[Andy Evan]

Andy Evan asked the Naked Scientists:

For hundreds of years the musical octave from C to C has been understood. Yet its only in recent science that we understood that every higher octave is a doubling in frequency.

How was an octave defined those hundreds of years ago? How did it end up being exactly a doubling of frequency?

Perhaps it is to do with the beat notes you can hear within harmonics?

What do you think?

[DoctorBeaver]

There are certain combinations of notes that sound "pleasing" to the ear. They tend to be those whose frequencies have an integer ratio.

So, a major 3rd (C¹ & E¹) sounds pleasing; a diminished 9th (C¹ & Db²) doesn't.

An octave has a 2:1 ratio. That is the most basic integer ratio one can get.

There are exceptions to the rule, obviously. Take a maj 7th chord (C E G B). One would think the C & B would clash. However, B is the maj 3rd of G and so is in harmony that way. Some jazz chords use many apparent discordant combinations that when combined are actually pleasing to the ear. That is especially true of transitions where the intermediate notes are used to move from 1 chord to the next. Although the transition may seem discordant, reaching the target chord brings the whole sequence to a pleasing end.

One of the greatest exponents of using apparently discordant chords in an ear-pleasing way (to some, at least) was Bartok. You look through some of his scores and think "How the hell can that work!?". But it sounds "right".

Also, some apparently cordant combinations can be made to sound unpleasant. For instance, consecutive 5ths can sound very wrong when played in a high register; but, as any heavy metal fan can verify, when played in a low register they can add power & depth to the music (in heavy metal music they're referred to as "power chords"). That, though, has nothing to do with ratios.

[DoctorBeaver]

P.S. Sorry about my digression in my previous reply. I was on a roll & got a bit carried away  [:I]

[RD]

Andy Evan asked the Naked Scientists:

For hundreds of years the musical octave from C to C has been understood.
 Yet its only in recent science that we understood that every higher octave is a doubling in frequency.

How was an octave defined those hundreds of years ago? How did it end up being exactly a doubling of frequency?

If you halve the length of a plucked string its frequency doubles: its pitch goes up an octave.
http://en.wikipedia.org/wiki/Vibrating_string 

[BenV]

If you halve the length of a plucked string its frequency doubles: its pitch goes up an octave.
http://en.wikipedia.org/wiki/Vibrating_string 

Exactly.  So 12th fret on a standard guitar/bass should be exactly halfway between bridge and nut. (For those of you that don't play the guitar, 12th fret sounds an octave above the open string)

[DoctorBeaver]

Andy Evan asked the Naked Scientists:

For hundreds of years the musical octave from C to C has been understood.
 Yet its only in recent science that we understood that every higher octave is a doubling in frequency.

How was an octave defined those hundreds of years ago? How did it end up being exactly a doubling of frequency?

If you halve the length of a plucked string its frequency doubles: its pitch goes up an octave.
http://en.wikipedia.org/wiki/Vibrating_string 

That's basically what I was getting at.

[Bored chemist]

Also, it sounds "right".
If you play a note and the octave above the higher note will (except in the weird case of a pure sine wave) already contain the higher note as one of the overtones.
http://en.wikipedia.org/wiki/Overtones

[lyner]

There is a subtle difference between a harmonic and and overtone.
A harmonic is an integer multiple of a fundamental but an overtone is the frequency of the which a musical instrument will actually produce in a higher register. Strings behave pretty well and the two tend to coincide.(n.b. the second harmonic is actually the first overtone and so on)
BUT wind instruments can go very much adrift because of the 'end effects' at the mouthpiece and 'bell'. The frequencies of overtones get further and further from the harmonics as you go higher. To get the notes to sound right, you have to 'pull the notes' with lips and strength of blow.

[DoctorBeaver]

Also, it sounds "right".
If you play a note and the octave above the higher note will (except in the weird case of a pure sine wave) already contain the higher note as one of the overtones.
http://en.wikipedia.org/wiki/Overtones

That has caused me problems a few times. The overtone comes out louder than the root note. I've had to mess around with filters and narrow-band compression to alleviate it.

It's not just pure sine waves. I've managed to get some quite strange effects mixing diferent wave types on my synths.

