This document contains various diversions that would have cluttered up the Tuning for Beginners page. You can think of it as the footnotes, only it's unusual for the footnotes to be longer than the main text, and some topics aren't linked to. Some things are pedantic, most involve some mathematics, although that's nothing to be frightened of. If your room has been invaded by a pack of lions, you should be frightened about that. A differential equation, on the other hand, is unlikely to hurt you. There are some exceptions, of course. Lions made out of wood are unlikely to hurt you unless you get a splinter off one of them or something like that. A differential equation governing the motion of an anvil flying towards your head may be the indirect result of some degree of physical pain. As a general rule, though, it's lions: bad and differential equations: good. Bear that in mind as you read this page. I'll be testing you afterwards. How did that bear get in here? Oh, it's hiding from the lion. Now I understand. Where was I?
Pitch and frequency
Frequency is a physical quantity equal to the number of events in a given time. In music, this means the rate at which an instrument vibrates, measured in cycles per second or Hz.
Pitch is a subjective quantity related to frequency and the overtone series. Pitch can be measured as the logarithm of frequency. Then, intervals can be added instead of multiplied. The ear roughly perceives this logarithmic frequency scale as a linear pitch scale.
I'm reliably informed that there is not a one to one relationship between pitch and frequency. The pitch also depends on the loudness and timbre and things like that. However, for simplicity, I will continue to use the simplification of pitch being the logarithm of frequency. This is similar to loudness being the logarithm of intensity.
The simplest unit for pitch is the octave. The equation for pitch to frequency conversion is p=2^f, or f=ln(p)/ln(2) or f=log2(p). This measures frequencies relative to 1Hz. For frequencies relative to an arbitrary f0, p = 2^(f/f0) + p0 and f=f0*log2(p-p0).
The frequency of a note corresponds to the reciprocal of the length of the string or air column producing it. Hence, dividing a string in two means raising its frequency by 2/1 or an octave.
Integer fequency ratios were originally discovered as integer string length ratios, as it's easier to measure the length of a string than the number of hundreds of vibrations per second.
The usual units for measuring intervals are cents. These were introduced by Ellis in one of the appendices to his translation of Helmholtz's On the sensations of tone.
There are 100 cents to a tempered semitone, hence 1200 to an octave. Cents are quite useful in defining tuning tables, because they show how much each note has to be tuned from its default value. Then again, semitones would be just as useful. For theories independent of 12-equal, octaves or millioctaves (moct) would be more useful. Unfortunately, the rest of the world uses cents.
In the instrucions to my Zoom 505 multi-effects unit, a fourth is consistently called "4 steps" and a fifth "5 steps". Really, they should be 3 and 4 steps respectively. The error is repeated in the French, German, Spanish and Italian translations. I'm assuming intervals are named differently in Japanese.
Intervals are described according to the number of notes they contained. Hence, C-D is a second, C-D-E a third, and so on. When you come to add intervals, it gets really confusing. A second plus a third is a fourth, but 2+3=5. Obviously, the terminology was not developed by mathemeticians, or at least not very sophisticated ones. The problem would probably have been that a unison would have to be called 0 steps, and it took a while for 0 to be recognised as a number.
The first recorded use of zero as a number is in India. Sure enough, the word "saptaka" roughly corresponds to the English "octave" and implies the number 7.
Mathematical definition of 12-equal
12-equal has 12 equal steps to an octave. "Equal steps" means the same frequency ratio or pitch difference. Each step must, therefore, be 1/12 oct. Really easy. As a frequency ratio, this must be 2^(1/12) or the twelfth root of 2.
All equal temperaments work the same way. They are easy to think of on a pitch scale. However, they are difficult to implement on an acoustic instrument with out mechanical aids, as the frequency ratios are irrational.
Meantones are defined such that 5t+2s=oct. Algebraic symbols always mean intervals on a pitch scale. If you define t, s must be (oct-5t)/2 where "oct" may not be a perfect octave.
A negative linear temperament is where the fifth is flat of that in 12-equal. That means 3t+s<7/12 oct. Set oct=5t+2s and you have 3t+s<7(5t+2s)/12 => 36t+12s < 35t+14s => t < 2s. Obviously, 12-equal is the case where t=2s.
