
Circle of fifths and roots of two - ColinWright
http://www.johndcook.com/blog/2009/09/30/perfect-fifths-equal-temperament/
======
tunesmith
The thing that might not be clear from this article is that if you tune with
Pythagorean Tuning, you'll run into cases where an Ab reached from one route
is different than an Ab reached from another route. Since instruments like the
piano need to have only one Ab in a particular octave, that is why they'll be
tuned as equal temperament, or some other method that guarantees only one Ab
for a particular octave.

Otherwise, you'll run into phenomena where an instrument will sound totally
awesome if played in one key, but really awful if they try to play the same
tune transposed to a different key.

This is also why those with sensitive ears are frustrated by their inability
to make a guitar sound in tune for both G major and E major chords.

~~~
baddox
Is that really why G and E chords are that way? I think it's more an issue of
fret intonation. Guitars are typically tuned to equal temperament, just like a
piano, so such a guitar with perfect intonation should sound just as good on
every chord as a perfectly tuned piano.

~~~
acjohnson55
Pianos aren't actually tuned to equal temperament, due to the inharmonicity of
their very tense and thick strings.

[http://en.wikipedia.org/wiki/Stretched_tuning](http://en.wikipedia.org/wiki/Stretched_tuning)

~~~
baddox
Yes, but that's a technicality. The goal is to have it _sound_ equal tempered.

~~~
acjohnson55
Of course. I wasn't criticizing, just mentioning, in case anyone found that
tidbit interesting.

------
stephencanon
The usage of “perfect fifth” to indicate only the just fifth (3:2 ratio) is
slightly idiosyncratic (but not unheard of); more commonly in modern usage,
“perfect fifth” refers to both the just (3:2) and equal-tempered (2^(7/12))
fifths, or to any other tempered fifth, for that matter (for reasons that
aren’t entirely clear—or at least differ depending on who you ask—the unison,
fourth, fifth, and octave are collectively called the “perfect” intervals in
common parlance, regardless of temperament).

~~~
mtinkerhess
A little more info for those who are curious: The perfect intervals (unison,
4th, 5th, 8ve) can be modified by being diminished or augmented, which shrinks
or expands the interval by one half step. The other intervals (2nd, 3rd, 6th,
7th) can be either diminished, minor, major, or augmented. The major interval
is what you'd get if you're measuring the intervals on a major (Ionian) scale;
minor is -1 half step, dimished -2, augmented +1.

My intuition about which intervals are major / minor and which are diminished
comes from the intervals that create the differences between the major and
minor scales. The major scale uses all major intervals; natural minor uses
minor 3rd, 6th and 7th; Dorian minor has a major 6th, and harmonic minor a
major 7th. That still doesn't account for why the 2nd is called major / minor
-- you could argue that you use it in Phrygian mode, but then you'd have to
acknowledge that you use the diminished 5th in Locrian mode, so why not call
that the major / minor 5th? (But almost no music is written in the Locrian
mode).

Another intuition is that the fourth and fifth are the most consonant
intervals (because the ratio of their frequencies is simplest) and so an
adjustment of a half step is more of an increase in dissonance than a similar
adjustment on another interval.

Finally, non-perfect 4th and 5th intervals are simply encountered much less
frequently than other intervals (with the exception of the tritone between the
3rth and 7th of a dominant V chord, the characteristic dissonance that sets up
the V-I authentic cadence).

~~~
mrbrowning
It's worth noting that the perfect fourth, when it involves the bass, is
considered a dissonance in Common Practice music,* and even in the freer
harmonic contexts of twentieth century music it's a bit of a cipher: it acts
as a dissonance in relatively consonant contexts and as a consonance in
relatively dissonant contexts.

