progressions, and the mathematical underpinnings of musical intervals and scales, which are core to the curriculum at Yerevan Komitas State Conservatory. Understanding how different types of sequences relate to musical structures allows for deeper analytical insights and compositional techniques. For instance, analyzing the harmonic series in acoustics or the construction of certain scales might reveal underlying arithmetic or harmonic relationships that influence their perceived consonance or dissonance. The ability to identify and extend such sequences is a foundational skill for any aspiring musician or theorist.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a common area of inquiry in advanced music theory and composition. A harmonic progression is a sequence of numbers where the reciprocals form an arithmetic progression. If \(a\), \(b\), and \(c\) are in harmonic progression, then \(1/a\), \(1/b\), and \(1/c\) are in arithmetic progression. This means that the difference between consecutive terms is constant: \(1/b – 1/a = 1/c – 1/b\). To find the missing term \(b\) when \(a = 1/2\) and \(c = 1/6\), we can rearrange the arithmetic progression property: \(2/b = 1/a + 1/c\). Substituting the given values: \(2/b = 1/(1/2) + 1/(1/6)\) \(2/b = 2 + 6\) \(2/b = 8\) \(b = 2/8\) \(b = 1/4\) Therefore, the harmonic progression is \(1/2, 1/4, 1/6\). The reciprocal sequence is \(2, 4, 6\), which is an arithmetic progression with a common difference of 2. This demonstrates the core concept of harmonic progressions. Understanding this relationship is crucial for analyzing and constructing harmonic sequences in musical composition, particularly in contexts that explore intervallic relationships beyond simple diatonic harmony, such as in certain 20th-century compositional techniques or in the study of acoustical phenomena that inform musical intervals. The ability to identify and manipulate such progressions reflects a sophisticated grasp of mathematical underpinnings in music theory, a hallmark of advanced study at institutions like the Yerevan Komitas State Conservatory.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a fundamental concept in music theory and composition, particularly relevant to the analytical skills fostered at Yerevan Komitas State Conservatory. A harmonic progression is a sequence of numbers where the reciprocals form an arithmetic progression. If \(a, b, c\) are in harmonic progression, then \(1/a, 1/b, 1/c\) are in arithmetic progression. This means \(1/b – 1/a = 1/c – 1/b\), which simplifies to \(2/b = 1/a + 1/c\). Let the three consecutive terms of the harmonic progression be \(h_1, h_2, h_3\). Their reciprocals form an arithmetic progression: \(1/h_1, 1/h_2, 1/h_3\). The common difference of this arithmetic progression is \(d = 1/h_2 – 1/h_1\). Therefore, \(1/h_3 = 1/h_2 + d = 1/h_2 + (1/h_2 – 1/h_1) = 2/h_2 – 1/h_1\). Rearranging this, we get \(1/h_1 + 1/h_3 = 2/h_2\). This equation defines the harmonic mean relationship. Now, consider the scenario presented: a composer is exploring harmonic relationships for a new piece at Yerevan Komitas State Conservatory. They have identified three pitches whose frequencies, when considered as a sequence, exhibit a harmonic progression. Let these frequencies be \(f_1, f_2, f_3\). According to the definition of a harmonic progression, their reciprocals are in an arithmetic progression. The question asks about the relationship between the middle term and the outer terms in a harmonic progression. The core principle is that the reciprocal of the middle term is the arithmetic mean of the reciprocals of the outer terms. This is equivalent to stating that the middle term is the harmonic mean of the outer terms. The harmonic mean \(H\) of