GNU Octave is a mathematics program, mostly compatible with Matlab. It runs Matlab's scripts, and interfaces nicely with gnuplot. It has a powerful macro language.

Very good choice if you need to do big or small calculations...

Here's a geekier take on it.

An octave is an exact doubling of pitch, a.k.a. frequency. A above middle C is canonically defined as 440 Hz1 (two octaves above the A string on a guitar which is 110 Hz). What that "440 Hz" means in plain English2 is that the thing producing the sound is making four hundred and forty complete back-and-forth motions per second3, and pushing the air back and forth at the same rate.

Within an octave, the frequency of each note (semitone, I should say; we're using both black and white keys) is greater than the next lower one by a factor of 1.0594631 (4), the 12th root of 2. Why? Because that way, you double your frequency every twelve semitones, and the pitches you get happen to be at useful ratios from each other. Two pitches will sound nice together5 if the ratio of their frequencies is relatively simple: An octave (A + A, 2:1) sounds very pleasant, or "consonant"; a "perfect fifth" (A + E, 2:3) sounds nice, and a perfect fourth (A + D, 3:4) also sounds nice. Weird or "dissonant" ones like the augmented fourth (A + D#, 32:45) sound like crap, but they can be useful as flavoring: You wouldn't eat a handful of salt, but a little bit is a good thing.

The important thing here is that the ratio between two notes in one octave (say, A 110 and E ~164.8) will be the same as the ratio between the same two notes in a different octave (like A 440 and E ~659.3). And if A 110 sounds good with E ~164.8, you can safely assume that it'll sound good with E ~659.3 because that's the way fractions work.

I've left out some of the most interesting stuff, like harmonics, but this is probably enough for one day. Anyhow I'm not gonna try to explain harmonics without illustrations. Maybe when I hit level 6 I'll put a bitmap with some waveforms in my home node. Aren't you excited?!

1 Hz == Herz == cycles per second

2 There's no such thing.

3 This assumes that you've got it tuned perfectly, and in the real world that never quite happens, but it should be reasonably close. If it's a stringed instrument and you tune it much too high, you'll wreck the thing. The crucial thing is to agree with the other strings on the same instrument, and with any other instruments you may be playing along with.

4 This is tempered intonation. The first volunteer to explain just intonation vs. tempered intonation gets a cupcake and my undying admiration. In brief, tempered intonation is a pragmatic fudge factor thing: The ratios between notes are all made to come out slightly wrong, instead of some being perfect and others being lousy.

5 von Helmholtz spent hundreds of pages on the gory details, but "they sound nice together" will do fine :)

themusic is right about subjectivity and harmony: Music is full of math, but let us not forget that you can't reduce it to math. There's a lot of squishy stuff in there too, which I tend do discuss only with words like "whoa" and "cool". I'm just a fuckin' punk with a guitar; he's a musician. :)

That bit about F# and Gb scares me. With frets, it's the same note, so I can avoid thinking about it. And I will, too.

In music notation, an octave symbol is the characters "8va" or "8bassa" in a small italic font with a dashed bracket continuing to the right of it, drawn just above or below a number of notes and the staff. This symbol indicates that the notes under or over it should be played one octave (twelve half steps) higher ("8va") or lower ("8bassa") than indicated. It's used to keep the number of ledger lines from getting unreadable when more than one or two notes require them.

On the staff, octave symbols look something like this:

```
8va - - - - -
|
--*-
/\                * |
---| /-------------*-|--|---------------------------------
|/           * |  |  |
---/-----------|--|--|--|---------------------------------
/|           |  |  |                  |
-/-|/\---------|--|------------------|--|-----------------
|  |  |        |                  |  |  |
|--|--|------------------------|--|--|--|-----------------
\ |  |                        |  |  | *
--\|_/-------------------------|--|-*---------------------
|                           | *
\|                         -*--        |
8bassa - - -
```

Also, the eighth album by The Moody Blues. It was released in 1978 and marks the start of the band's decline from incredible concept album masterpiece rock into sometimes-lucid horrendous 80's synth pop. It was keyboardist Mike Pinder's last project with the band; he was only heavily involved with the writing process of one song, "One Step Into The Light." Octave was released by Threshold Records, and produced by Tony Clarke, who might as well have been in the band.

Tracks:
• Steppin' In A Slide Zone (this was the "big hit" of the album)
• Under Moonshine
• Had To Fall In Love
• I'll Be Level With You
• Driftwood
• Top Rank Suite
• Survival
• One Step Into The Light
• The Day We Meet Again

Oc"tave (?), n. [F., fr. L. octava an eighth, fr. octavus eighth, fr. octo eight. See Eight, and cf. Octavo, Utas.]

1.

The eighth day after a church festival, the festival day being included; also, the week following a church festival.

"The octaves of Easter."

Jer. Taylor.

2. Mus. (a)

The eighth tone in the scale; the interval between one and eight of the scale, or any interval of equal length; an interval of five tones and two semitones.

(b)

The whole diatonic scale itself.

⇒ The ratio of a musical tone to its octave above is 1:2 as regards the number of vibrations producing the tones.

3. Poet.

The first two stanzas of a sonnet, consisting of four verses each; a stanza of eight lines.

With mournful melody it continued this octave. Sir P. Sidney.

Double octave. Mus. See under Double. -- Octave flute Mus., a small flute, the tones of which range an octave higher than those of the German or ordinary flute; -- called also piccolo. See Piccolo.

4.

A small cask of wine, the eighth part of a pipe.

Oc"tave (?), a.

Consisting of eight; eight.

Dryden.