* If you read Schoenberg's _Harmonielehre_ , he has some pretty idiosyncratic but somewhat plausible theories about why the 6/4 chord is dissonant, involving the overtones of the bass being diatonic (say we're talking about C 6/4 and the relevant overtones being B and D) but also dissonant with regards to the nominal root of C. That the 6/3 chord is relatively consonant is explained by the fact that the overtones of the bass in that arrangement suggest a sufficiently distant key so as not to plausibly oppose the sounding fundamentals. I think the more likely explanation is the historical one, which treats the 6/4 as the reification of the frequently used motif of a suspension from the pre-dominant harmony over the dominant bass that then passes downward to the goal tones (in C major, the archetypal lines would be A -> G -> F -> E and F -> E -> D -> C over a bassline like F (or D) -> G -> G -> C). The 6/4 chord's dissonance, then, is merely a result of the fact that its appearance came to unequivocally imply a certain resolution.

------
leephillips
Article 2nd paragraph: "Suppose a string of a certain tension and length
produces an A when plucked. If you make the string twice as tight, or keep the
same tension and cut the string in half, the string will sound the A an octave
higher."

This is not even dimensionally correct. The frequency is proportional to the
square root of the tension.

Later: "The ratio of 3/2 is called a “perfect” fifth to distinguish it from
the ratio 1.498."

I don't think so. The interval is called a "perfect" fifth to distinguish it
from the "diminished" or "augmented" fifth. Nothing to do with tuning.

~~~
stephencanon
The usage of “perfect” to mean “just” is uncommon, but not really wrong. It’s
somewhat more common in older (19th century), AFAIK.

------
brownbat
Examples of four temperaments using Mozart's Piano Sonata No. 11:
[http://www.youtube.com/watch?v=XbGq43Ol0tk](http://www.youtube.com/watch?v=XbGq43Ol0tk)

I also found this short essay on Bach and tuning informative:
[http://historicaltuning.com/Overview.html](http://historicaltuning.com/Overview.html)

~~~
_nato_
Wow. I went my entire life thinking with complete error that J.S. Bach's The
Well-Tempered Clavier was a musical dossier celebrating modern tuning (of
keyboards). I was 100% wrong. Thank-you for sharing that essay!

------
osteele
Related is that going four steps around the circle of fifths, for a total of
five pitch classes, produces the _pentatonic_ scale (e.g. the black keys).
This is the longest sequence that doesn't contain any half steps. Continue for
a total of seven notes and it's the _diatonic_ scale (e.g. white keys†). This
is the longest sequence that doesn't contain consecutive half-steps. Finally,
continue for a total of twelve notes, as the article describes; the resulting
_chromatic_ scale is either (in just temperament) the longest sequence where
every step is at least a half step, or (in even temperament) the longest
sequence before the pitch classes repeat.

† Annoyingly, the tonic (tonal center) of the diatonic scale is the _second_
note in the sequence of fifths (e.g. C in FCGDAEB). I believe this is so you
depart the tonic by a fifth in either direction, since fifths sound so good.

~~~
acjohnson55
Yep. In other words, the common scales are the number of consecutive perfect
5ths to "almost" land back on the first note, with "almost" defined
progressively stricter. This exercise could be extended to make longer scales,
but the additional notes would be close enough to the first 12 to just sound
to human ears more like knockoffs of notes in the first 12 than new notes in
their own right.

Instead, the standard of modifying the 11 notes after the first in various
ways to make instruments sound good in more keys proved to be more worthwhile.

------
dmlorenzetti
I once went with a friend to price electric pianos. The models she was looking
at sampled traditional pianos to get their notes, and had a switch to toggle
between different sampled instruments.

She asked why there was such a price difference between models, based on the
number of keys (unfortunately I don't remember the numbers anymore, but
basically some of the models had shorter keyboards).

The response was that, for models with a smaller number of keys, they could
just sample one octave of the original instrument, and scale the frequencies
electronically. However, once you go beyond a certain range, straight scaling
didn't work anymore, so they had to sample the whole range of a traditional
piano-- hence more electronics needed in the electric piano.

