A.Diederich – International University Bremen – USC – MMM – Spring 2005 Scales Roederer, Chapter 5, pp. 171 – 181 Cook, Chapter 14, pp. 177 – 185 Cook, Chapter 13, pp. 150 – 152, 157 – 162

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Definition (For a purely practical purposes)  A scale is a discrete set of pitches arranged in such a way as to yield a maximum possible number of consonant combinations (or minimizes possible number of dissonance) when to or more notes of the set are sounded together.  (Here a scale is a set of tones with mathematically defined frequency relationships. This is to distinguish from the various scale modes, defined by the particular order in which whole tones and semitones succeed each other.)

A.Diederich – International University Bremen – USC – MMM – Spring 2005  The solfeggio notation do-re-mi-fa-sol-la-ti-do is used in the following to indicate relative positions on the scale (i.e, croma), not actual pitch.

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Just scale, first steps Consonant interval The notes do-mi-sol constitute the major third, the building stone of Western music harmony.

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Continuing to "fill in" tones, in each step trying to keep the number of dissonance to a minimum an the number of consonance to a maximum, yields Just diatonic scale

A.Diederich – International University Bremen – USC – MMM – Spring 2005  The intervals 9/8 and 10/9 define whole tones.  9/8: just diatonic major whole tone  10/9: just diatonic minor whole tone  The interval 16/15 defines a semitone.  8 notes – 28 possible pairs  16 consonant intervals  10 dissonant intervals  2 "out-of- tune" consonances

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Within the just diatonic scale we can form  three just major triads  do –mi – sol  do – fa – la  re – sol – ti  two just minor triads  mi – sol – ti  do – mi – la  one out-of-tune triad  re – fa – la

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Pythagorean scale  The Pythagorean scale is build up from the so-called perfect consonances, the just fifth, the just fourth, and the octave.  Only one whole tone interval, the Pythagorean whole tone (9/8)  Pythagorean diatonic semitone (265/243)

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Problems with these scales  Only a very limited group of tonalities can be played with these scales without running into trouble with out-of-tune consonances. That is  Both scales impose very serious transposition and modulation restrictions. (recognized in the 17 th century)  The type of music that can be played is extremely limited.

A.Diederich – International University Bremen – USC – MMM – Spring 2005 The equally tempered scale  In the tempered scale the frequency ratio is the same for all 12 semitones lying between do and do'.  Call this ratio s.  This is the frequency ratio for a tempered semitone.

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Frequency ratios and values in cents of musical intervals

A.Diederich – International University Bremen – USC – MMM – Spring 2005

Example: C major triads in temperaments 1.just temperament 2.mean tone in C 3.just temperament 4.equal temperament 5.just temperament 6.mean tone in C 7.mean tone in C# 8.mean tone in C Track 61

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Other scales, e.g., the Bohlen Pierce Scale Track 62

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Mel scale  A psychological scale for pitch is the mel scale proposed by Stevens, Volkman, and Newman (1937).  A unit of that scale – a 1000 Hz tone at 40 dB has a pitch of 1000 mel.

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Relation between Pitch and Frequency Pitch of 1000 Hz at 40 dB : 1000 mel (mels = 2410 log[1.6 ¢ f +1])

A.Diederich – International University Bremen – USC – MMM – Spring 2005 A piano keyboard normalized to mel scale, that is the keyborad is warped to match steps which are equal "distance" on the mel scale

A.Diederich – International University Bremen – USC – MMM – Spring 2005  At least two different kinds of pitch: the mel scale measurements and the musical pitch  The perceived difference between two notes decreases at the extreme ends of the keyboard.  Pitches, and differences between them, are not as clear at low and high frequencies.

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Scales with equal steps on the mel scale  Example:  Down the chromatic mel scale  Diatonic mel scale in the midrange  Diatonic mel scale in high range  Diatonic mel scale on low range Track 48

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Mel scale music  Tune in middle mel scale  The same but in lower mel scale  Original Bach tune  Mel version of Bach's tune Track 49

A.Diederich – International University Bremen – USC – MMM – Spring 2005  The mel scale is not an appropriate musical scale.  Experiments are usually done with pure sine tones and other sounds that had no standard musical interval relationships.  With musical material the psychological scale resulted in a log frequency scale within the common musical frequency range.

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Separating height from chroma  Two components of pitch:  the pitch height (vertical position on the pitch helix)  the chroma (position within an octave around the cylinder  Either of these components can be suppressed 1 octave

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Example  Pitches are rising by major seventh, but can be heard going down chromatically  Pitches are falling by seventh, but chroma is rising by a major scale. Track 50

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Example: Suppress chroma while leaving height (To construct: pass noise through a band-pass filter. The center frequency of the filter determines the perceived height of the sound)  height only, without chroma change, noise  height only, without chroma change, string timbers  height only, no change of chroma, sine tones  noise example Track 51

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Suppress height while leaving chroma  More difficult: Construct tones with an ambiguous spectrum, e.g., with harmonics lying only on octaves, and with a spectrum that decreases from a maximum near the center to zero at extremely low and high frequencies.  After ascending or descending a complete octave, the final tone is located at the same place where the scale began.

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Example Shepard tritone paradox  tritone can be heard going up or down  diminished thirds going upward give the context of "upwardness"  tritones: now likely heard as going up  diminished thirds going downward give the context of "downwardness"  tritones: now likely heard as going down Track 52

A.Diederich – International University Bremen – USC – MMM – Spring 2005 Confusing chroma and height  If chroma underlies pitch, can a melody be scrambled in terms of height while retaining chroma?  If this could be done, would the melody still recognizable?  It would seem that if the listener were able to attend only to chroma and ignore height, the answer would be "yes".

A.Diederich – International University Bremen – USC – MMM – Spring 2005  Our perception of melody actually depends quite critically on height as well on chroma.  Melodies scrambled in height while retaining chroma are not readily recognized.  A melody constructed with incorrect chroma, but with the right shape and height contour, can be more recognizable than one that has been completely randomly height scrambled Track 53