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KVRist
 
282 posts since 14 May, 2002, from feather island
 

Postby Taika-Kim; Thu Dec 13, 2012 2:20 am

Something I didn't see yet discussed here is the effect of the ratios of the overtone series has on the instinctive habit of tuning the instruments.

I wholeheartedly recommend Sethares's book Tuning, timbre, spectrum, scale to everybody, it's the only book in acoustics I've read one one sitting, too fascinating stuff!
http://sethares.engr.wisc.edu/prelude.html

The idea is, that because of practically all instruments that we have in the west have an overtone structure following an 1:2 octave, so we have instinctively approximated certain ratios of intervals from this.
And the instruments that by nature would have different overtone ratios (like free bar instruments like xylophones) are tuned by overtones to fall in more nicely with the octave-based instruments.

He writes a lot about gamelan, and how the instruments in that family are tuned to achieve close to the same ratios that we have in our music too, altough by fundamentals the scales look different.

He also presents formulas to calculate scales for instruments with any overtone content, and the other way around.

This 20 second video presents his idea very well:
http://eceserv0.ece.wisc.edu/%7Esethare ... alloct.avi

"Challenging the octave", where a 2.1 overtone ratio sound plays dissonant in a 1:2 octave, but consonant when the octave is stretched.
KVRist
 
448 posts since 5 Mar, 2005, from USA
 

Postby AnalogGuy1; Thu Dec 13, 2012 8:12 am

Michael1985 wrote:Why we use exactly 7 notes, not 8, 9, 10 or even 11 or 12???


There are plenty of scales that use other numbers of notes. Pentatonics use 5, some blues scales use 6, 7, or 8, diminished scales use 8, chromatics use 12... I think you mean "why do diatonic scales use 7?". The real question is not "why do major scales have 7 notes", but "why are you asking about a 7 note scale, rather than a 5 note scale, or a 12?". It's like asking "why do septagons have exactly 7 sides, not 8, 9, 10, or even 11 or 12????"

It would be different if you asked "why where the particular whole and half step sequences chosen for the 7 notes of the major scale taken from the chromatic scale, rather than some other sequence, like 6 half steps in a row followed by a big jump?". That answer is: the ratios between frequencies of the root note and any of the notes from the major scale (except the 7th degree) we are biologically hardwired (not just culturally) biased to find sonorous. It was first observed by Pythagoras that we find low-order ratios of note values pleasing, and every culture recognizes the importance of the octave.

Not every culture recognizes a 12 note chromatic scale, but most scales in all cultures recognize scales in which note frequencies are close to being a simple frequency ratio from the root. Scales in which more ratios are more nearly simple frequency ratios, like the major scale, tend to be more popular than scales in which fewer frequencies are simple ratios from the root, like the whole tone scale.

I don't know why humans are biologically wired to find simple ratios (the simplest being 1/2 and 2, ie octaves), although I would guess it is because our brains find it simpler to process regularly-spaced harmonics (in a logarithmic scale) than irregularly spaced ones because our cochlea is in a logarithmic spiral. If that is correct, then your question really is: "why did we evolve ears in a logarithmic spiral, and not in some other shape?".

I think I'll make some music now.
KVRian
 
1097 posts since 10 Oct, 2004

Postby JumpingJackFlash; Thu Dec 13, 2012 8:59 am

AnalogGuy1 wrote:That answer is: the ratios between frequencies of the root note and any of the notes from the major scale (except the 7th degree) we are biologically hardwired (not just culturally) biased to find sonorous.


This is rather simplistic, and certainly not true in equal temperament.
In any case, it doesn't explain why we have a 7-note scale, nor why the intervals in the scale are arranged as they are.

