| THE PIANO STUDIO, Hamilton, ON.
piano lessons to APPRECIATE... |
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Piano Lessons in Hamilton, Ontario:
OUR RESOURCES
The Samick JS115 makes piano playing a real treat! The Samick's tone is warm and mellow. The Yamaha DGX 500 keyboard enables orchestral accompaniments and modern band arrangements to come to life, adding a fresh dimension to piano instruction. It is also used extensively in our Lab Work* [see below].
The Piano Studio, Hamilton, Ontario sets a higher standard for piano studios in Hamilton, and everywhere! Every music lesson becomes an adventure!
Students will enjoy the Piano Studio's recent addition, a Zoom H2 MP3 recorder. Student performances can be digitally recorded, enjoyed on the Mirage speakers of our outstanding sound system, and even sent to the student by email!
We begin the new season with the addition of a webcam. Students of The Piano Studio will be able to SEE and HEAR their own performances, with the video being reviewed at the lesson by teacher and pupil!
Add an exhaustive library of printed music of all genres, educational texts on every musical subject, and a large and varied collection of recorded piano music spanning more than a century of outstanding pianists, then top it off with a state-of-the-art sound system and recording studio...The Piano Studio, Hamilton becomes a learning and lesson centre of virtually unlimited potential.
The Piano Studio, Hamilton receives many questions, regarding piano lessons, piano teachers, pianos, etc., and offer dependable, free advice. While all questions are welcome, here are a few of the most frequently asked questions:
1. At what age should my child start piano lessons? Am I too old to start?
2. Should we start our child in Group Lessons, or is Private Instruction better?
3. Are on-line piano lessons a smart alternative?
4. Do we need a real piano, or will a "keyboard" be adequate?
5. What kind of piano should I buy?
6. How often should my piano be tuned?
7. How much practice time is expected?
8. Are Conservatory exams, scales, exercises and studies necessary?
9. Is "playing by ear" a help or a hindrance to piano progress?
10. Should one study piano to learn to play for others, or for one's own fulfilment?
11. How can I tell if my child has a gift for music?
12. I am an advanced pianist and/or piano teacher, but I feel "burnt out". How can I renew my own enthusiasm?
At the Piano Studio, Hamilton, we believe we can guide you on an exciting and rewarding musical journey.
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SOME RESULTS OF OUR LAB WORK...
Copyright 2002-2009, Fredrick Pritchard
INTRODUCTION: Some Basic Principals of Acoustics
Copyright 2002-2009, Fredrick Pritchard
INTRODUCTION: Some Basic Principals of Acoustics
It is known that when a given musical tone is sounded, it generates a series of higher tones, known as Harmonics. These may seem inaudible, being overpowered by the strength of the sounded tone (the Fundamental). The frequency of Harmonics is determined by multiplication of the Fundamental frequency (Harmonic 1) by any integer (whole number). For example, if the Fundamental has a frequency of 100 Hz (100 cycles per second), Harmonic 2 will have a frequency of 200 Hz, which is exactly one octave higher than the Fundamental. Harmonic 3 will have a frequency of 300 Hz, exactly a Perfect 5th above Harmonic 2. Harmonic 4 at 400 Hz, is exactly a Perfect 4th above Harmonic 3. Harmonic 5, with a frequency of 500 Hz, is exactly a Major 3rd above Harmonic 4. Harmonic 6, with a frequency of 600 Hz, is exactly a Minor 3rd above Harmonic 5.
Therefore, if "C" is a given Fundamental, it's Harmonic Series will be as follows (ascending):
The present system of tuning used in Western Music is Equal Temperament. In this system, the octave is divided into twelve equal portions, in which each semitone is defined as the "12th root of 2". All intervals except the octave are compromised; only the octave is integer-based. Therefore Equal Temperament was regarded by many musicians as "unnatural", "contrived", or "a necessary evil". However, Equal Temperament allows the use of all 12 major and 12 minor keys. Here are a few examples of the compromise:
Perfect Fifth: In Just Intonation, 1.5 (3:2); in Equal Temperament, 1.4983...
Major Third: In Just Intonation, 1.25 (5:4); in Equal Temperament, 1.2599...
Major Sixth: In Just Intonation, 1.666 (5:3); in Equal Temperament, 1.6818....
The Human Ear, in childhood, can hear frequencies over a range of about 10 octaves; this range diminishes with age. The human ear can detect pitch differences as small as about 0.05 semitones in mid-range frequencies, but is far less discerning in the extreme high and low ranges. Keep this in mind, as you assess our Lab Work!