~~~
colanderman
I don't buy it. Sampling isn't typically done for an entire octave and then
scaled 2× or ½×; that would sound terrible. Rather, usually every 3rd note (or
so) is sampled, and the "missing" notes are scaled from those.

More likely the reason is that keyboard size is used as a proxy to
discriminate between amateurs (who can play most anything they care to on,
say, a 61-key keyboard) and professionals (who would look like fools trying to
fit certain pieces into 61 keys when they are meant for the full 88).

~~~
baddox
You might be surprised at the results you can get these days. There are even
"soft pianos" that are almost completely synthesized rather than sampled. They
can sound amazing:
[http://www.truepianos.com/demos.php](http://www.truepianos.com/demos.php)

~~~
coldtea
Analog modelling ("the soft pianos" you mention) don't have much to do with
stretching samples for several octaves the OP described.

------
beefman
See the xenharmonic wiki for more music math than you can shake a stick at

[http://xenharmonic.wikispaces.com/Mathematical+Theory](http://xenharmonic.wikispaces.com/Mathematical+Theory)

------
esquivalience
This is very interesting indeed. They say music goes hand in hand with maths,
but it's never taught that way... and musical theory (at least in the UK)
teaches what exists but not the principles behind it.

~~~
fyrabanks
Lots of information out there, but you have to have an idea of what to look
for.

Sit down with this 63 part series and you'll have a very good understanding of
the mathematics and physics behind sound:
[http://www.soundonsound.com/sos/may99/articles/synthsec.htm](http://www.soundonsound.com/sos/may99/articles/synthsec.htm).
You'd probably come to a lot of those realizations just working with computer
audio software.

Tons of papers out there on the maths behind, say, phase-vocoder techniques
(same maths behind autotune):
[http://www.ee.columbia.edu/~dpwe/papers/LaroD99-pvoc.pdf](http://www.ee.columbia.edu/~dpwe/papers/LaroD99-pvoc.pdf)

The average music student is more interested in the music than what's behind
it. Not at all a knock on them, just how it is.

------
danelectro
C, E-flat, and G walk into a bar. The bartender says, "Sorry, but we don't
serve minors."

So E-flat leaves, and C and G have an open fifth between them. After a few
drinks, the fifth is diminished, and G is out flat.

F comes in and tries to augment the situation, but is not sharp enough. D
comes in and heads for the bathroom, saying, "Excuse me; I'll just be a
second." Then A comes in, but the bartender is not convinced that this
relative of C is not a minor.

Then the bartender notices B-flat hiding at the end of the bar and says, "Get
out! You're the seventh minor I've found in this bar tonight."

E-flat comes back the next night in a three-piece suit with nicely shined
shoes. The bartender says, "You're looking sharp tonight. Come on in, this
could be a major development." Sure enough, E-flat soon takes off his suit and
everything else, and is au natural.

Eventually C sobers up and realizes in horror that he's under a rest. C is
brought to trial, found guilty of contributing to the diminution of a minor,
and is sentenced to 10 years of D.S. without Coda at an upscale correctional
facility.

------
jessaustin
Really clear explanation! Linked in the comments is a natural sequel, which I
wouldn't have understood without reading this first:

[http://www.slate.com/articles/arts/music_box/2010/04/the_wol...](http://www.slate.com/articles/arts/music_box/2010/04/the_wolf_at_our_heels.single.html)

------
throwaway13qf85
It's tough for someone who's not a professional musician to tell the
difference between a just fifth (3:2) and an equal tempered fifth (2^(7/12):1)
but it's not impossible.

Probably the easiest way is to go to a piano and play a fifth with the base
note at middle C. The sound noticeably changes after a couple of seconds.
That's because the ratio isn't exactly 3:2 and the frequencies of the pure
notes start to get out of alignment (the same phenomenon causes 'beats' when
two similar but not exactly equal frequencies are played together).

It's even more pronounced when you play a C major chord (C-E-G) because the
major third (C-E) isn't exactly 5:4 and the minor third (E-G) isn't exactly
6:5 either.