The "linear temperament" terminology is another of Ellis's. I maintain that meantones are planar as they are defined by two intervals.
In JI theory, t is an approximation to 9/8 or 10/9 and s to 16/15. ln(9/8)/ln(16/15)=1.825 and ln(10/9)/ln(16/15)=1.633. So, t<2s is in line with acoustics, which is why meantones work.
See also my matrix definition of meantone temperament
Staff notation and meantone temperaments
Staff notation is based around a 7 note scale which can obviously be defined as a diatonic scale. Sharps and flats arise from transpositions of that scale. For example, Bb is the perfect fourth above F. Bb is flatter than B by an interval of t-s, which I call r. This is generally true of all sharps and flats: whenever you see a # symbol, raise the note by r and whenever you see a b symbol lower the note by r.
Staff notation is inconsistent with JI as it does not distinguish the D a perfect fifth above G and that a major sixth above F.
Provided you keep to 5-limit harmony, the usual diatonic rules mean the chords will be specified correctly in staff notation. So, a lot of music written ostensibly in 12-equal can be converted painlessly into meantone.
The problem comes with 7-limit harmony. The 7/4 above C should be written as A# in meantone. It is closer than Bb for all negative linear temperaments. However, the C7 chord (C-E-G-Bb) can be interpreted as a 7-limit tetrad (C-E-G-A#). This chord is so important that there's a whole page on the Internet devoted to it.
To avoid this confusion, I always say that the ratio 7/4 should be described as an augmented sixth or subminor seventh, rather than simply a minor seventh. The term "minor seventh" as it arises from staff notation only has a meaning in meantones, and so should be defined accordingly.
As 7/4 is considerably flat of 10 steps of 12-equal, it is debatable whether a 12-equal interval can represent the 7-limit at all. However, the tritone 7/5 is tuned that bit better. So, the chord C7 contains enough close to 7-limit intervals that it may be interpretable as 7-limit. The consonant (IMHO) 4:6:7 chord is much rarer in 12-equal music than the more dissonant diminished triad, implying that the absence of the tritone spoils any 7-limit resemblence.
Saying whether chords can be considered equivalent is fairly arbitrary. Changing the pitch of one note doesn't radically affect the sound of the chord. However, in some temperaments, major and minor triads are very similar, so are they really the same chord? It is probably that the 12-equal C7 gains some of its consonance from the 7-limit. It's much more common that you'd expect from any 5-prime-limit interpretation. In the same way, the Pythagorean major triad functions as a 5-limit chord, although it has a different sound to a JI major triad.
Diminished triads in 12-equal are profoundly ambiguous. Probably, the two minor triads should approximate 6/5 and the tritone 7/5. How you tune that in meantone is very context dependent.
I use "5-limit" and "7-limit" to mean the odd limit. That is the largest odd number in the octave reduced frequency ratios. This concept originates with Harry Partch. It's probably what he meant by "5 limit" but he never specifically defines it as such. Page 110 of Genesis of a Music is the biggest clue.
The term "5-limit" is also used to mean ratios involving prime numbers no larger than 5. I always specify this as the 5-prime-limit or extended 5-limit. Helmholtz or Ellis used a term with the same meaning. I'm not sure if it's "quintal" or "tertian" though. I'll have to look it up. "Quintal" is a good word for the 5-prime-limit, but I don't like "tertian" for third-based music. Firstly, I see no point in ossifying the "n+1" interval names by inventing new names derived from them. The term "septimal" could follow either meaning, so I'll stick with that.
The 5-limit intervals within an octave are:
It's usually stated that 5-limit harmonies are the strongest consonances. I'm not sure about this. I find 4:6:7 to be more consonant than a minor triad, at least with some timbres in 31-equal. Minor triads are more common because they exist naturally in a diatonic scale. Also, when tuned to 12-equal, they are closer approximations to 16:19:24 or something.
The overtone series
Mathematically, the overtones above a fundamental frequency f are n*f where n is an integer. That means the pitch above p is p+log2(n) oct.