2. Simple Harmonic Ratios are found within our Solar System. The rotational and orbital periods of the planet Mercury form a 3:2 ratio. Transposed through many octaves, they form the Perfect 5th, F# to C#. Neptune and Pluto also enjoy a 3:2 ratio in their orbital periods...in fact, once again transposed through many octaves, the Perfect 5th, F# to C#. Three of the principal satellites of Saturn enjoy a 4:5:6 ratio...a Major Triad in root position. (In other words, if we had the means to do so, we could play the first line of "O Canada" on the moons of Saturn!)
We have studied all major bodies of our Solar System and, by transposing through many octaves, have arrived at frequencies for all the planets and their principal satellites. (A recording of our results is available upon request.)
3. Although we have just begun to explore atomic resonance frequency ratios, it is already apparent that harmonic relationships are as prevalent in the microcosmos as in the macrocosmos.
Conclusions...
It should be noted that the purpose of our Lab Work is NOT to reduce Music to a Science. Instead, we seek to recognize Music as an expression of the surrounding Universe, and the Universe itself an expression of Music.
Therefore, if "C" is a given Fundamental, it's Harmonic Series will be as follows (ascending):
C..............C......G....C..E..G etc.
The Harmonic Series continues virtually endlessly, the number of Harmonics doubling in each octave. It was once believed that all "pure" intervals must be based upon relationships between harmonics, and therefore result only from frequency ratios of integer-based fractions, such as "6/5". This method of defining intervals is known as "Just Intonation". However, any tuning system based upon Just Intonation severely restricts the composer, allowing the use of an extremely limited number of keys.The present system of tuning used in Western Music is Equal Temperament. In this system, the octave is divided into twelve equal portions, in which each semitone is defined as the "12th root of 2". All intervals except the octave are compromised; only the octave is integer-based. Therefore Equal Temperament was regarded by many musicians as "unnatural", "contrived", or "a necessary evil". However, Equal Temperament allows the use of all 12 major and 12 minor keys. Here are a few examples of the compromise:
Perfect Fifth: In Just Intonation, 1.5 (3:2); in Equal Temperament, 1.4983...
Major Third: In Just Intonation, 1.25 (5:4); in Equal Temperament, 1.2599...
Major Sixth: In Just Intonation, 1.666 (5:3); in Equal Temperament, 1.6818....
The Human Ear, in childhood, can hear frequencies over a range of about 10 octaves; this range diminishes with age. The human ear can detect pitch differences as small as about 0.05 semitones in mid-range frequencies, but is far less discerning in the extreme high and low ranges. Keep this in mind, as you assess our Lab Work!
OUR OBSERVATIONS:
PART I: Music and Math
PART I: Music and Math
The simplest Pythagorean right-angle triangle has sides with the proportions 3:4:5. As Harmonics, the 3:4:5 ratios produce the Second Inversion of a Major Triad. This is of particular interest, since Pythagoras (as far as we know) conducted the first exploration of Harmonics with his monochord!
If an isosceles right-angled triangle is constructed such that the equal sides forming the right-angle are each "1", the hypotenuse is "sqrt 2". If "1" is regarded as a Fundamental Frequency, the hypotenuse (sqrt 2) is exactly 6 semitones, or one-half octave lower. (This can be demonstrated by placing 3 nails in a piece of wood, and wrapping them in one string, all segments being of equal tension. Pluck each of the 3 string segments.)
An Experiment In The Perception of Tuning
If the human ear is sensitive to pitch differences as small as 1/20 of a semitone, we must ask how any musician can accept Equal Temperament, with its Major 3rds being stretched by about 1/7 of a semitone, and its Major 6ths by about 1/6 of a semitone. This led to a simple experiment.
On one of our studio pianos a C major triad was tuned to Just Intonation, and on another, to Equal Temperament. More than 50 musicians were asked to compare, and express their preference. Unanimously, they agreed that the Just Intonation tuning sounded "dull and dead", while the Equal Temperament tuning sounded "vibrant and alive". Perhaps more than "accustomization" is involved; one would expect the "natural" to be preferred to the "contrived".
This initiated a search for a possible "natural basis" to Equal Temperament.
The ratio of Harmonics 139:138 almost perfectly coincides with 1/8 of a semitone of Equal Temperament. (The 1/8 semitone was explored by the Mexican composer, Julian Carrillo, in the late 19th century.) In fact, "(139/138) to the 8th power" (1.059463...) almost perfectly coincides with one Equal Tempered semitone (1.059463...). The discrepancy is 0.00000487 semitones per semitone, or 0.00005843 semitones per octave. The accumulated discrepancy over the complete range of a piano is 0.00042362 semitones, and over the entire range of human hearing (about 10 octaves), 0.00058431 semitones. Remember, the human ear can detect a pitch discrepancy of about 0.05 semitones. Therefore, for all practical purposes, the miniscule discrepancy may be disregarded, and the two systems considered one and the same. We conclude: Equal Temperament, although more complex, is as natural as Just Intonation, since, for all practical purposes, it can be constructed from an integer-based fraction, and therefore from natural harmonic ratios.
