#1August 21st, 2006 · 01:56 PM
1 threads
2 posts
United States of America
Just Intonation Help
Hello, my name is Do. I have been teaching guitar for 1 year now. I am 16, and i just recently, in the past year encountered microtunings. I was immediately stumped.... My only problem is

the numbers on either side of the ratio..... I just cant figure it out. Is it like this?

a=440Hz, so a 1:1 ratio is: one cent to one cent, and 4:3 would be the D in the next octave?
If so, How would you set a root note?

I probably made a complete fool of myself, My father understands it my old theory teacher understands it, but i am 99% stumped....


#2August 26th, 2006 · 01:30 AM
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#3May 24th, 2007 · 06:16 PM
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#4May 25th, 2007 · 08:59 AM
1 threads
2 posts
United States of America
thanks, i get it now, I got it then also. But wasnt sure   I figured out my problem was the notation. Does anyone have the notation?


#5May 25th, 2007 · 09:50 AM
37 threads / 19 songs
618 posts
United States of America
Guess what? Except for octaves, none of the intervals and chords played with those pitches are precisely in tune. Musicians normally don't notice that their music is minutely out of tune, because they have become accustomed to the 12 pitches during the past 200 years.

To play intervals and chords that are completely in tune, the precise pitches of many notes must be shifted slightly from their normal frequencies. Microtuning is the term used to describe those tiny frequency adjustments. Trained singers, wind-instrument players, and fretless stringed-instrument players constantly perform those shifts to produce intervals that are as in tune as possible. On the other hand, keyboards, fretted strings, and mallet-percussion instruments can play only fixed frequencies and therefore are never perfectly in tune.

Why did Western music settle on a set of notes that is always out of tune? How can electronic musicians overcome the tyranny of such a limited palette of pitches? To answer those questions, you must understand the nature of musical intervals and what it means to be in tune.

A note is defined by its pitch, which corresponds directly to its fundamental frequency. Intervals consist of two notes sounding at the same time or sequentially, and chords consist of several simultaneous intervals. The relationship between those notes is often expressed as the ratio of their frequencies. In the interval of an octave, for example, the frequency of the higher note is exactly twice the frequency of the lower note; the ratio of the two frequencies is 2:1. Intervals with ratios of two whole numbers are called pure intervals. The common pure intervals include the octave (2:1), the perfect fifth (3:2), the perfect fourth (4:3), the major third (5:4), and the major second (9:8). There are many other intervals, but some can be one ratio or another, depending on the tuning system. For example, the ratio of a minor second is 16:15 in one tuning system and 17:16 in another system.

Other tuning systems, including that used in Western music, use intervals that cannot be expressed as ratios of two whole numbers. Such intervals are called impure, and their ratios are called irrational. Those intervals are impossible to represent with whole-number ratios, so a different interval-measuring system was developed.

The octave was divided into 1,200 equal intervals called cents, which let you measure pure and impure intervals in the same way. For example, the pure major third is approximately 386 cents, whereas the impure major third used in Western music is exactly 400 cents. As a result, modern major thirds are sharp with respect to the pure variety.

All music students encounter the circle of fifths in their studies The graphic includes all 12 notes of the standard Western tuning system in a sequence of perfect fifths. In that tuning system, the circle closes on itself, because B# is just a different name for C. Those two notes are called enharmonic equivalents. But if you use pure perfect fifths in the exercise, the final B# is 23.46 cents higher than the starting C (discounting octaves). Under those conditions, the circle of fifths becomes a spiral of fifths.

That 23.46-cent discrepancy is called the Pythagorean comma, named after the ancient Greek scholar Pythagoras, who did a lot of fundamental research of musical intervals. Because most tuning systems are octave based (that is, they include a set of intervals that repeats in each octave), the Pythagorean comma must be placed in the scale to preserve the pure octave. Exactly how that is done is the art of creating a tuning system.

Constructing a tuning with nothing but pure intervals, you must specify each interval individually. Such a system is generally called just intonation Each interval with the root note sounds perfectly in tune. However, like most scales other than the common Western tuning, the notes in just intonation are not equally spaced. As a result, you can play only in the key defined by the root note and a few closely related keys. For example, in just intonation with a root of C, the major third from C to E is 386 cents, but the major third from B to D# is 428 cents (42 cents sharp with respect to a pure major third). So in the key of C, everything sounds fine, but modulating to the key of B sounds terrible. One of the first tunings to allow modulating into other keys is called meantone temperament Temperament refers to the fact that some or all intervals are tempered, or adjusted, from their pure forms to allow performances in different keys. In meantone temperament, some perfect fifths are shortened slightly to accommodate the comma. However, they are not shortened by the same amount, so some keys sound distinctly better than others.

By the beginning of the 18th century, Western music was becoming more complicated and modulating into increasingly distant keys. Many musicians and theorists devised temperaments to allow modulation into any key. Among the most successful was Andreas Werckmeister whose temperaments were used by J. S. Bach and others. The notes were still not equally spaced in the scale, so each key had a distinct character. In fact, Bach wrote The Well-Tempered Clavier to demonstrate the character of each key in a temperament.