The idea of being "biologically hardwired" is also rather debatable.
Unfamiliar words can be looked up in my Glossary of musical terms.
Also check out my Introduction to Music Theory.
KVRist
 
448 posts since 5 Mar, 2005, from USA
 

Postby AnalogGuy1; Thu Dec 13, 2012 9:43 am

JumpingJackFlash wrote:
AnalogGuy1 wrote:That answer is: the ratios between frequencies of the root note and any of the notes from the major scale (except the 7th degree) we are biologically hardwired (not just culturally) biased to find sonorous.


This is rather simplistic, and certainly not true in equal temperament.In any case, it doesn't explain why we have a 7-note scale, nor why the intervals in the scale are arranged as they are.


Yes, it THE explanation for why intervals are arranged as they are in the diatonic scale. For example, three very common chords in modern music are the tonic, the fourth, and the fifth. Compare these intervals with a note interval not in the diatonic scale, for instance the flatted fifth.
Code: Select all
             Exact   Rat approx to 0.0015  In equal tempered diatonic
unison          1             1/1          YES
fourth        1.334           4/3          YES   
fifth         1.498           3/2          YES   
flat fifth    1.414        1393/985        NO

JumpingJackFlash wrote:The idea of being "biologically hardwired" is also rather debatable.


Well, there's about 2,500 years of published scientific literature here, from Pythagoras who first reasoned the human appreciation of harmony stems from a mathematical basis that appeals to something innate in the brain ("Manual of Harmonics" as quoted by Nicomachus, p66) to very modern works (e.g. Professor David Schwartz from Duke Universities well-known Purves Lab, who says: "...evidence suggested this particular ordering of tone is much the same among people of different societies, even among infants. So it seems to be somewhat independent of the musical environment in which one grows up. It seems to be telling us something fundamental about the human auditory system."
KVRist
 
448 posts since 5 Mar, 2005, from USA
 

Postby AnalogGuy1; Thu Dec 13, 2012 10:01 am

JumpingJackFlash wrote:In any case, it doesn't explain why we have a 7-note scale, nor why the intervals in the scale are arranged as they are


Sure they do. Try working out an example for yourself. Take a simple one: a 4 note equally tempered scale. To the same rational approximation the intervals there are:

Code: Select all
interval  exact   rat approx
unison      1         1/1
first       1.892     1785/1501
second      1.414     1393/985
third       1.618     3002/1785


You'd only have one sonorous note out of the whole scale, the root. Yuck.

As vurt said at the very start of the thread, it is all ratios.

Tappermike? Jancivil?
KVRian
 
1097 posts since 10 Oct, 2004

Postby JumpingJackFlash; Thu Dec 13, 2012 10:24 am

AnalogGuy1 wrote:Yes, it THE explanation for why intervals are arranged as they are in the diatonic scale.


I meant why the major scale goes tone-tone-semitone-tome-tone-tone-semitone.
The perfect fifth and perfect fourth might be very simple rations, but the major second or major sixth significantly less so.

AnalogGuy1 wrote:Sure they do. Try working out an example for yourself. Take a simple one: a 4 note equally tempered scale. ... You'd only have one sonorous note out of the whole scale, the root. Yuck.


The same is true of the 7-note equally tempered scale which we use now. Yuck!

AnalogGuy1 wrote:As vurt said at the very start of the thread, it is all ratios.


That's too simplistic. Ratios are part of it yes, but there's a lot of historical and cultural factors too.

But we've been through this several times on here already.
Unfamiliar words can be looked up in my Glossary of musical terms.
Also check out my Introduction to Music Theory.
KVRist
 
448 posts since 5 Mar, 2005, from USA
 

Postby AnalogGuy1; Thu Dec 13, 2012 10:29 am

Do you want me to work out the tables for every note of the major scale, and compare it to every note of a different scale to show you? I thought the point was clear by choosing some common examples.
KVRian
 
1097 posts since 10 Oct, 2004

Postby JumpingJackFlash; Thu Dec 13, 2012 11:37 am

If our major scale really was superior, mathematically perfect, biologically wired and all the rest of it, then every culture in the World would use it. But they don't, only ones who have been influenced by European history/culture.

