Lecture 10Musical Notes and ScalesScales and TimbrePythagorean ScaleEqual Temperament ScaleUnorthodox Scales
Instructor: David Kirkby ([email protected])
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MidtermThe average score on the midterm was 64%. The average on the multiple choice section (73%) was higher than on the written sections (59%).
This average corresponds to C+/B-, which is most likely where the final course average will end up and is normal for an intro physics course.
I will be checking that the grades for the two versions are consistent, and make adjustments if necessary when calculating your final grade.
Remember that the midterm contributes 25% to your final grade for the course (homework is 40%, the final is 35%).
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Students Hector Aleman and Claire Dreyer should see me after class today.
Here are somedistributions fromthe midterm grades:
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Drop DeadlineThe deadline to drop this course is Friday.
For Drop Card signatures, see the Physics Undergrad Affairs Coordinator:
Kirsten [email protected] MSTB
Homework and midterm scores are posted on the web for your reference:
http://www.physics.uci.edu/undgrad/coursescores.html
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Review of Lecture 9In the last lecture, we covered:
•The perception of combination tones (difference tones)
•Different modes of hearing (analytic/synthetic, harmonic/inharmonic)
•The physical basis for dissonance•Theories of pitch perception (the relative importance of wavelength and frequency cues)
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Why does a pianohave 7 white notes and 5 black notes
per octave?
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Musical ScalesThere is an infinite continuum of possible frequencies to use in music.
But, in practice, most music uses only a small (finite) number of specific frequencies.
We call each of these special frequencies a musical note, and call a set of notes a musical scale.
Different cultures have adopted different scales. The choice of scale is primarily aesthetic, but some aesthetic judgments are heavily influenced by physical considerations (e.g., dissonance).
What can physics tell us about musical scales?
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Harmonic TimbresMost musical sounds have overtones that are approximately harmonic (ie, equally spaced on a linear frequency axis).
This is most likely due to a combination of two related factors:
•The resonant frequencies of many naturally occurring resonant systems are approximately harmonic.
•Your brain is optimized for listening to timbres that are approximately harmonic.
Note that there are examples of naturally occurring inharmonic sounds (eg, a hand clap) but we do not perceive these as being musical.
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Octaves RuleTwo notes played together on instruments with harmonic timbres sound most consonant (least dissonant) when their fundamental frequencies are an exact number of octaves apart:
In this sense, an octave is a special interval that we can expect will play a special role in any “natural” scale (although it is certainly possible to invent un-natural scales).
Try this demonstration to see if you can pick out octaves.
frequency
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The correct answer to the octave test was #4, although most people prefer a slightly bigger octave with a frequency ratio of about 2.02:1 that corresponds to #6.
This preference for slightly stretched octaves may be due to our familiarity with listening to pianos which are usually deliberately tuned to have stretched octaves (more about this in Lecture 14).
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Subdividing the OctaveIn practice, this means that if a particular frequency is included in a scale, then all other frequencies that are an exact number of octaves above or below are also included.
Therefore, choosing the set of notes to use in a scale boils down to the problem of how to subdivide an octave.
Is the choice of how to subdivide an octave purely aesthetic, or are there physical considerations that prefer certain musical intervals?
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Scales and TimbreThe choice of a scale (subdivisions of an octave) is intimately related to the timbre of the instrument that will be playing the scale.
The scale and timbre are related by dissonance: the notes of a scale should not sound unpleasant when played together.
For example, most people listening to an “instrument” with no overtones (ie, a pure SHM sine wave) will have no preference for how to subdivide an octave (and the octave is no longer a special interval).
Try these demonstrations to learn more about the connection between scales and timbre.
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However, most people listening to an instrument with harmonic timbre (ie, most “musical” instruments) will have a definite preference for certain intervals where overtones coincide exactly.
Different instruments with harmonic timbres have different strengths for the various harmonics. These differences affect how consonant the preferred intervals are but do not change their frequencies.
