Musical Notes and Scales

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Lecture 10

• Musical Notes and Scales
• Scales and Timbre
• Pythagorean Scale
• Equal Temperament Scale
• Unorthodox Scales

Instructor David Kirkby (dkirkby_at_uci.edu)
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Midterm

• The 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 Deadline

• The deadline to drop this course is Friday.For
  Drop Card signatures, see the Physics Undergrad
  Affairs Coordinator
• Kirsten Lodgard
• klodgard_at_uci.edu
• 137 MSTB
• Homework and midterm scores are posted on the web
  for your reference

http//www.physics.uci.edu/undgrad/coursescores.ht
ml
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Review of Lecture 9

• In 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 Scales

• There 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 Timbres

• Most 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 Rule

• Two notes played together on instruments with
  harmonic timbres sound most consonant (least
  dissonant) when their fundamental frequencies are
  an exact number of octaves apart

frequency
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.
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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 2.021
  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 Octave

• In 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 Timbre

• The 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 Notation

• The 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.

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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.

Gb
Ab
Bb
Db
Eb
Gb
Ab
Bb
Db
Eb
F
G
A
C
D
F
G
A
C
D
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 Scales

• The usual choice of 5 notes in a pentatonic scale
  corresponds to the black notes on the piano

This scale includes the dissonant whole tone
(98) interval, but leaves out the less dissonant
major (54) and minor (65) 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 Scales

• The 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,

C
A
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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 Scales

• Although 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?

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The Circle of Fifths

• We can reach all 12 notes of the chromatic scale
  by walking up or down the piano in steps of a
  fifth (32)

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

Steps downin octaves
Steps up in fifths
startingnote
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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 upin octaves
Steps down in fifths
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 f0We end up
close but not exactly where we started!
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Pythagorean Scale

• If 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 Scales

• Since the major (54) and minor (65) 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 Scale

• The 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 Scales

• Instead 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 arent 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 Instruments

• Some 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 Instruments

• Although 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 Standardization

• Most 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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Summary

• We covered the following topics

• Musical Notes and Scales
• Scales and Timbre
• Pythagorean Scale
• Equal Temperament Scale
• Unorthodox Scales