http://www.slate.com/id/2250793/pagenum/all/#p2
explains so much
evolution of tuning systems
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- KVRAF
- 2217 posts since 15 Jul, 2003
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- KVRAF
- Topic Starter
- 2217 posts since 15 Jul, 2003
I had often wondered where the notion of differnt keys having certain character (not just the Spinal Tap reference) and this offers the best explanation I've come across
I'd also heard in interviews how violin players had to adjust to playing with piano and had a vague notion of what that was about, but this is very clear.
It's not just me, there are historical reasons why most music was written in the keys between 2 flats to 3 sharps
it's also interesting from a science perspective of how we use math as a tool to model the world and in this very observable case, it falls a bit short
makes it even more of a challenge to understand how some concert pianist like to have their concert grand tuning stretched
I'd also heard in interviews how violin players had to adjust to playing with piano and had a vague notion of what that was about, but this is very clear.
It's not just me, there are historical reasons why most music was written in the keys between 2 flats to 3 sharps
it's also interesting from a science perspective of how we use math as a tool to model the world and in this very observable case, it falls a bit short
makes it even more of a challenge to understand how some concert pianist like to have their concert grand tuning stretched
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- KVRist
- 441 posts since 30 Apr, 2007
I don't think the article touches at all on stretched tuning, but it is a different example of the simplified model not meshing with reality. We would be taught that the "ideal" string has a harmonic that twice the fundamental, another thrice, another four times, and so on. But a real wire inside a piano has other issues at play which make the overtones a bit sharper than that. Stretched tuning matches the fundamental of one note with the real overtones of the notes that are octaves below it, instead of the simplified model.wrench45us wrote: it's also interesting from a science perspective of how we use math as a tool to model the world and in this very observable case, it falls a bit short
makes it even more of a challenge to understand how some concert pianist like to have their concert grand tuning stretched
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- KVRist
- 149 posts since 27 Jan, 2007 from Eyeth
Equal Temperament makes all keys (and their respective chords) to have the same character, since all intervals are identical.
Only the timbre of ideal strings corresponds exactly to the mathematical model of the overtone series, as you would describe it with Fourier's. Real strings have different stiffness, tension, width, etc. Piano strings are somewhat stiff, which results in inharmonicity (shift of the overtones from their ideal positions), so stretching is needed in order to compensate for that and match the overtones thus making the interval in tune.
Only the timbre of ideal strings corresponds exactly to the mathematical model of the overtone series, as you would describe it with Fourier's. Real strings have different stiffness, tension, width, etc. Piano strings are somewhat stiff, which results in inharmonicity (shift of the overtones from their ideal positions), so stretching is needed in order to compensate for that and match the overtones thus making the interval in tune.
Last edited by Km7 on Sat Jun 12, 2010 4:16 pm, edited 1 time in total.
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- KVRAF
- 26033 posts since 20 Oct, 2007 from gonesville
Good work!Nystul wrote:I don't think the article touches at all on stretched tuning, but it is a different example of the simplified model not meshing with reality. We would be taught that the "ideal" string has a harmonic that twice the fundamental, another thrice, another four times, and so on. But a real wire inside a piano has other issues at play which make the overtones a bit sharper than that. Stretched tuning matches the fundamental of one note with the real overtones of the notes that are octaves below it, instead of the simplified model.wrench45us wrote: it's also interesting from a science perspective of how we use math as a tool to model the world and in this very observable case, it falls a bit short
makes it even more of a challenge to understand how some concert pianist like to have their concert grand tuning stretched
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- KVRer
- 15 posts since 2 Feb, 2009 from Sweden
That article is full of fundamental misunderstandings...
The reason for tempering out the difference between these two notes (the pythagorean comma), is simply that we want to play them using the same key. So there is no mysterious anomaly in nature, as the author seems to think - we deliberately detune intervals for purely practical reasons.
Since all fifths are tempered by the same amount, all other intervals also sound the same in all keys. So what about the "wolf fifth"? It is not a fifth! It is really a diminished sixth, like G#-Eb. The interesting thing with meantone is that it does not temper out the pythagorean comma (only the syntonic comma) so there actually is a difference between G# and Ab, Eb and D# etc. That means we have to choose between them (or split the keys) and the most common way to tune meantone is to tune the black keys to C#, Eb, F#, G# and Bb. Keys like A flat major will then be unplayable, simply because there is no A flat note, but there will be no difference in character between the usable keys.
According to music theory, that would be a B#, so there really is no reason to "expect" to return to C.if you start with a C at the bottom of a piano keyboard and tune a series of 12 perfect 3:2 fifths up to the top, you discover that where you expect to have returned to a perfect high C...
The reason for tempering out the difference between these two notes (the pythagorean comma), is simply that we want to play them using the same key. So there is no mysterious anomaly in nature, as the author seems to think - we deliberately detune intervals for purely practical reasons.