The Golden Ratio (Golden Mean) and Music
The "Golden Ratio", or "Phi" is derived from the Fibonacci Number Series, and is usually expressed as "[(sqrt 5) plus 1] / 2". [The first numbers of the Fibonacci Series (after 0) are 1, 2, 3, 5, and 8. When taken as frequency ratios, they form a beautifully voiced major chord, found, for example, as the final chord of Chopin's "Harp" Etude, Op. 25, No. 1.]
Taken as a frequency ratio, Phi is almost exactly 8.33 semitones above a given musical frequency. Because every third number in the Fibonacci Series is an even number, the relationship between two consecutive even numbers in the series approaches "Phi cubed" (with rapidly increasing accuracy, as the Number Series is ascended.) When "Phi-cubed" is applied to any given musical frequency, the resultant tone is almost exactly 25 semitones higher, for example, C to C#)
Two Methods of Tuning by the Golden Ratio
Method 1:
The result of "Phi cubed" (C#) is transposed down two octaves, and again "Phi cubed" is applied, resulting in a D. By continuing to apply the same principle, we can arrive at a 12-tone chromatic scale. While the deviation from normal E.T. semitones is just 0.00729111 semitones per semitone, this accumulates to 0.08749331 semitones per octave, or 0.63443265 semitones per "piano-range", or 0.87493312 semitones per "range of human hearing". Thus, such a tuning system would be practical only for musical instruments of extremely limited range.
Method 2:
In this method, the semitone is defined by the Golden Ratio. "Phi cubed", instead of being regarded as almost exactly 25 semitones of conventional Equal Temperament, is seen as exactly 25 "Golden Semitones". ONE Golden Semitone is therefore defined as the "25th root of Phi-cubed". The discrepancy between one Golden semitone and one conventional semitone of Equal Temperament is 0.00029164 semitones per semitone, accumulating to 0.00349973 semitones per octave; 0.02537306 semitones per "piano range"; 0.03499733 semitones per "range of human hearing". The difference between this method of Golden Ratio Tuning and conventional Equal Temperament rests well beyond detection by the human ear. For all practical purposes, the miniscule discrepancy may therefore be disregarded, and the two systems considered one and the same.
Next, consider the cochlea of the inner ear, which distinguishes various pitches. Looking rather like a snail, it is a logarithmic spiral, proportioned according to the Golden Ratio.
In consideration of the relationship of the Golden Ratio to both the semitone and to the cholea, the Golden Ratio may possibly explain our acceptance of intervals based upon the semitones of Equal Temperament. (Further investigation of the Golden Ratio is strongly recommended.)
If division of the octave into 12 equal parts is natural, perhaps other equal divisions are equally natural. We have experimented with many possibilities, but find a division of 5 equal parts especially appealing. This division is the basis of the Slendro Scale, used in the Gamelan music of Java and Bali. Although tunings of the Slendro scale vary slighly from region to region, each step can essentially be regarded as the "5th root of 2". Although the number 5 is also critical in the definition of Phi, we have not established a direct relationship between Phi and the Slendro scale.
1. The "Primary Colours of Light" possess a frequency ratio of 4:5:6, the same as the the Root Position of a Major Triad.If an isosceles right-angled triangle is constructed such that the equal sides forming the right-angle are each "1", the hypotenuse is "sqrt 2". If "1" is regarded as a Fundamental Frequency, the hypotenuse (sqrt 2) is exactly 6 semitones, or one-half octave lower. (This can be demonstrated by placing 3 nails in a piece of wood, and wrapping them in one string, all segments being of equal tension. Pluck each of the 3 string segments.)
An Experiment In The Perception of Tuning
If the human ear is sensitive to pitch differences as small as 1/20 of a semitone, we must ask how any musician can accept Equal Temperament, with its Major 3rds being stretched by about 1/7 of a semitone, and its Major 6ths by about 1/6 of a semitone. This led to a simple experiment.
On one of our studio pianos a C major triad was tuned to Just Intonation, and on another, to Equal Temperament. More than 50 musicians were asked to compare, and express their preference. Unanimously, they agreed that the Just Intonation tuning sounded "dull and dead", while the Equal Temperament tuning sounded "vibrant and alive". Perhaps more than "accustomization" is involved; one would expect the "natural" to be preferred to the "contrived".
This initiated a search for a possible "natural basis" to Equal Temperament.
