12 note equal temperament

Meantone temperaments

5-limit triads

The overtone series

Just intonation

Quarter comma meantone

Equal temperaments

Schismic temperament

Tunings from around the world

The tuning mini-FAQ

Some books to read

Here I will outline the realm of alternative tunings. No prior knowledge of music is required, so you can skip anything you already know. There is a bit of keyboard bias, but I aim to show the general principles that can be applied to any instrument. You don't even have to know everything here to use alternative tunings. Just find a scale, and tune it up!

There is a strong bias toward those things that can be explained mathematically. You'll find that over the whole site. That isn't a value judgment, but it reflects the things I think I understand.

Kyle Gann's history of European tunings covers similar material, but assumes more musical knowledge.

There are a few links to another page on my site, that covers the same things in more detail, or related issues.

This is the white notes on a piano keyboard. If you don't know what a keyboard looks like, here's a picture of one. If it doesn't come out right, or you've got Java but choose not to use it, click here for a GIF in a window all of its own.

The scale is
defined in terms of *tones(t)* and *semitones(s)*. For
example, C major is like this:

C D E F G A B C t t s t t t s

The scale repeats every 7 steps. That interval
is called an *octave*.

So why are there only 7 notes in an octave? Because intervals on a
diatonic scale are always over counted. So, two
diatonic steps are a
*third*, three steps are a *fourth* and so on. No diatonic
steps are a *unison*. Be aware that the people
who write or translate instruction manuals for Japanese equipment
don't seem to understand this.

In 12-equal, t is set to 2 steps and s to 1, hence there are 12 equal steps to the octave. This is how keyboards are pretuned. The diatonic scale can be transposed (moved) to start on any note and the intervals stay the same. As synthesizers can be retuned in a fraction of a second, that advantage of 12-equal disappears. Most people don't bother to retune, though. My aim here is to explain why you should.

Where t and s have unique values, you have a
meantone temperament.
12-equal is a special case of meantone temperament. For t and s to
have any meaning, t must be the larger. That is, t>s. Usually, 2s>t
as well, giving a *negative linear temperament*.
On the Internet, meantones are generally spoken of as historical scales.
The exception to this is Charles Lucy
who is a very strong advocate for one meantone in particular.

I've now got a page dedicated to meantones.

Three or more notes sounding together make a chord. A three note chord is a triad.

In root position, a 5-limit triad spans a
*perfect fifth*. A perfect fifth is
3 tones plus a semitone, or 3t+s. The fifth is divided into a *major*
and *minor third*. A major third is 2t, and a minor third t+s.

If the lowest two notes are separated by a major third, you have a
*major triad*. Examples from the white notes are C-E-G, F-A-C and
G-B-C. If the chord starts with a minor third, you have -- and pay
attention, this is quite tricky -- a *minor triad*. Examples are A-C-E,
D-F-A and E-G-B.

Triads are still triads when you independently transpose the notes
up or down in octaves, but they stop being in root position. This is
called *octave invariance of harmony*. Some people take it very seriously.

Midi files are provided of major and minor triads.

The difference between an octave and a perfect fifth is a *perfect
fourth*, 2t+s. You also get *major sixths* (4t+s) and *minor
sixths*. Along with the thirds and perfect fifth previously mentioned,
these are *5-limit consonances*. A chord is a 5-limit cosonance
if it contains no other intervals than these, or octave equivalences.
In traditional harmony, only the 5-limit is recognised, so such chords are
simply defined as consonances. Other chords are dissonances.

Consonances have a smoother, or more restful sound than dissonances. At least, that's partly true. Try a variety of chords in a variety of tunings and timbres to decide for yourself The diatonic scale has 6 5-limit triads in it. You could say that the diatonic scales is defined by having 6 5-limit triads, but that's dodgy historically.

A complex note is made up of *overtones* or *partials*.
These give an instrument its timbre or tone colour.
With many instruments, the overtones are at special intervals,
and this is the overtone series.

Overtones are also partials.The lowest partial, which isn't an
overtone, is called the *fundamental*. The overtones are all
higher than the fundamental, hence the over bit.
In older versions of this page, I got the numbering wrong on this,
so there may still be wrong numberings.
The second
partial is an octave above the fundamental. The third and fourth
partials are a perfect fifth and octave respectively above the
second partial.

