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12

You are exactly correct that it is the logarithmic nature of pitch that causes this effect. In cases like this, I find that a picture is helpful. Here I've labeled equally spaced octaves (1200 cents) along the x-axis (representing pitch). I've then labeled the corresponding frequencies on the y-axis as multiples of some arbitrary base frequency f. Note that ...


10

A trivial answer : yes. When I was quite young I wrote a computer program to spit out a succession of 'beeps' at random frequencies not related to any musical scale; I suspect many people who have a computer and a bit of an interest in music have done the same. In practice how close you could get to infinity (!) would be limited by the resolution at which ...


7

A pithy way of saying it is that intonation is the process by which a temperament is achieved. Intonation is what is done in order that the sound is produced at the desired/intended pitch. This can be done as part of instrument setup, e.g. "setting the guitar intonation", or as an integral part of performing the music, e.g. as in expressive intonation. ...


6

As some of the other answers have eluded to, there are two basic problems with your question: The first is the question of how you generalize a "tritone" in a non-12-TET based system. One possibility is to interpret it literally as three whole tones (which then begs the question as to how you define a whole tone in a non 12-tone system). Another ...


6

"Harmonies" by Gyorgy Ligeti is an interesting example of microtonal music. It's written for organ, but it's intended to be played with reduced air and manipulation of the stops, so the pipes don't play at their designed frequency. With (mainly) slow chord changes and wide voicings the overall effect is a slowly evolving harmony and dissonance through the ...


5

It depends on how you define the 12-tone octave. Up until at least the 19th century, several theories did not use equal temperament to conceptual musical space. Imagine that you ascend from C by perfect fifth; if you do this twelve times in equal temperament, you will end on a C that is perfectly in tune with the original starting C. If, however, you do ...


5

While 1.5 lies perfectly in the arithmetic middle of 1 and 2, the arithmetic middle is not relevant for music. The intervallistic middle of two frequencies is their geometric mean. Go two octaves up, and your frequency goes from 1 to 4 times its value. But one octave up is not 2.5 times the frequency, but rather 2 times. So if you have two notes with ...


5

I would just add the (possibly obvious) answer that a capella choirs can also drift off because of singing out of tune. Typically, they tend to get flatter if the music has lots of jumps to high notes, which they don't quite get up to, and they sometimes get sharper if they are nervous about getting flat. I've experienced both in concerts.


5

Well, how about the starting clarinet solo in Gershwin's "Rhapsody in Blue"? It's been almost a century ago. Granted, doing the glissando continuously on its last part was not written into the score originally but was rather an impromptu trick by the clarinetist that the composer then insisted on incorporating into the premiere, but it has been very much ...


4

I would like to add the point that the comma pump may happen in either direction, resulting ascending or descending drift. However, tonal music is such that the intervals between the roots (the fundamentals of the chords) usually appear in one direction and not in the other: descending fifths or ascending fourths and ascending seconds, mainly. As a result, ...


4

Note: For the sake of discussion, I'm limiting myself here to equal temperaments, which is the most common way of tuning keyboards. Other systems exist, of course, but would probably only confuse the matter. Why do B and C and E and F not have a sharp note between them? Simply because, acoustically speaking, there is no room in our current system for ...


4

The MIDI Tuning Standard allows for arbitrary remapping of all 128 note values. It was ratified in 1992, and can be implemented by both GM and GM2 devices. (Very few do, however.) There are also Scale/Octave Tuning messages, which allow slight adjustments to the 12 tones in an octave. Only these are required by GM2.


4

Yes, many old organs were built with only some of the 12 chromatic keys in some octaves (particularly the lowest one, since the biggest pipes are the most expensive ones). The reason is that the more remote chromatic tones were rarely used in compositions of the time, and so this saved a lot of money for only a little inconvenience. The same was also done ...


3

The octave can be split into more intervals than 12. There is 19 equal temperament, and other temperaments based on octave division into 31, 41 or 53 equal intervals. Some 30 years ago I had to work on a mathematical problem where it was proved that some divisions were "better" than others. Better in the sense that they better approximated simple fractions. ...