This is wrong - all keys sound exactly the same in meantone.in meantone each key had an audible personality
Since all fifths are tempered by the same amount, all other intervals also sound the same in all keys. So what about the "wolf fifth"? It is not a fifth! It is really a diminished sixth, like G#-Eb. The interesting thing with meantone is that it does not temper out the pythagorean comma (only the syntonic comma) so there actually is a difference between G# and Ab, Eb and D# etc. That means we have to choose between them (or split the keys) and the most common way to tune meantone is to tune the black keys to C#, Eb, F#, G# and Bb. Keys like A flat major will then be unplayable, simply because there is no A flat note, but there will be no difference in character between the usable keys.
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- KVRAF
- 26033 posts since 20 Oct, 2007 from gonesville
No, that article isn't, your post is.
Meantone does not result in 'all keys have the same intervals'.
you entirely refute yourself in the next paragraph,
Note: 'unplayable'. 'usable keys'
Wait, 'C' etc = playable, with all keys usable. what just happened.
Meantone does not result in 'all keys have the same intervals'.
you entirely refute yourself in the next paragraph,
Note: 'unplayable'. 'usable keys'
Wait, 'C' etc = playable, with all keys usable. what just happened.
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- KVRer
- 15 posts since 2 Feb, 2009 from Sweden
That you fail to understand my post doesn't make it wrong.jancivil wrote:No, that article isn't, your post is.
Meantone does not result in 'all keys have the same intervals'.
Try to understand that there is no enharmonic equivalence in meantone.
Please, name one major third that is not pure in meantone!
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- KVRist
- 149 posts since 27 Jan, 2007 from Eyeth
I didn't even waste my time reading the article, when I saw how long it is, but I am not surprised it isn't entirely correct. Please, don't learn this stuff from articles, but from books. And by the way, jof knows his tuning theory. It's the well temperaments that give different key colours. You could even view 12-TET as a kind of meantone, where the fifth is flattened by a different amount (only 2 cents). Speaking of that, there are different meantone temperaments (1/4, 1/5, 1/6... of the syntonic comma), which results in different purity of the thirds.
To see what's going on in MT, you could proceed as follows:
1. Tune four perfect fifths: 3/2 * 3/2 * 3/2 * 3/2 = 81/16.
2. Divide by 4 (two times by 2) to take it two octaves down to a major third, which gives you 81/64, which is the Pythagorean major third.
3. 5/4 is the pure or just major third. 81/64 : 5/4 = 324/320 = 81/80, which is the syntonic comma, or the difference between a just major third and the sharper Pythagorean major third.
4. 1/4 meantone temperament means to flatten the pure fifth by 1/4 syntonic comma (the 4th root of 81/80), which is approx. 5 (5.4) cents, which gives you approx. 697 (696.6) cents for the fifth (~3 cents flat compared to ET).
5. Now use this fifth to generate the scale.
At this point you are close to understanding (by trying it yourself) why there is no enharmonic equivalence in MT, where the wolf is and which keys are unusable.
To see what's going on in MT, you could proceed as follows:
1. Tune four perfect fifths: 3/2 * 3/2 * 3/2 * 3/2 = 81/16.
2. Divide by 4 (two times by 2) to take it two octaves down to a major third, which gives you 81/64, which is the Pythagorean major third.
3. 5/4 is the pure or just major third. 81/64 : 5/4 = 324/320 = 81/80, which is the syntonic comma, or the difference between a just major third and the sharper Pythagorean major third.
4. 1/4 meantone temperament means to flatten the pure fifth by 1/4 syntonic comma (the 4th root of 81/80), which is approx. 5 (5.4) cents, which gives you approx. 697 (696.6) cents for the fifth (~3 cents flat compared to ET).
5. Now use this fifth to generate the scale.
At this point you are close to understanding (by trying it yourself) why there is no enharmonic equivalence in MT, where the wolf is and which keys are unusable.
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- KVRer
- 3 posts since 24 Jun, 2010 from Kyiv, Ukraine
anyone used Scala tuning software?
Makes big difference you read @about@ tuning or you re-tune your instruments in practice. It's like reading @about@ someone's swimming and swimming yourself.
12-ET is just one of the historical systems nowadays
. Still widespread, and still useful for performing XX century music.
Makes big difference you read @about@ tuning or you re-tune your instruments in practice. It's like reading @about@ someone's swimming and swimming yourself.
12-ET is just one of the historical systems nowadays
. Still widespread, and still useful for performing XX century music.a bit of salo in cold cyberspace
http://www.myspace.com/viktorpushkar
http://www.myspace.com/viktorpushkar
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- KVRAF
- 26033 posts since 20 Oct, 2007 from gonesville
'No enharmonic equivalence' means the same thing as "all keys have the same intervals'???. I quoted you. Thirds are not the end-all of intervals in a key. Do the math. Do a proof of concept.jof wrote:That you fail to understand my post doesn't make it wrong.jancivil wrote:No, that article isn't, your post is.
Meantone does not result in 'all keys have the same intervals'.
Try to understand that there is no enharmonic equivalence in meantone.
Please, name one major third that is not pure in meantone!
"Note: 'unplayable'. 'usable keys'
Wait, 'C' etc = playable, with all keys usable. what just happened." Whether or not you know your stuff doesn't fix that one.