The fourth, fifth and sixth partials form a major triad in root position. Hence, this is known as a 4:5:6 chord. There's also a major triad formed from partials 3, 4 and 5. In Western Classical theory, this chord is considered a dissonance because it has a fourth in the bass. Guitar chords don't follow this rule, so ignore it if you like. Anyway, it's probably not a coincidence that consonant triads are found in the overtone series. Some people think otherwise, and it's best not arguing with them.

The 7th overtone leads to 7-limit hamony and the 8th overtone is an octave above the 4th.

Partials that are out of tune with the idealized overtone series
described so far are called *inharmonic*.
Bells have a lot of inharmonic partials.
A piano has a slightly stretched overtone series and so
octaves are usually tuned sharp to compensate for this.

There's lots about Just Intonation on the Web. Try The Just Intonation Network for starters.

With just intonation(JI), you tune chords according to the overtone series.
This means integer frequency ratios. Recall that overtone 3 is a fifth above
overtone 2. So, the perfect fifth can be a frequency ratio of 3/2. That
is, the frequency of the higher note is one and a half times that of the
lower note. **3/2** can also be written **2:3**.

A just octave is a ratio of 2/1. Two octaves are then 4/1. Most temperaments keep just octaves, for simplicity if nothing else.

I said above that a major triad in root position is 4:5:6. This describes the frequency ratios as well as the overtones. So, a major third is a frequency ratio 5/4 and a minor third is 6/5.

Adding intervals means multiplying frequency ratios. So, adding a
major and minor third gives 6/5*5/4=3/2, a fifth. A fourth is the
difference between a fifth and an octave, so 2/(3/2)=4/3. A tone is
the difference between a fourth and fifth. (3/2)/(4/3)=9/8. But,
a major third is two tones. 9/8*9/8 is not 5/4. So, there are two
different tones in just intonation. 9/8 is a *major tone*, and a
*minor tone* is 10/9. That comes from (5/4)/(9/8)=(10/9).

Chords in Just Intonation do have a particular sound. At least, if you use harmonic timbres without vibrato and everything else. A lot of people must like this sound, as there's quite a bit of interest in just intonation. Try it out yourself to see if you're one of them.

All this manipulation of fractions is quite unwieldy. To avoid this, you can use ratio space vectors. My matrix ideas follow on from this. Some people no doubt think they're intended to confuse poor artists, but the real aim is to clarify inherently difficult concepts.

The diatonic scale can then be written:

C D E F G A B C T t s T t T s

Where T is a major tone and t is a minor tone. The semitone, s, must be the difference between a major tone and minor third, or (6/5)/(9/8)=16/15.

The problem with this scale is that the minor triad D-F-A is wrong. The minor third should be T+s, but is t+s. For all triads to be in tune, you need to set T=t. This defines a meantone temperament, and is how I originally defined the diatonic scale. All this used to be explained in more detail by John Starrett but it looks like the link is broken.

As there are two different sizes of tones, the best compromise for a meantone is to average them out. This is called quarter comma meantone. The major third is the same as for just intonation, but the fifth is sightly flat.

The tone in 1/4 comma meantone comes out as a frequency ratio of sqrt(5)/2. This is the geometric mean of 9/8 and 10/9. Originally, "meantone" referred purely to 1/4 comma meantone for this reason. It would be more logical to use a different word for general meantones, and "mesotonic" is the best I've heard. However, musical terminology has little to do with logic. Hence abreveis such a long note it's rarely used, and thecor anglaisis neither English nor a horn.

Quarter comma meantone is used primarily as an approximation to just intonation. There are plenty of other meantones to choose from. Traditional staff notation is effectively geared towards meantone, although few of the people who use it understand this. Most Western Classical Music can be interepreted as a meantone approximation of just intonation.

With meantones, there are two different sizes of intervals: t and s. With equal temperaments (ETs), there is only one interval. As stated above, in 12-equal t=2s. A good approximation to 1/4-comma meantone is 31-equal, where t is 5 steps and s is 3 steps.

A whole range of ETs are used by microtonalists. Equal tempering simplifies scale construction. If an instrument can play all the notes in an ET, a lot more scales will be possible than if it had the same number of notes from a meantone, let alone just intonation. Also, a chord or tune can be transposed to start on any note of an ET. This becomes more important the harder it is to retune the instrument.

Not that equal temperaments need have anything to do with just intonation. Only when they don't, there isn't much I can say about them.

Another thing is that it's easy to describe scales on ETs. Each interval is described by one number, and it's obvious roughly how large an interval is.

Equal temperaments need not be octave based.