two numbers \(a\) and \(b\) is given by \(H = \frac{2ab}{a+b}\). In our case, \(h_2\) is the harmonic mean of \(h_1\) and \(h_3\). So, \(h_2 = \frac{2 h_1 h_3}{h_1 + h_3}\). This relationship is crucial in understanding how intervals and chord voicings can create specific sonic textures and resolutions, a key area of study in advanced music theory and composition at Yerevan Komitas State Conservatory. The ability to recognize and manipulate these underlying mathematical relationships in musical parameters like frequency is a hallmark of sophisticated musical understanding. The question tests this foundational knowledge by asking for the direct expression of this relationship. The correct answer is that the reciprocal of the middle term is the arithmetic mean of the reciprocals of the outer terms. This is the defining characteristic of a harmonic progression and directly leads to the harmonic mean formula. Understanding this concept allows students to analyze and create music with greater depth and intentionality, aligning with the Conservatory’s commitment to rigorous musical scholarship.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a fundamental concept in music theory and composition often explored in advanced music studies. A harmonic progression is a sequence of musical chords. A sequence of numbers \(a_1, a_2, a_3, \dots\) is in harmonic progression if their reciprocals \(1/a_1, 1/a_2, 1/a_3, \dots\) are in arithmetic progression. Let the sequence of frequencies be \(f_1, f_2, f_3, f_4\). If these frequencies form a harmonic progression, then their reciprocals \(1/f_1, 1/f_2, 1/f_3, 1/f_4\) form an arithmetic progression. We are given that the first term \(f_1 = 440\) Hz and the fourth term \(f_4 = 110\) Hz. Let the common difference of the arithmetic progression of reciprocals be \(d\). The terms of the arithmetic progression are \(1/f_1, 1/f_1 + d, 1/f_1 + 2d, 1/f_1 + 3d\). So, \(1/f_4 = 1/f_1 + 3d\). Substituting the given values: \(1/110 = 1/440 + 3d\) To find \(d\): \(3d = 1/110 – 1/440\) \(3d = (4 – 1) / 440\) \(3d = 3 / 440\) \(d = 1 / 440\) The second term of the harmonic progression is \(f_2\), which corresponds to the second term of the arithmetic progression of reciprocals, \(1/f_2 = 1/f_1 + d\). \(1/f_2 = 1/440 + 1/440\) \(1/f_2 = 2/440\) \(1/f_2 = 1/220\) Therefore, \(f_2 = 220\) Hz. The third term of the harmonic progression is \(f_3\), which corresponds to the third term of the arithmetic progression of reciprocals, \(1/f_3 = 1/f_1 + 2d\). \(1/f_3 = 1/440 + 2(1/440)\) \(1/f_3 = 1/440 + 2/440\) \(1/f_3 = 3/440\) Therefore, \(f_3 = 440/3\) Hz. The question asks for the frequency of the second harmonic in this sequence, which is \(f_2\). The concept of harmonic progressions is crucial in understanding the relationships between frequencies in musical intervals and the construction of scales and chords. In Western music theory, the overtone series, which is a natural phenomenon, exhibits harmonic relationships. Understanding how these relationships can be manipulated or represented mathematically, as in harmonic progressions, is vital for composers and theorists. At Yerevan Komitas State Conservatory, a deep appreciation for the physics and mathematics underlying musical phenomena is fostered, enabling students to engage with both historical compositional techniques and contemporary sonic exploration. This question tests not just the ability to perform a calculation but also the conceptual grasp of how musical relationships are encoded in mathematical sequences, a core tenet of musicological and compositional studies at the conservatory. The ability to identify and manipulate such progressions is fundamental to analyzing complex musical textures and creating new ones.