Therefore, there is a universal set of preferred subdivisions of the octave for instruments with harmonic timbres (based on a physical model of dissonance).
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How Finely to Chop the Octave?Minimizing the dissonance of notes played together on instruments with harmonic timbres gives us some guidance on how to create a scale with a given number of notes, but not on how many notes to use.
Some of the conventional choices are:•Pentatonic: octave is divided into 5 notes (eg, Ancient Greek, Chinese, Celtic, Native American music)
•Diatonic, Modal: octave is divided into 7 notes (eg, Indian, traditional Western music)
•Chromatic: octave is divided into 12 notes (modern Western music)
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A Primer on Musical NotationThe white notes on the piano are named A,B,C,D,E,F,G.
After G, we start again at A. This reflects the special role of the octave: we give two frequencies an octave apart the same note name.
C D E F G A B C D E F G A B C
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Going up in frequency (towards the right on the keyboard) from a white note to its adjacent black note gives a sharp: C goes to C#, D goes to D#, etc.
Similarly, going down in frequency gives a flat: D goes to Db, E goes to Eb, etc.
C D E F G A B C D E F G A B C
F# G# A# C# D# F# G# A#C# D#
Gb Ab Bb Db Eb Gb Ab BbDb Eb
C# and Db are necessarily the same note on the piano, butthis is not generally true for all possible scales!
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Pentatonic ScalesThe usual choice of 5 notes in a pentatonic scale corresponds to the black notes on the piano:
This scale includes the dissonant whole tone (9:8) interval, but leaves out the less dissonant major (5:4) and minor (6:5) third intervals. Why?
Presumably because music limited to just 5 notes would be boring without some dissonance to create tension.
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Other choices of 5 notes are also possible.
Examples:•Indian music•Chinese music•Celtic music: Auld Lange Syne, My Bonnie Lies Over the Ocean
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Diatonic ScalesThe major and minor scales of Western music are diatonic scales, in which the octave is divided into 7 steps.
The notes of the major scale correspond to the white notes on a piano, starting on C. The (natural) minor scale corresponds to the white notes starting on A.
Diatonic scales can also start on any other white note of the piano. The results are the modes with names like Dorian, Phrygian, Lydian, …
CA
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Most Western music since the 17th century is based on major and minor scales.
Earlier music was primarily modal.
Example: Gregorian chants
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Chromatic ScalesAlthough most Western music is based on diatonic scales, it frequently uses scales starting on several different notes in the same piece of music (as a device for adding interest and overall shape).
A major scale starting on C uses only white notes on the piano, but a major scale starting on B uses all five black notes.
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The main reason for adopting a chromatic scale is to be able to play pieces based on different scales with the same instrument.
An octave divided into twelve notes includes all possible seven-note diatonic scales.
Not all instruments adopt this strategy. For example, harmonicas are each tuned to specific diatonic scales. To play in a different key, you need a different instrument (or else to master “bending” techniques).
What exactly should be the frequencies of the 12 notes that make up a chromatic scale?
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Is there an obvious way to subdivide an octave into twelve notes?
Yes: the notes should be equally spaced and include all of the special consonant frequency ratios (3:2, 4:3, …).
To what extent is this actually possible?
After the octave, the fifth (3:2) is the most consonant interval for harmonic timbres. The fourth (4:3) is really just a combination of the octave and fifth:
4/3 = (3/2) x (1/2)
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The Circle of FifthsWe can reach all 12 notes of the chromatic scale by walking up or down the piano in steps of a fifth (3:2):
Going up, we reach all white notes of the piano except F, and then go through the sharps.
Going down, we hit F first and then go through the flats.
Either way, we eventuallyget back to a C (7 octaves away) if we start on a C.
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Using the circle of fifths, we can calculate the frequency of any note we reach going up as:
f = f0 x (3/2) x (3/2) x … x (3/2) / 2 / 2 / … / 2
startingnote
Steps up in fifths Steps downin octaves
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A similar method works for each step down by a fifth:
f = f0 / (3/2) / (3/2) / … / (3/2) x 2 x 2 x … x 2
Steps down in fifths Steps upin octaves
What happens when we get back to our original note?