The usual mnemonic for n equal steps to an octave is **n-TET**.
I prefer **n-equal** because it's easier to say. Also, the first
"T" stands for "tone" which is an Americanism in this context. And
the other "T" stands for temperament. Well, single interval scales
needn't be temperaments. That is, they needn't imitate integer
frequency ratios. So, "n-equal" covers the full generality.

*Schismic temperament* is an alternative to meantone. It's also known
as *skhismatic* and various things in between. A third is associated
with 8 fourths instead of 4 fifths. So, it's still compatible with 12-equal.

Schismic temperaments in general are a much better approximation to 5-limit just intonation than meantones. The price is that they do have sizes of tone, so they're almost as complex as well!

I define schismic scales in terms of intervals called s and r. They should be the Greek letters sigma and rho. A diatonic semitone is s, a minor tone is 2r and a major tone is s + r. An octave is 5s + 7r.

A diatonic scale comes out like this, with alternative tone sizes:

s+r 2r s s+r 2r s+r 2r s+r s s+r 2r s+r

One particularly good ET that is consistent with schismic temperament is 53-equal. r is 4 steps an s is 5. 53-equal is so close to 5-limit just intonation that it can be considered identical within the tuning limitations of most instruments.

See my fourth based tuning for a way of implementing schismic temperament on a conventional 7+5 keyboard.

Schismic scales aren't that well known. It was purely a whim of mine to put them in this introduction. If you want to know more, I've dedicated a whole page to them.

I used to have a big section here, but decided to get rid of it. All you need to know is that most of the scales in use in traditional musics around the world have little or nothing to do with anything on this page. So, you're free to ignore all this if you want.

Every now and then, someone on the Tuning List says we should have a FAQ. But, nobody ever gets round to writing one, so the same questions keep on getting asked. However, a lot of web pages do answer these questions, so here I am listing them.

- Where do I get a synthesizer that supports <tuning x>?
- Try The Microtonal Synthesizer Home Page maintained by John Loffink.
- What does <obscure tuning term x> mean?
- Try Joe Monzo's tuning dictionary
- What sort of tunings would <dead composer x> have used?
- Try Kyle Gann's history of European tunings. And I've already linked to that higher up. You weren't paying attention, where you?
- What tunings are commonly used in <exotic part of the world x>?
- Hmm, this is a tricky one. I think Joe Monzo's putting something together.

Books aren't really that useful on tuning theory. There are some which cover historical temperaments, but I haven't read them and so can't really comment. Try the "Temperament" entry of New Grove. Otherwise, because there's a lot of it online and there isn't a dogma, or even a universal terminology, to learn you're generally better experimenting for yourself. But if you like books, here are some

William Sethares, *Tuning, Timbre, Specrum, Scale*
is an excellent introduction to acoustics, some historical
and exotic tunings, along with heaps of original theory.
If you're interested in non-harmonic timbres, this is one
book you probably do need.
If not, so long as you don't mind the scientific approach,
give it a try.
On Springer Verlag, quite expensive but worth it.

Herrman von Helmholtz's *On the Sensations of Tone as a Psychological
basis for the Theory of Music* translated from German by A. J.
Ellis, is the classic tuning text. Although it's old, and all
subsequent books (as well as this page) have been influenced by it,
it's still something a lot of people expect you to have read.
As many appendices were added by Ellis, the translation
may even be better than the original. Reprint by Dover
Publications, 1954.

Harry Partch's
*Genesis of a Music*
is a generally recognised as a classic. It's currently out on
Da Capo Press.
It isn't always reliable, but is inspirational.

Nicola Vicentino's *Ancient Music Adapted to Modern Practice*
is another idiosyncratic approach to microtonality.
Because of it's age, it gives a different perspective to most
modern treatises.
Translation by Maria Rika Maniates, available on Yale University
Press.

W. A. Mathieu's *Harmonic Experience* is an introduction
to traditional harmony, but from a perspective of just intonation.
I like it, but still haven't finished reading it.
Others have quibbles.
It partly depends on how advanced you already are in harmony and
singing.

Other than these, there may well be books on musical acoustics or some such at your local library. Or look at Manuel Op De Coul's bibliography which looks definitive, although he insist's it's woefully incomplete, particuarly for sources in Asian languages. There are some interesting things in academic journals which I don't have time to list let alone read.

Remember, whatever you read, to check that you can hear what you're supposed to hear. The historical difficulty of producing alternative tunings has meant a lot of weird theories end up floating around -- like this website for example! As long as you've got a tunable synthesizer, and go by your own preferences, you should be okay.