The ratio of Harmonics 139:138 almost perfectly coincides with 1/8 of a semitone of Equal Temperament. (The 1/8 semitone was explored by the Mexican composer, Julian Carrillo, in the late 19th century.) In fact, "(139/138) to the 8th power" (1.059463...) almost perfectly coincides with one Equal Tempered semitone (1.059463...). The discrepancy is 0.00000487 semitones per semitone, or 0.00005843 semitones per octave. The accumulated discrepancy over the complete range of a piano is 0.00042362 semitones, and over the entire range of human hearing (about 10 octaves), 0.00058431 semitones. Remember, the human ear can detect a pitch discrepancy of about 0.05 semitones. Therefore, for all practical purposes, the miniscule discrepancy may be disregarded, and the two systems considered one and the same. We conclude: Equal Temperament, although more complex, is as natural as Just Intonation, since, for all practical purposes, it can be constructed from an integer-based fraction, and therefore from natural harmonic ratios.
The Golden Ratio (Golden Mean) and Music
The "Golden Ratio", or "Phi" is derived from the Fibonacci Number Series, and is usually expressed as "[(sqrt 5) plus 1] / 2". [The first numbers of the Fibonacci Series (after 0) are 1, 2, 3, 5, and 8. When taken as frequency ratios, they form a beautifully voiced major chord, found, for example, as the final chord of Chopin's "Harp" Etude, Op. 25, No. 1.]
Taken as a frequency ratio, Phi is almost exactly 8.33 semitones above a given musical frequency. Because every third number in the Fibonacci Series is an even number, the relationship between two consecutive even numbers in the series approaches "Phi cubed" (with rapidly increasing accuracy, as the Number Series is ascended.) When "Phi-cubed" is applied to any given musical frequency, the resultant tone is almost exactly 25 semitones higher, for example, C to C#)
Two Methods of Tuning by the Golden Ratio
Method 1:
The result of "Phi cubed" (C#) is transposed down two octaves, and again "Phi cubed" is applied, resulting in a D. By continuing to apply the same principle, we can arrive at a 12-tone chromatic scale. While the deviation from normal E.T. semitones is just 0.00729111 semitones per semitone, this accumulates to 0.08749331 semitones per octave, or 0.63443265 semitones per "piano-range", or 0.87493312 semitones per "range of human hearing". Thus, such a tuning system would be practical only for musical instruments of extremely limited range.
Method 2:
In this method, the semitone is defined by the Golden Ratio. "Phi cubed", instead of being regarded as almost exactly 25 semitones of conventional Equal Temperament, is seen as exactly 25 "Golden Semitones". ONE Golden Semitone is therefore defined as the "25th root of Phi-cubed". The discrepancy between one Golden semitone and one conventional semitone of Equal Temperament is 0.00029164 semitones per semitone, accumulating to 0.00349973 semitones per octave; 0.02537306 semitones per "piano range"; 0.03499733 semitones per "range of human hearing". The difference between this method of Golden Ratio Tuning and conventional Equal Temperament rests well beyond detection by the human ear. For all practical purposes, the miniscule discrepancy may therefore be disregarded, and the two systems considered one and the same.
Next, consider the cochlea of the inner ear, which distinguishes various pitches. Looking rather like a snail, it is a logarithmic spiral, proportioned according to the Golden Ratio.
In consideration of the relationship of the Golden Ratio to both the semitone and to the cholea, the Golden Ratio may possibly explain our acceptance of intervals based upon the semitones of Equal Temperament. (Further investigation of the Golden Ratio is strongly recommended.)
If division of the octave into 12 equal parts is natural, perhaps other equal divisions are equally natural. We have experimented with many possibilities, but find a division of 5 equal parts especially appealing. This division is the basis of the Slendro Scale, used in the Gamelan music of Java and Bali. Although tunings of the Slendro scale vary slighly from region to region, each step can essentially be regarded as the "5th root of 2". Although the number 5 is also critical in the definition of Phi, we have not established a direct relationship between Phi and the Slendro scale.
PART 2: Music and the World Around Us
2. Simple Harmonic Ratios are found within our Solar System. The rotational and orbital periods of the planet Mercury form a 3:2 ratio. Transposed through many octaves, they form the Perfect 5th, F# to C#. Neptune and Pluto also enjoy a 3:2 ratio in their orbital periods...in fact, once again transposed through many octaves, the Perfect 5th, F# to C#. Three of the principal satellites of Saturn enjoy a 4:5:6 ratio...a Major Triad in root position. (In other words, if we had the means to do so, we could play the first line of "O Canada" on the moons of Saturn!)
We have studied all major bodies of our Solar System and, by transposing through many octaves, have arrived at frequencies for all the planets and their principal satellites. (A recording of our results is available upon request.)
3. Although we have just begun to explore atomic resonance frequency ratios, it is already apparent that harmonic relationships are as prevalent in the microcosmos as in the macrocosmos.
Conclusions...
It should be noted that the purpose of our Lab Work is NOT to reduce Music to a Science. Instead, we seek to recognize Music as an expression of the surrounding Universe, and the Universe itself an expression of Music.
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