Los Senderos Studio
Tuning Systems
 Issue 87 December 2016 

Words from the Glossary

At the risk of boring you to death, this month we discuss tuning systems, a subject that is not easily understood, and not exactly easy to explain. But I'm going to give it a stab.

This month's terms: tuning system, temperament, Pythagorean tuning just intonation, mean-tone temperament, equal temperament. (Note: Click on the term to view its definition in the glossary.)

Tuning Systems

Many musicians play their music without giving much thought to what tuning system we use. In our system of music, we use the equal temperament. Temperament is how we define the intervals between notes to achieve a tuning system. This usually means slightly compromising the pure intervals to produce the best overall sound.

The Greek mathematician Pythagoras is often credited with working out the first tuning system in the third century BC, but some historians say the Egyptians were doing it 2000 or 3000 years before that. Pythagoras discovered that notes with certain mathematical relationships produced sounds that were pleasing to the ear. For example, notes with the numerical ratio of frequencies of 1:2 or 2:3 are pleasant sounding. For example, two notes whose frequencies are 400 and 800 Hz have the frequency ratio of 1:2. Today we call such an interval an octave, and we give the notes the same name, but one octave apart. The interval between two notes whose frequencies are 400 and 600 Hz have a frequency ratio of 2:3 and is called a perfect fifth.

The problem is how to divide an octave into eight mostly whole steps or 12 half-step notes. (European and western music uses 12 notes per octave, but other cultures use other numbers). Pythagoras said the intervals should be simple ratios because they sound the best. So he defined his scale using the ratios of 3:2 (the fifth) and 4:3 (the fourth). The difference between these two had the ratio of 9:8, which he defined as a whole step. He then tried to divide the octave into eight whole steps. However, to get the mathematics to add up, he wound up with two half steps with the ratio of 256:243, which we call the Pythagorean comma thath is not very melodic.

 Musical
Interval
Frequency Ratio
to Tonic
Interval
Ratio
Interval
in Cents
Tonic 1:1
2nd 9:8 9:8 204
3rd 81:64 9.8 204
4th 4:3 256:243 90
5th 3:2 9:8 204
6th 27:16 9:8 204
7th 243:128 9:8 204
Octave 2:1 256:243 90

To improve upon Pythagorean tuning, just intonation was devised. Just intonation (also called Ptolemaic tuning, just temperament, pure temperament, or pure intonation) makes the major third intervals perfect by changing one of the fifths. This makes all the major sixths (except for F-D) perfect. Unfortunately, the triad D-F-A is not very consonant, while all the others are perfectly in tune. The just intonation scale uses steps of two different sizes having the ratios 9:8 and 10:9. The results were not very satisfactory, and just intonation was not used very much.

 Musical
Interval
Frequency Ratio
to Tonic
Interval
Ratio
Interval
in Cents
Tonic 1:1
2nd 9:8 9:8 204
3rd 5:4 10:9 182
4th 4:3 16:15 112
5th 3:2 9:8 204
6th 5:3 10:9 182
7th 15:8 9:8 204
Octave 2:1 16:15 112

With the mean-tone temperament, the major thirds are made exact. This slightly flattens the fifths, but this spreads the error of the just intonation over four fifths. This reduces the dissonance, making the fifths more acceptable. The mean-tone temperament has whole steps that are the same size and are equal to square root of 1.25. which equals 1.11803 (half of a major third). This Results in the mean-tone temperament being more melodic than just intonation.

 Musical
Interval
Frequency Ratio
to Tonic
Interval
in Cents
Tonic 1:1
2nd 1.118:1 193
3rd 5:4 193
4th 1.338:1 117
5th 1.495:1 193
6th 1.672:1 193
7th 1.869:1 193
Octave 2:1 117

The equal temperament divides the octave into 12 equal half steps with each half step equal to the twelth root of 2, which equals about 1.059463. All the intervals are half steps. For example a fifth is 7 half steps and a third is 5 half steps. All the half steps in the equal temperament chromatic scale equal 100 cents. That means there are no perfectly tuned intervals. However, the mistuning of fifths is only 2 cents and for thirds is 14 cents. Apparently this is close enough to sound consonant to the ear.

 Musical
Interval
Frequency Ratio
to Tonic
Interval
in Cents
Tonic 1:1
2nd 1.223:1 200
3rd 1.260:1 200
4th 1.335:1 100
5th 1.449:1 200
6th 1.682:1 200
7th 1.889:1 200
Octave 2:1 100

With Pythagorean tuning, just intonation, and mean-tone temperament, instruments (especially keyboard instruments, which could not adjust their tuning) would be in tune in one key, but would be dissonant in other keys. The big advantage of equal temperament is that all keys are in tune.


Although Elvis only recorded twenty Christmas songs, his holiday albums have sold more than twenty-five million copies in the US alone.

Before settling on Ringo Starr, the Beatles had four other drummers: Norman Chapman, Tommy Moore, Pete Best, and Jimmy Nichol.

Three of the members of the studio group that performed on most of Boz Scaggs hits would later form a band that would have four top ten hits of there own. The band was Toto and their hits included "I Won't Hold Back" (#10), "Hold The Line" (#5), "Rosanna" (#2), and "Africa" (#1).

In the Studio

Dos Amigos Band
Dos Amigos Band

Richard Vidmer and Judy Moore are the Dos Amigos Band. Richard grew up in the West Texas town of Marfa. Being raised in a cattle ranching community, he learned first hand the ways of the cowboy, and he has many fond memories and stories about west Texas. Many of these are reflected in his songwriting.

He developed a passion for music early on. His mother was a wonderful pianist and singer, and he inherited much of that talent. His mother's native American ancestry was also a big influence on him and his music.

Realizing that being a performing artist can be a long, hard road, Richard dropped out of the music world in his early twenties choosing instead to become a carpenter. He later formed his own home-building business. His success in that business along with his real estate investments allowed him to return to his passion as a singer/songwriter later in life.

Judy Moore grew up in California, and graduated from Palmdale High School. She was a court reporter in San Antonio when she met Richard several years ago. They have essentially been inseparable ever since. Together they have an incredibly diverse dimension of harmony, weaving their voices together like a fine tapestry and have an amazing vocal range. With Richard's accomplished guitar styling and Judy's diverse drumming and percussion work, they perform regularly as Dos Amigos Band.

Richard recorded a solo album with the title Under Texas Skies at Los Senderos Studio several years ago. Although he recorded and produced it under his name, Judy can be heard singing backup on several tracks. Los Senderos Studio also recorded a live album of the duo entitled Live from the Friendly Bar Bistro. Although the Friendly Bar is no longer around, the CD is still available. Currently they are working in the studio finishing up the album Dos Amigos Band Unplugged. It is intended to duplicate the sound of their live shows.

Richard and Judy perform all around the Hill Country at restaurants, bars, and vineyards. Check out their schedule at dosamigosband.com.


Los Senderos Studio, LLC


A Recording Studio in the Hill Country.


8409 N US Highway 281 ★ Blanco, TX 78606-5024
Phone: 512-565-0446 ★ Email: Larry@LosSenderosStudio.com


< Previous Issue | Newsletter Index | Next Issue >

Home


Copyright © 2008-2016
Los Senderos Studio, LLC | Los Senderos Ranch | 8409 N US Highway 281 | Blanco | TX | 78606