# VCF.MTH.2

Lesson Summary

Formulate intervals using ratios.

Vocabulary
A440 - the musical note A above middle C, serves as a general tuning standard for musical pitch and has a frequency of 440 Hz.

Logarithm -  the logarithm of a number is the exponent to which another fixed value, the base, must be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 10 to the power 3 is 1000: 1000 = 10 × 10 × 10 = 103

Equal Temperament - The system of tuning, in which every pair of adjacent notes (semitone) has an identical frequency ratio. Since pitch is perceived as the logarithm of frequency, this means that the "distance" from every note to its semitone is the same for every note.  Western musical notation uses twelve tone equal temperament. Example: The fourth of an A note is D.  D is 5 semitones up from A so for A440 we can find our D frequency by multiplying 440 by 2 to the power of 5/12. Pythagorean tuning - Historically the first tuning system in the western world. This tuning system is based around the ratio 3:2, or the Perfect Fifth.  All other frequencies in a scale are derived from that ratio.

Just Intonation - A tuning system in which the frequencies of the scale notes are related to one another by simple numeric ratios. In tunings such as 1:1 9:8 5:4 3:2 7:4 2:1, all the pitches are chosen from the harmonic series (divided by 2 to ensure they are the same octave), so all the intervals are related to each other by simple numeric ratios.  An extension of Pythagorean tuning used historically before Equal Temperament.

Interval - the difference between two pitches. In western notation they are referenced ascending by 1 semitone as follows: Unison (Tonic, Root), Minor 2nd, Major 2nd, Minor 3rd, Major 3rd,  Fourth, Tritone (Augmented Fourth, Diminished Fifth), Perfect Fifth, Minor 6th, Major 6th, Minor 7th, Major 7th, Octave.

Ratio - A ratio represents, for every amount of one thing, how much there is of another thing.  Ratio's are used in music to determine pitch intervals and frequencies within tuning systems.

Exercise

Visit the Werkstatt interval applet to explore the various relationships of notes and frequencies available with the Werkstatt.

Materials Jumper Cables
Hardware

All analog synthesizers are arranged in twelve-tone equal temperament.  However by turning the KB TRIM pot on the Werkstatt's PCB we can hear the widening of the frequency ratio between the octave note keys.  The KB TRIM pot is located at VR10 and is highlighted in Figure 1.

Figure 1. KB TRIM pot on the PCB.

Start by unscrewing all four screws on the top panel and carefully removing it.  Plug the Werkstatt in making sure that no wires are laying across the PCB that could connect any wrong parts of the board. Using a screwdriver carefully move the KB TRIM pot up and down and notice the difference in makes to the frequency relationships between the keys on the Werkstatt.  How does this change the playability of the Werkstatt? Why do you think equal temperament was chosen as the default tuning system for the Werkstatt?

Practice

1. The ratio between the the root (tonic) and the perfect fifth varies between tuning systems. Compute the perfect fifth of A440 in Just, Pythagorean, and Equal Temperament tunings.

2. The ratio between between the the root (tonic) and the minor 3rd varies between tuning systems. Compute the minor 3rd of A440 in Just, Pythagorean, and Equal Temperament tunings.

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