~~~
theOnliest
The equal-tempered major third is 14 cents higher (1 semitone = 100 cents)
than a just-tempered third, which is pretty noticeable even for non-trained
musician.

------
pflanze
I've calculated the values according to the described circle of fifths and
drawn a plot comparing it to equal temperament (x is the pitch steps 1..13, y
is the frequency 0..2, the curve at the bottom is the difference, magnified 10
times, the gray line at the bottom is the x axis). Done quick and dirty, i.e.
I manually filled in the slots for the 12 pitches. I felt a bit disappointed
when I reached the end of the article and was told that this isn't actually
what's normally done...

[http://christianjaeger.ch/scratch/octave/](http://christianjaeger.ch/scratch/octave/)
(The code is written in Scheme with some libraries that I haven't published so
far.)

------
ChuckMcM
There is a great book, "Musical Applications of Microprocessors" [1] which has
an excellent treatment of this. In my early experiments the floating point
accuracy of 8bit micros was poor enough that even tempered scales were pretty
hard to do. That said, the article describes the different techniques
exceptionally well.

[1] [http://www.amazon.com/Musical-Applications-
Microprocessors-H...](http://www.amazon.com/Musical-Applications-
Microprocessors-Hayden-microcomputer/dp/0810457539)

------
ArkyBeagle
Equal Temperament is an abstraction - different instruments will have an
entire discipline of temperament. One of the more complex suites of
temperament regimes are the temperament regimes used on pedal steel.

Getting a steel to "do" ET is actually an achievable engineering problem, but
players may prefer the sound of something more akin to Just intonation and
older guitars may not be able to be in ET. Guitars with all the parts to
achieve ET are relatively obscure.

------
pervycreeper
Can anyone give a mathematical explanation of how to find n such that equally
tempered divisions of the octave into n parts give rise to consonant
intervals? I have read that 53 works even better than 12, but I wonder if
there are more (infinitely more, I would imagine) exponents, and how this
sequence (the next number is the lowest n that is better than the previous
one) can be found.

~~~
analog31
I have played with this question in the past, and never found a better
"formula" than a search along the following lines:

For each n, find the closest interval to a perfect fifth (3/2 ratio), and
write down the error in octaves. Graph error as a function of n, and you will
see certain values of n with low error. On the same graph, do the 4/3 ratio,
and so forth, for a small handful of small-numbered ratios.

Somehow 19 springs to mind as having reasonably consonant intervals, but I
don't remember for sure.

12 is the smallest, and is also an auspicious number in ancient cultures.

~~~
malexw
I built a tool to compare the relationship of different n-note equal
temperment tunings a few years ago[1]. The UI is a bit obtuse, but it compares
the frequencies of the n-note system with the frequencies of the older just
intonation system. In 19 Tone Equal Temperment, the Eb and A are within 1 cent
of what they would be in a just intonation tuning, which is pretty cool. I
find 22 TET interesting as well.

[1]
[http://www.eng.uwaterloo.ca/~mawillia/music/](http://www.eng.uwaterloo.ca/~mawillia/music/)

------
icecreamguy
I had a music teacher who told us that when one of Pythagoras' disciples
pointed out this discrepancy, Pythagoras threatened to kill him if he should
ever reveal it to anyone else. Obvoisly hard to verify but a funny story
nonethless!

~~~
zb
[http://en.wikipedia.org/wiki/Hippasus_of_Metapontum](http://en.wikipedia.org/wiki/Hippasus_of_Metapontum)

------
sopooneo
I expect it is obvious, but why is the ratio of 3/2 called a "fifth"? It's not
because 3 + 2 = 5 is it?

~~~
salmonellaeater
"In classical music from Western culture, a fifth is the interval from the
first to the last of five consecutive notes in a diatonic scale."[1]

CDEFG -> CG is a fifth

[1]
[http://en.wikipedia.org/wiki/Perfect_fifth](http://en.wikipedia.org/wiki/Perfect_fifth)