3

theory that works on a continuous (or non-quantized octave). In musique concrète, the theoretical framework is audio waveforms, not octaves or tunings or scales. You’re not limited to a system of writing down musical notes so that the next person can play them on a standard instrument with a standard octave and standard tuning, because you write down ...


3

In terms of frequency ratios "flattening" is not "subtracting" at least not mathematical subtraction. In the way that you are expressing it the mean tone fifth would be (3/2)/[ (81/80)**(1/4)]=1.495... The reduction is achieved by division. Often you want to think about things in terms of cents: a logarithmic measure of pitch. By working with these the ...


3

First of all, putting a capo on does not change the temperament of your guitar. Your guitar temperament is equal temperament so and that's not changing so forget about the temperament aspect of this.. The only thing it does is change what strings are "Open". So without a capo, your open strings are the typical E-A-D-G-B-E. For every fret you move it, all ...


3

There's scattered support for other 12-tone temperaments, but MIDI just isn't going to be able to work with tuning systems with more notes. It's an issue of the amount of information that a MIDI message can encode--the existing standard is for a 7-bit (128-value) note number, which is enough to encode over 10 octaves of 12-tone, but only 5 octaves of ...


3

It is significant when you are trying to tune an instrument by ear, using the purity of intervals as your guide. You (and the pages you link) refer to jumping up 7 octaves vs. 12 fifths, but don't forget that any notes you reach that way can also be brought down by one or more octaves as well. To illustrate this, let's bring all the notes down into the same ...


3

It is only significant to someone designing an instrument with a finite set of fixed pitches like a keyboard. Tuning such an instrument in a system based on pure fifths causes octaves to not be equivalent - which is terrible because the octave is the only truly pure interval, you can hear the beats from a mile away because it doesn't form a difference tone ...


2

If you are talking about microtonality - of which I know little, there will have to be a lot more than just changes to E/F and B/C. It's possible to have notes between any adjacent semitones. There could be as many extra notes between G and G# as between E and F. It just happens that it's accepted (and has been for centuries) that the note called F is ...


2

Any single string creates harmonics as it rings. Up to eight parts a C would generate C C G CEG Bb and C, essentially a dominant 7th chord. The lower partials are louder and than the upper partials due to string length influencing volume. A longer portion of the string vibrating will be louder. Tempered tuning requires that octaves and fifths sound good ...


2

Why do you require abbreviation? If there's a perfectly good term for this that doesn't use an abbreviation, will it be acceptable? "Notes per octave" or "pitches per octave" seem pretty widely used, universally understood, and tuning-agnostic. As an extension of this, scales themselves can be described as n-tonic, where n is a Greek number (as in, ...


2

A capo is a transposition device: everything gets moved to a higher pitch while retaining the same voicing that you have in the lower position. The resulting chord voicing may differ from playing the chord with the same "name" without capo. As an example, playing G major without capo will usually be voiced as G B d g b g', so 1-3-5-8-10-15 in scale steps. ...


2

For practical musical performance purposes, a capo simply transposes up by a semitone per fret. All notes are "theoretically" simply higher by the number of semitones times the number of frets you move the capo. In a perfect world, what the capo attempts to do is the same effect as re-tuning your guitar one half step sharp for each capo position. In the ...


2

Mathematically, everything about pitch is logarithmic. "Adding a perfect fifth" really means "multiply the pitch by 3/2. So "subtracting a syntonic comma" means multiplying by the reciprocal of the comma; and "a quarter of" means the fourth root of. So try working out: 3/2 x 4th-root(81/80) ...and see if this is more like the correct answer. (I haven't ...


2

John Luther Adams created a sonification for weather, astronomical and geological data in real time, called The Place Where You Go to Listen The sound parameters (mostly pitch, by I think others too) change "continuosly" (between comas, as of course we are talking about discrete digital events incrementally changing in time) according to the actual external ...


1

That ratio applies to the frequency which is an absolute measurement, not cents which more of a relative distance measurement between notes. They are different in nature. Just a simple example, the perfect fifth above A4 (440 Hz) is E5 (660 Hz Just intonation/ 659.26 Equal temperament). This is where it makes sense to describe the interval in a ratio. A4 ...



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