The overtone series arises from the wave equation:
d²a --- = -wt dt²
Which gives f=w/2p.
As many instruments are roughly one dimensional oscillators, they produce almost periodic waveforms which, through a Fourier series, give overtones. The overtones also have some independent acoustical life.
Any interval m/n can be rewritten 2^a * 3^b * 5^c * ... through all the prime numbers where a, b, c, ... are integers. Most ratios you use in music only use the first few prime numbers, so they can be specified in terms of a, b c, ... Ratio space is defined accordingly. Ratio space is three dimensional for octave specific 5-prime-limit harmony.
For example, 3/2 is 2^-1 * 3^1 * f^0. In ratio space, that is (-1,1,0). 16/15 is (4,-1,-1).
However, intervals are not really points in ratio space, they are vectors. The interval 16/15 is then (-4 1 1)H. H is defined in my matrix exposition.
Ratio space is discrete: the coordinates must be integers. So, distances are measured on a city block metric. The length of the vector (a b c)H is (|a| |b| |c|)H or |a|log(2) + |b|log(3) + |c|log(5). This is quite a good estimate if the inherent dissonance of the interval. It is also equal to the logarithm of the LCM of the ratio n/m, which is n*m.
The octave invariant version of (a b c)H is (b c)h. Octave invariance is a good way of simplifying ratios. As 2 is the smallest prime, factors of it can safely be ignored. Another way of thinking of this is that the 2-direction is the smallest dimension. If scales are defined to repeat every octave, as they usually are, all permutations of the 2 will occur. When ratio space is 2-dimensional, it can be drawn on a screen:
D---A---E---B---F#--C#--G#--A#--E# | | | | | | | | | Bb--F---C---G---D---A---E---B---F# | | | | | | | | | Gb--Db--Ab--Eb--Bb--F---C---G---D
In octave specific ratio space, the length of the vectors corresponding to a major third 5/4 and minor third 6/5 are about the same. In octave invariant space, though, 5/1 is smaller than 3/5. To correct for this, you rate the interval m/n by the larger of m and n, rather than their product. This is the Partch limit.
When major and minor thirds are equidistant, it makes sense to use a triangular lattice:
D---A---E---B---F#--C#--G#--A#--E# / \ / \ / \ / \ / \ / \ / \ / \ / Bb--F---C---G---D---A---E---B---F# / \ / \ / \ / \ / \ / \ / \ / \ / Gb--Db--Ab--Eb--Bb--F---C---G---D
There's an interesting connection between octave invariant ratio space and time invariant physical space. For this reason, I suggest making the 2- direction the zeroth dimension of ratio space, like time is the zeroth dimension of spacetime. Then, the numbering of the other dimensions is the same whether or not octaves are involved.
What's so great about quarter comma meantone?
A meantone major third is four octave reduced steps on the spiral of fifths. A minor third is the difference betweeen a fifth and a major third, so must be 1-4=-3 steps. For every cent you change the fifth, the minor third changes by 3 cents and the major third by 4 cents. So, the average of these intervals will be better the better the major third is and best when the major third is just.
The tone of ¼-commma meantone is half of a just major third, or ½log2(5/4) octaves. This is 0.16096 oct or 193.16 cents. The semitone is then (1-5*0.16096)/2=0.09759 oct or 117.11 cents. The fifth is 3*0.16096+0.09759=0.58042 oct, which is 4.48 moct or 5.38 cents flat. The minor third is flat by the same amount. So, this is the maximum deviation from just intonation for a 5-limit interval. A sharper fifth will be better in tune, but the minor third will be worse. A flatter fifth improves the tuning of the minor third, but means the fifth is worse. Hence, this is the minimun value for the maximum deviation of 5-limit intervals from just intonation for any meantone temperament. (Detuning the octave will also favour some intervals over others, worsening the worst case).
If you take the RMS of the pitch deviations, a flatter fifth comes out as ideal, because the minor third improves faster than the fifth deteriorates. It's still fairly close to 1/4-comma meantone, though. Quarter comma meantone comes out better in the 9- and 11-limits.