For example, after going 12 fifths up, we get back to a Cthat is 7 octaves up which corresponds to a note:
f = f0 x (3/2)12 / (2)7 = f0 (531441/524288) = 1.014 f0
We end up close but not exactly where we started!
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Pythagorean ScaleIf we ignore this problem of not getting back to where we started, we end up with the set of notes corresponding to the Pythagorean scale.
The Pythagorean scale has the feature that all octave and fifth intervals are exact (and therefore so are fourths).
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But the Pythagorean scale also has some shortcomings:
•The frequencies we calculate for the black notes depend on whether we are taking steps up or down, so C# and Db are different notes!
•The semitones from E to F and B to C are bigger than the semitones from C to C# and Db to D.
•Frequency ratios for intervals other than 8ve, 4th, 5th depend on which note you start from, and can be far from the ideal ratios.
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Alternative ScalesSince the major (5:4) and minor (6:5) 3rd intervals are important for diatonic music, several alternative scales have been proposed that have these intervals better in tune (ie, closer to their ideal frequency ratios) without sacrificing the octave, fourth, and fifth too much.
Some alternatives that I will not discuss are the meantone and just intonation scales (see Sections 9.3-9.4 in the text for details).
These scales both improve the tuning of intervals but sound differently depending on the choice of starting note for a diatonic scale (Beethoven described Db-major as “majestic” and C-major as “triumphant”). They also give different frequencies for C# and Db, etc.
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Equal Temperament ScaleThe scale that is most widely used today is the equal temperament scale.
This scale is the ultimate compromise for an instrument that is tuned infrequently and for which the performer cannot adjust the pitch during performance.
The equal temperament scale gives up on trying to make any intervals (other than the octave) exactly right, but instead makes the 12 notes equally spaced on a logarithmic scale.
Listen to the difference between equally-spaced notes on linear and logarithmic scales:
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Mathematically, each semitone corresponds to a frequency ratio of 21/12 = 1.059, so that 12 semitones exactly equals an octave.
The equal temperament scale has the main advantage that all intervals sound the same (equally good or bad) whatever note you start from.
Therefore, diatonic scales played from different notes (eg, C-major, D-major, …) are mathematically identical except for their absolute frequency scale (which most people have no perception of).
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Unorthodox ScalesInstead of dividing the octave into 12 equally spaced notes, we can divide it into any number of equally spaced notes.
Listen to these scales with different numbers of notes:•12 notes (standard equal-tempered chromatic scale)
•13 notes•8 notes
Why aren’t 13 and 8 note scales popular? Because they are more dissonant than 12 note scales when two or more notes are played together with harmonic timbres.
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Unorthodox InstrumentsSome instruments designed to play unorthodox scales have actually been built and played:
Fokker organ designedto play a 31-note scale
http://www.xs4all.nl/~huygensf/english/index.html
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Unorthodox InstrumentsAlthough most real acoustic instruments have approximately harmonic timbres, artificial instruments can be electronically synthesized to have any timbres.
In particular, we can create instruments that are less dissonant when played in non-standard scales.
The results are interesting and easy to listen to (compared with the Fokker organ). For example:
•11-note scale:•19-note scale:
http://eceserv0.ece.wisc.edu/~sethares/mp3s/
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Frequency StandardizationMost people have no perception of absolute pitch so it is not surprising that we managed for a long time without any standard definition of the frequency of middle C.
In 1877, the A4 pipes on organs reportedly ranged from 374 - 567 Hz (corresponding to the modern range F-C#).
The modern standard is A4 = 440 Hz and was adopted in 1939.
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SummaryWe covered the following topics:
Musical Notes and ScalesScales and TimbrePythagorean ScaleEqual Temperament ScaleUnorthodox Scales