[DoctorBeaver]

BUT wind instruments can go very much adrift because of the 'end effects' at the mouthpiece and 'bell'. The frequencies of overtones get further and further from the harmonics as you go higher. To get the notes to sound right, you have to 'pull the notes' with lips and strength of blow.

I remember well the first time I tried to play a saxophone. Oh dear  [xx(]

I had a friend, Willy Garnett, who was a sax/clarinet session musician (he was the sax player for Zoot in The Muppets). I remember him buying a new sax and the first thing he did was drop it upright onto the floor. I asked him why he did it and he said that a little dent helped with the overtones so he didn't have to work so hard while playing it.  []

[lyner]

All this 'reality' must be what makes instruments sound so interesting.
It also applies in the design of quartz crystals and crystal oscillators. Some crystals are specified to operate on an overtone and are supposed to be used in an appropriate circuit. Others are supposed to operate on their fundamental frequency and for the following circuit stages to select the appropriate harmonic. You gotta read the small print on the side!
The crystal is supported like a glockenspeil bar and the supports are not exactly at the ends, the exact position being appropriate for only one particular mode of oscillation to be accuracte.

[rosalind dna]

http://en.wikipedia.org/wiki/Octave
I can't really tell the difference but do know that an Octave is the
8th note as it's latin.

[BenV]

http://en.wikipedia.org/wiki/Octave
I can't really tell the difference but do know that an Octave is the
8th note as it's latin.

The easy way to spot an octave is to sing "Somewhere Over the Rainbow"

Some -> Where is an octave jump.

[lyner]

The next question is "why do we use an eight note scale when there are twelve possible semitones available to choose from?"

[DoctorBeaver]

The next question is "why do we use an eight note scale when there are twelve possible semitones available to choose from?"

AN 8-note scale? There are lots of scales (OK, most are technically modes) and, strictly speaking, they are 7-note not 8-note. When playing a scale, the octave is actually the start of the scale in the next register. If you play 2 consecutive scales there are 15 notes; 3 scales, 23 notes, etc.

A mode differs from a scale only insofar as it uses the same notes as a given scale but each mode starts on a different note. Each note in a scale has a mode, so there are 7 modes.

Taking the scale of Cmaj as an example, we have the notes CDEFGAB. So, the notes of the Cmaj modes use only those from the Cmaj scale. Starting with each note consecutively we get:-

mC Ionian     - CDEFGABC
mD Dorian     - DEFGABCD
mE Phrygian   - EFGABCDE
mF Lydian     - FGABCDE
mG Mixolydian - GABCDEF
mA Aeolian    - ABCDEFG
mB Locrian    - BCDEFGA

Then there are the pentatonic (5-note) modes. These are based on the 7-note modes but with the 4th & 7th notes removed.

Most Western music is based on these scales or modes.

Then there are the more exotic scales & modes such as Inverted Diminished, Ultra Locrian, Neapolitan Minor, Maj.Phrygian (Dom), Marva, Locrian Natural 2nd or Mixolydian Aug. - but I won't bore you with the technicalities.  []

[lyner]

When you turn on the radio (in the UK, of course), you tend to get the well known one I'm referring to.
The question is still there, though. Why only pick some of the notes once you have chosen 12 semitones? And why choose 12 semitones?

[DoctorBeaver]

Blame the ancient Greeks. The tuning of Greek lyres was based on what is known as diatonic tetrachords. A tetrachord, as the name implies, is a chord made up of 4 notes. A diatonic tetrachord consisted the root note, 2 whole tones & a semitone. So, consecutive diatonic tetrachords would span an octave (e.g. CDEF & GABC). Although other tunings were used, this is the 1 that gained favour and became the basis of Western music.

I once heard the theory put forward that the diatonic tetrachord reflected how Alexander the Great's mother would say his name. I don't put much credence in that theory although the development of tetrachords was contemporaneous with Alexander's later military exploits. I think it more likely that Alexander's conquests would have inspired many Greek songs praising him and, if the diatonic tetrachord were gaining favour at that time, it stands to reason that there would be a great increase in its use.

[lyner]

are you calling me a lyre?

[DoctorBeaver]

I don't like your tone!