The question probes the understanding of harmonic progressions and their relationship to arithmetic progressions, a concept fundamental to advanced music theory and composition, particularly relevant for students at Yerevan Komitas State Conservatory. A harmonic progression is a sequence of numbers where the reciprocals form an arithmetic progression. If \(h_1, h_2, h_3, \dots\) is a harmonic progression, then \(1/h_1, 1/h_2, 1/h_3, \dots\) is an arithmetic progression. Let the harmonic progression be \(H_1, H_2, H_3, H_4\). The reciprocals form an arithmetic progression: \(A_1 = 1/H_1, A_2 = 1/H_2, A_3 = 1/H_3, A_4 = 1/H_4\). The common difference of the arithmetic progression is \(d = A_2 – A_1 = A_3 – A_2 = A_4 – A_3\). We are given the first term of the harmonic progression \(H_1 = 1/2\) and the third term \(H_3 = 1/6\). Therefore, the first term of the arithmetic progression is \(A_1 = 1/(1/2) = 2\). The third term of the arithmetic progression is \(A_3 = 1/(1/6) = 6\). In an arithmetic progression, the \(n\)-th term is given by \(A_n = A_1 + (n-1)d\). For the third term: \(A_3 = A_1 + (3-1)d\). Substituting the known values: \(6 = 2 + 2d\). Solving for \(d\): \(6 – 2 = 2d\) \(4 = 2d\) \(d = 4/2 = 2\). Now we need to find the second term of the harmonic progression, \(H_2\). This corresponds to the second term of the arithmetic progression, \(A_2\). Using the formula for the \(n\)-th term of an arithmetic progression: \(A_2 = A_1 + (2-1)d\) \(A_2 = A_1 + d\) Substituting the values of \(A_1\) and \(d\): \(A_2 = 2 + 2 = 4\). Since \(A_2 = 1/H_2\), we can find \(H_2\): \(H_2 = 1/A_2 = 1/4\). The question asks for the second term of the harmonic progression. The calculation shows this to be \(1/4\). This type of problem requires understanding the inverse relationship between harmonic and arithmetic progressions, and applying the formulas for arithmetic progressions. Mastery of these concepts is crucial for analyzing harmonic structures and developing compositional techniques, aligning with the rigorous curriculum at Yerevan Komitas State Conservatory. Understanding such relationships allows students to delve deeper into the mathematical underpinnings of musical harmony and to explore advanced theoretical frameworks.

The question probes the understanding of harmonic progression and its relationship to arithmetic progression, a common concept in music theory and composition, particularly relevant to foundational studies at Yerevan Komitas State Conservatory. A harmonic progression is a sequence of musical chords or intervals that are perceived as a logical progression. The question asks to identify the characteristic that is *not* shared between a harmonic progression and an arithmetic progression. Let’s consider the properties of each: Arithmetic Progression (AP): A sequence of numbers such that the difference between consecutive terms is constant. For example, 2, 4, 6, 8… where the common difference is 2. Harmonic Progression (HP): A sequence of numbers such that their reciprocals form an arithmetic progression. For example, 1/2, 1/4, 1/6, 1/8… is a harmonic progression because their reciprocals (2, 4, 6, 8…) form an arithmetic progression. Now, let’s analyze the options in the context of musical harmony and mathematical sequences: 1. **Constant Difference/Ratio:** An AP has a constant *difference* between terms. An HP, by definition, does not have a constant difference between its terms; rather, the difference between the reciprocals of its terms is constant. A geometric progression (GP) has a constant *ratio* between terms. While musical intervals can be related by ratios, the fundamental definition of an HP is tied to the reciprocal relationship with an AP, not a direct constant ratio within the HP itself. Therefore, a constant difference or ratio between consecutive terms is not a shared characteristic of HP and AP. 2. **Ordered Sequence:** Both AP and HP are sequences, meaning they consist of terms arranged in a specific order. This is a fundamental property of both. 3. **Reciprocal Relationship:** An HP is *defined* by its reciprocal relationship with an AP. This is a core concept. 4. **Predictable Pattern:** Both AP and HP exhibit predictable patterns. In an AP, the next term is found by adding the common difference. In an HP, the next term is found by taking the reciprocal of the term that follows the corresponding AP term. This predictability is a shared characteristic. The question asks what is *not* shared. The most distinct difference in their fundamental structure is the nature of the progression itself. While both are ordered and predictable, the *mechanism* of progression differs. An AP progresses by addition, while an HP progresses by a reciprocal relationship to an additive progression. The idea of a “constant ratio” is characteristic of a GP, not an AP or HP directly in their primary definitions. An HP does not have a constant ratio between its terms; its defining feature is the constant difference between the reciprocals of its terms. An AP has a constant difference, not a constant ratio. Therefore, the characteristic of having a constant ratio between consecutive terms is not shared by both AP and HP. The correct answer is the characteristic that is *not* common to both. An AP has a constant difference. An HP does not have a constant difference, nor does it have a constant ratio between its terms. Its defining feature is the constant difference of the reciprocals of its terms. Therefore, the concept of a constant ratio between consecutive terms is not a shared characteristic.