The 7-note scale goes back at least as far as the ancient Greeks who combined tetrachords to form their modes.
Unfamiliar words can be looked up in my Glossary of musical terms.
Also check out my Introduction to Music Theory.
KVRist
 
448 posts since 5 Mar, 2005, from USA
 

Postby AnalogGuy1; Thu Dec 13, 2012 8:26 pm

I'm with you, dude; the Pythagoras work I referenced said just that.

But there is a misunderstanding; I did not say the only scale that has some nice ratios is our major scale. Plenty of other non-Western cultures have other scales with nice ratios that are also popular, even though they may use different numbers of notes. The Middle Eastern Hejaz scale, for instance, consists of just five notes, the most commonly used of which form simple frequency ratios. The Pelog scale of Bali and Java in Indonesia are perhaps the scales from a culture as different as I can think of from the Western world; they are not even close to being equally tempered, and yet, the notes they chose are within a few percent of being simple frequency ratios.

So I agree with you completely that not all scales come from the Western world, and that many have different numbers of notes, and temperaments. Yet, there seems to be a consistent gravitation towards a preference of simple frequency ratios, in a manner that transcends culture, and these preferences have been described by many, from the ancients to modern well-regarded scientists, as I earlier gave referenced examples.

There are exceptions. There is an Indian concept of intervals, called Shruti, which were once regarded as being the smallest perceptible difference between two notes. These do not have a small frequency ratio separating them. There are 23 Shrutis in an octave. But even here, notes that are 9 and 13 Shrutis from each other are considered by the system to be consonant. Although the exact ratios are irrational (2^(9/22) and 2^(13/22)), they work out at just a few cents away from 4/3 and 3/2.
KVRist
 
448 posts since 5 Mar, 2005, from USA
 

Postby AnalogGuy1; Thu Dec 13, 2012 8:49 pm

It just struck me that the OP may want to check out some of the peer-reviewed literature that is the result of many thousands of hours of research and experiment in the field, since he has asked the question on KVR a couple of times. Besides the work I quoted earlier, read some of the papers by Bill Sethares. It is accessible, well-regarded, and explains just what the OP asks, and in the context of the multiple-cultures / non-Western question JumpingJackFlash posits. Vurt's already nicely summarized the findings, if a bit briefly.
KVRist
 
81 posts since 26 Nov, 2009

Postby anomandaris1; Fri Dec 14, 2012 1:41 am

The major scale has got variations - just change the notes in the upper tetrachord. If your melody doesn't sound good, the problem is your phrasing. And even random collection of notes can sound great, if it is played great. I heard once two clarinetists improvising in two voices over simple major chords - they were using double harmonic major with chromatic ornamentations. The tempo was 220 to 300 bpm, 7/4. Even with the beating between the melody and the harmony - and between 1st and the second clarinet which was playing 3rd above the first - it was great sounding.
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KVRAF
 
3946 posts since 28 May, 2005, from Netherlands

Postby Nielzie; Fri Dec 14, 2012 2:03 am

I'd like to think of it like they are colors. "Colors" for the ear that is. Some colors blend very well with each other, sometimes making new pretty colors, and some just don't. Some colors can influence certain emotions. In some cultures or religions some colors have more meaning then others and are therefore more liked then others. Etc. Etc.

And it's often the drawings and the lightfall that makes the colors really shine (as in composition and production).

But why? Dunno, something to do with how the brain works I guess.. :shrug:
RIP Reason Lahalla
KVRAF
 
3447 posts since 19 Jan, 2008

Postby tapper mike; Fri Dec 14, 2012 7:52 pm

Pythagoras's work was part scientific and part wishful thinking. We've come to know the 7 tone diatonic scale from him however it was a mean as opposed to justified system and not wholly based on the 12 tone chromatic scale we know today. he started off with 7 repeatable tones that were very easy to identify and build modes from. However upon later inspection he found 5 more repeatable tones and somehow had to justify them to the system he had already created. Had he derived the 12 tone chromatic system first music itself would be much different today.