Quarter-comma meantone is also the optimal minimax meantone in the 7-limit.
So, quarter comma meantone is an optimal approximation to just intonation. That doesn't in itself make it a good tuning, of course. However, there is a distinctive sound to well tuned chords which I, for one like. I can't speak for anyone else. Also, if you want a dissonance, there are plenty of them in most scales. Consonances you have to work for. The other thing is that scales that approximate JI, and meantones in particular, will play familiar tonal tunes well. If there's a load of existing music you want to play, it'll probably work fairly well in a meantone. And, if it's got lots of triads, they'll sound "smoothest" in quarter comma meantone.
It isn't a perfect temperament, though. There isn't a perfect temperament. One thing is that it's useless for Pythagorean scales, which is one reason I use schismic temperament as well. Also, it has relatively large semitones. This can be a problem as some people like small leading tones for melodic purposes. Ivor Darreg, in Xenharmonikon 4 (I hope to be able to link to the article soon) suggests using an officially "out of tune" chromatic semitone in such situations. I've given this a little try out, and it works surprisingly well. It doesn't sound as if there's anything odd going on, which suggests that this sort of thing could be going on all the time in real music. The cadential effect is noticeably stronger. The problem with this sidestep is that you need to have a lot of notes for the choice to be there.
I got my guitar retuned to a meantone very close to 1/4 comma that I decided optimised most 11-limit intervals. So, by existential logic, I obviously believe this to be the best temperament, at least for a guitar. If I were fretting it now, though, it'd be to a subset of 31-equal for the difference it'd make, so that all of the 11-limit is optimised.
11-limit harmony and neutral thirds
See also 5-limit harmony
11-limit harmony uses ratios containing odd numbers no greater than 11. One such interval is 11/9, which is called a neutral third. The name means it is midway between a major and minor third. Two such neutral thirds make up 2.02 perfect fifths, so this is a good description.
Lots of traditional musics use neutral thirds. Whether that implies 11-limit harmony is a matter of opinion. If you think 11-limit harmony works, all neutral thirds are evidence for it. However, if you think melody is more important, a neutral third is simply used to be ambiguous between a major and minor third.
In the same way, a piece played on a 12 note meantone keyboard that deliberately avoids diatonic scales will usually make sense in the 7-limit.
The Arabian schismic connection
Historical Arabic scales are usually specified as Pythagorean. That does not, however, assume that they are Pythagorean. They clearly imply a schismic approximation. As such, many of them are obviously 5-limit. In fact, the JI major scale is one of them.
The Pythagorean notation may have been an aid to tuning, or an attempt to follow Pythagoras's theories, which had quite a following at the time. Helmholtz, or maybe it was Ellis, demonstrated the schismic interpretation. However Partch, who actually quotes from Ellis on the subject, ignores this in the text of Genesis of a Music (although he gets it right ibn the graph on p.430). He goes as far as to say "... the work of Alfarabi and subsequent Arabic theorists seems to be of value simply as a repository of Greek learning and a source of information." (p.371)
Obviously, Arabic theory will be of little value if you ignore its primary achievement. The error is compounded on p.404 when he says "And as each new degree in a Pythagorean-inspired scale is added, the number of intervals ungratifying to the ear is disproportionately increased..." not, apparently, noticing that, when you get to (13 -8), or 8192/6561, you have a very good (to 2 cents) approximation to the just major third, 5/4. Did Partch tune this up before declaring it ungratifying? He always criticises other theorists for not doing this with 11-limit scales.
The only reason for using such a large interval can be to imply a schismic approximation. Without cents, this is as good as you can get to specifying that exact 5-limit tuning should be used.
Bear this in mind when someone tells you that Arabian music is Pythagorean.
Oh, it's that bear again! Got rid of the lion yet? Oh good.
Genesis of a Music
Pedants will note that the full title is Genesis of a Music: An Account of a Creative Work, Its Roots and Its Fulfillments but this isn't on either the cover, the spine or the first title page. It isn't in the catchily titled "Library of Congress Cataloging in Publication Data" either.
For more Partch related stuff, see Corporeal Meadows.