During the same period in history, other musicians experimented with equal temperament, in which the 12 notes were equally spaced within the octave. That “equality” is achieved by shortening each perfect fifth in the spiral of fifths by about 2 cents, making each one exactly 700 cents. The interval between consecutive notes in the chromatic scale is exactly 100 cents, which collapses the spiral into the circle of fifths. With that compromise, you can play in any key with equal ease. Each key sounds identical, with no change in character from one to another. Unfortunately, they also sound equally out of tune. Compared with their pure forms, perfect fifths are 2 cents flat, major thirds are 14 cents sharp, and minor thirds are 16 cents flat. The other intervals are similarly out of tune compared with their pure forms.

Other scales with equal steps come closer to producing pure intervals. Some musicians divide the octave into 19, 31, or 53 equal steps, and those scales include many almost-pure intervals. Wendy Carlos has taken a slightly different approach, assembling a series of equal steps that doesn't repeat in each octave. Her alpha scale    includes steps of 78 cents each. The tuning produces nearly pure thirds, fourths, fifths, and minor sevenths, though there is no pure octave.

As Western musicians converged on 12-tone equal temperament, the rest of the world was using many different tunings, some of which survive to this day. The musics of Indonesia, India, Asia, and the Middle East sound exotic and foreign because they are based on intervals different from those in Western music. For example, Indonesian music primarily uses one of two scales: Pelog or Slendro 


One primary reason to adopt 12-tone equal temperament is the historical tendency toward music that is intended to be played on a fixed-pitch keyboard and that modulates into diverse keys. With early tunings that are highly key dependent, you must retune the keyboard instrument each time you play in a different key. That is not something you'd want to do with a harpsichord or an acoustic piano in the middle of a piece of music. Equal temperament eliminates that requirement, so it found favor among Western musicians. Retuning digital synthesizers is easy. All it takes is the appropriate software to recalibrate the oscillators to produce any set of frequencies you desire. The Yamaha DX7II was the first widely available synth to offer that capability. Since then many electronic-keyboard manufacturers have included the ability to use tunings other than equal temperament.

Most of those instruments — which include models from E-mu, Korg, and Kurzweil — can retune only the 12 notes in an octave, and those tunings are repeated in all octaves. For key-dependent tunings, you can usually specify the desired root note. In a few instruments, you can retune each note in the entire MIDI range independently. That capability lets you construct larger tunings, such as 53-tone equal temperament or the Indian 22-note scale from which ragas are derived.

Synthesizers with alternate tunings usually can't share their tuning data with dissimilar instruments or retune on the fly, so Robert Rich and Carter Scholz developed the MIDI Tuning Standard (MTS), which was added to the official MIDI specification. The standard includes two major parts: bulk dumps and single-note retuning. It outlines the messages by which an instrument can be retuned during a performance. The specified resolution is 0.0061 cent, which is fine for most researchers and musicians.


Alternate tunings can be used in many ways, particularly with synths. Early and ethnic music can be played with more authenticity, and you can achieve better consonance in all forms of music, particularly if you don't modulate into widely divergent keys. Even if you do modulate, you often can change tunings at the same time. For example, you might create two synth patches with the same sound and different tunings, such as just intonation in the keys of C and B, and select the patch that is tuned to the key you are playing in.

Another important application of microtuning is education. If you're a music teacher, you can impart a greater sense of historical perspective to your students by playing music with appropriate tunings from different periods and locations. For example, play a sequence with equal temperament followed by the same sequence in just intonation. The difference is startling. You also can explore the world of sound and acoustics with greater ease and precision.
Excerpts from Electronic Musician  = http://emusician.com/mag/emusic_microtuning/
#6May 25th, 2007 · 10:04 AM
37 threads / 19 songs
618 posts
United States of America
Hope this might help a little.. check with the web
page link.. you'll find a few useful charts to give
references...  this came from a book that I had
studied from a little myself.  but to be honest,
the main thing I remember from the whole book,
"Home Studio: How Everything Really Works",
was that I kept it way to long and ended up paying
almost  $8 in overdue fees on it... (lol) what can
I say, I didn't quite 100% grasp micro-tuning either..
I did barely get past it in that week of classes.. but
that was over 20years ago bro..  but at least I knew
right where to go find it.. did a google on my big
overdue book.. or primer as they called it! I still
have a couple of midi books I got on clearance
sale table from the college book store.. one of
them I know has a section on micro-tuning..
will have to get the book when it gets unpacked.
there are a couple dozen boxes of books just
waiting to be put out on our shelves here.
but the thing is there is a hesitation because
we still don't know if we're going to be able
to purchase this house or not.. from where I
sit right this second, it would tend to lean more
towards "not".... but don't tell ld (wife) that, she
believes we can do it.. if they'll give us the time
we need to get back leveled off from what  I have
put us through with my health issues and all..
just another reason I have to find some manual
labor help to get the studios up and making a few
dollars..  I offered someone 18 hours of studio
time for three hours of help... but he hasn't called
me back yet.. still out of town... ok thanks guys
and hope the article helped!

Dan - Bluey
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