Temperment, tuning and intonation have had a long journey to get to where we are today. Justified tuning is a compromise so that all twelve tones resonate fairly consistently. Equal Temperment is a compromise a common, effective compromise especially for playing in tune across multiple keys.

http://en.wikipedia.org/wiki/Equal_temperament
http://en.wikipedia.org/wiki/Well_temperament
http://en.wikipedia.org/wiki/Just_intonation
http://en.wikipedia.org/wiki/Meantone_temperament
http://en.wikipedia.org/wiki/Regular_temperament

Here is a nice video on different temperaments
http://www.youtube.com/watch?v=APtJsaPx ... F738EF9997
205 pitches per octave as opposed to 12

So it's not just ratio's and the 12 tones do not sit and play well with each other if you base it all on simple defined ratios.
So it's not universal and usually even using western musical standards it's a compromise from the start even before we get into intonation and defining what the value of A is. A@440 is concert pitch. It's long been the standard in the music industry and produces the most consistent results as well it is of medium resonance. Not to sharp or bright/brittle and not to flat or dull.
Well on piano's and guitars with relatively fresh strings.

Even the best of standard guitars have intonation issues when the intonation is set to the standard matching the octave at the twelfth fret with the harmonic at the twelfth fret. One would need to have a very complex setting of separate frets for each string at each location to provide a unified equal temperament across the guitar. While there are compound scale necks and bridge systems that try to accommodate a truer intonation
Image

Generally they are the exception not the rule. Guitarists deal with an even less then perfect equal temperament system then do keyboard players. We just come to accept the slightly more unstable harmonics when playing chords which is where it counts the most (the second most is when single notes are played against a harmony)

I've had the displeasure of playing a far from properly intonated guitar. and it drove me nuts. Anything after the 6th fret was way off the mark. What drives me even more nuts is when someone won't tune their guitar or tune it by ear when they don't have perfect pitch. USE A FREAKING TUNER THAT'S WHY THEY MAKE EM. When someone is in tune but not in A440 and the rest of the band is it sounds terrible almost as bad as when they are completely out of tune.

I'm slightly more forgiving (as most peoples ears are) when it comes to intonation on fretless instruments and wind/brass instruments. In regards to wind instruments resonance and therefore intonation has much to do with how the mouth is formed around the mouthpiece. As well instruments like the trumpet and trombone have note formation to center the note based on how they blow. When I hear a perfect brass harmony the first thing I think is...Fake/synth. That's because I expect it not to be perfectly intonated. Many softsynths accommodate for this unevenness to make it sound real by adjusting the notes a few cents above or below the mark.

Fretless string instruments in unision are much like a chorus effect. Thats because a chorus effect takes the original signal splits it and slightly detunes the secondary signal as it is returned in a wave shape. If two violins were perfectly playing the same piece it wouldn't amplify the feeling of a true two instrument presence it would just sound like one instrument or like a generic string synth. Unison in a string section actually smooths the sound for me as opposed to a single violin playing a single part.

The one fretless instrument I do take sever umbrage with when it's wildly out of tune or intotation is the double bass or fretless guitar especially in the upper range of the instrument.

Getting back to the Guitar for a moment. When you strike a sting with your fingernail, a pick or the flesh of your finger the string is initially pulled out of tune as the string enters it's path of vibration it becomes more in tune. It's more dramatic on a guitar with a tremolo bar then one with a fixed bridge. Uncharacteristically tapping is a more "in tune" method of approaching the note so as the string vibrates it's starting in on the proper cycle faster.

With the voice I feel I need to explain this. there is a very center of the notes tuning and there are sides flat and sharp that are close to the mark but not on the mark. The character of the voice can overshadow the note accuracy. Rod Stewart has a far from A440 voice but due to the character of his voice it can be slightly off the mark and still be close enough like horseshoes and hand grenades. One of my favorite singers of all times Al Green had a method where he would come in just below or above the target then move into the target zone. Having perfect pitch as rare as that is does not mean one can sing or even sing in key. It just means you can recognize notes without support of other notes and recognize how flat or sharp they are from the center. If there were such a thing as the universality of intonation and temperment we'd all have perfect pitch and anyone would be able to figure out any song just by listening once. Being able to play it with the correct inflection is a different matter. Usually those with perfect pitch can hit the notes because they do on a regular basis not because they don't and just imagine them.
KVRer
 
8 posts since 19 Dec, 2012

Postby djtrancendance; Wed Dec 19, 2012 2:28 pm

The reason a scale works isn't so simple as "it's a harmonic series" because, if it was, everyone would be using a straight harmonic series scale of 7:8:9:10:11:12:13:14 (where 14/7 = the octave) where the entire scale is one huge chord.

The point, when working with scales that span multiple octaves (e.g. most as music does) is to make the ratios in a scale contain several low-numbered factors, such as 2,3, and 5...that work well across many octave.

7:8:9:10:11:12:13:14...doesn't do so well as, over the second octave, it becomes

7:8:9:10:11:12:13:(octave)14:16:18:20:22:24:26:28...then you get high ratios such as 28:13 in the second octave and complex "chords" like 12:13:28 (not very consonant/"sweet sounding").
-----
One way to get around this problem is to simplify the scale by making it out of fractions that contain many common factors, such as 2,3,and 7...such as in the scale below (note how elegantly the ratios simplify).

The cool thing is (shocker)...you can actually make your own new very harmonious scales this way! You aren't limited to the diatonic scale (ALA C-major, D-minor, etc.) or the whole tone or pentatonic scales or the one called "Pythagorean" either (all of the above are only slight variations of the same 'boring' thing)... This is where xenharmony/'microtonality' comes into play...

12th Harmonic Scale
12/12 = 1/1
14/12 = 7/6
15/12 = 5/4
16/12 = 4/3
18/12 = 3/2
20/12 = 5/3
21/12 = 7/4
24/12 = 2/1

So, here you get chords like 1/1 7/6 4/3 AKA 6:7:8 and 4/3 6/3 2/1 or 4:6:8 AKA 2:3:4 (simplified/reduced)...which are very simple and harmonious.
--------
Also, for reference, here is what C-major looks like in harmonic series/fractional format:

The 7 tone diatonic scale (virtually the same as "C Major" on your keyboard)
24/24 = 1/1 (C)
27/24 = 9/8 (D)
30/24 = 5/4 (E)
32/24 = 4/3 (F)
36/24 = 3/2 (G)
40/24 = 5/3 (A)
45/24 = 15/8 (B)
48/24 = 2/1 (C)

And, if you ever wondered how the Middle Eastern scales (e.g. Rast) or Blues scales (complete with blue tones not available on standard keyboards) simplify, it's virtually the same method:

Middle Eastern / Blues scale based on the number 18
18/18 = 1/1 (C)
20/18 = 10/9 (a bit below D)
22/18 = 11/9 (E neutral)
24/18 = 4/3 (F) 8
27/18 = 3/2 (G) 9
30/18 = 5/3 (A)
33/18 = 11/6 (B neutral) 11
36/18 = 2/1 (C)


___________________________

For more check out my page:


http://xenharmonic.wikispaces.com/Mike+ ... eries+Page
KVRAF
 
3447 posts since 19 Jan, 2008

Postby tapper mike; Wed Dec 19, 2012 2:44 pm

If you follow early greek classicalism that is exactly what a chord is.
http://en.wikipedia.org/wiki/Tetrachord

As well if you follow Alan Holdsworth he too thinks in the context of chord scales.

Even further down the line if you study jazz extended chords as I have already explained you get the whole scale in the chord. Such as the mixolydian more while playing a 13th chord. 1-3-5-7-9-11-13
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