I am quite aware of the difference between all of the various temperaments of that era vs ET, thank you very much.
There are obvious factual problems in that post. That you failed to convey accurately your idea isn't my fault.
I don't do that critique to sound good or impress some people, I do it to clarify some mud.
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- KVRer
- 15 posts since 2 Feb, 2009 from Sweden
No, you didn't. Your "quotes" are made up and now you are quoting yourself...jancivil wrote:I quoted you.
What I'm saying is that all major thirds are pure, all fifths and minor thirds are 1/4 comma narrow and so on. Every interval has the same size in all keys.jancivil wrote: Thirds are not the end-all of intervals in a key. Do the math. Do a proof of concept.
That some keys are "unusable" doesn't mean that the intervals have different sizes in those keys; it means that some thirds are missing.
This is where enharmonic equivalence is important: in ET, you can use the same key to play both G# and Ab, but in meantone these notes have different pitches. Because of this, you can never play both the E-G# and Ab-C major thirds in meantone, unless you have two black keys between G and A. With only one black key, you will have one major third (E-G# or Ab-C) an one diminished fourth (G#-C or E-Ab, respectively). It is possible to play in any tonality if you add enough keys and every tonality will have the same character; that's my point.
The math is simple - here is a formula you can cut-and-paste into GNU Octave:
sort(1200*mod(log2((3/2/(81/80)^(1/4)).^[-15:19]),1))
It will give you a table of the pitches (in cents) of all nominals, flats, sharps, double flats and double sharps in one octave.
This is the same table made with Scala, where you can easily see that every interval has the same size in all tonalities:
Code: Select all
0: 1/1 0.000 C unison, perfect prime
1: 34.990 cents 34.990 B##
2: 128/125 41.059 Dbb minor diesis, diesis
3: 76.049 cents 76.049 C#
4: 117.108 cents 117.108 Db
5: 152.098 cents 152.098 C##
6: 193.157 cents 193.157 D
7: 234.216 cents 234.216 Ebb
8: 269.206 cents 269.206 D#
9: 310.265 cents 310.265 Eb
10: 625/512 345.255 D##
11: 351.324 cents 351.324 Fbb
12: 5/4 386.314 E major third
13: 32/25 427.373 Fb classic diminished fourth
14: 462.363 cents 462.363 E#
15: 503.422 cents 503.422 F
16: 538.412 cents 538.412 E##
17: 544.480 cents 544.480 Gbb
18: 579.471 cents 579.471 F#
19: 620.529 cents 620.529 Gb
20: 655.520 cents 655.520 F##
21: 696.578 cents 696.578 G
22: 737.637 cents 737.637 Abb
23: 25/16 772.627 G# classic augmented fifth
24: 8/5 813.686 Ab minor sixth
25: 848.676 cents 848.676 G##
26: 889.735 cents 889.735 A
27: 930.794 cents 930.794 Bbb
28: 965.784 cents 965.784 A#
29: 1006.843 cents 1006.843 Bb
30: 1041.833 cents 1041.833 A##
31: 1047.902 cents 1047.902 Cbb
32: 1082.892 cents 1082.892 B
33: 1123.951 cents 1123.951 Cb
34: 125/64 1158.941 B# classic augmented seventh
35: 2/1 1200.000 C octaveFunny how arrogant know-it-alls don't understand me, when people like Km7, who obviously know the math, understand me without clarifications...jancivil wrote: I am quite aware of the difference between all of the various temperaments of that era vs ET, thank you very much.
There are obvious factual problems in that post. That you failed to convey accurately your idea isn't my fault.
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- something special
- 8627 posts since 16 Mar, 2002 from Birmingham, Alabama
[mod voice]no reason to start calling names..keep it civil[/mod voice]jof wrote: Funny how arrogant know-it-alls don't understand me, when people like Km7, who obviously know the math, understand me without clarifications...
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- KVRAF
- 26033 posts since 20 Oct, 2007 from gonesville
here's some reality then -
EG: *Werckmeister III*:
Note Exact frequency relation Value in cents
C \frac{1}{1} 0
C♯ \frac{8}{9} \sqrt[4]{2} 96
D \frac{9}{8} 204
D♯ \sqrt[4]{2} 300
E \frac{8}{9} \sqrt{2} 396
F \frac{9}{8} \sqrt[4]{2} 504
F♯ \sqrt{2} 600
G \frac{3}{2} 702
G♯ \frac{128}{81} 792
A \sqrt[4]{8} 900
B♭ \frac{3}{\sqrt[4]{8}} 1002
B \frac{4}{3} \sqrt{2} 1098
Give me a transposition of this, a single one, which prooves the concept.
Enharmonic equivalance is about naming. It means nothing real to point to that concept, as if it's the same as 'all intervals are...' It's not actually even true that all (major) thirds will be as pure in every case in all 'well temperaments'; give me the proof of concept.
It's quite absurd to claim all intervals in all keys are the same, outside of actual equal temperament; in fact, ET would never have been necessary if this were true.
You may wish to pretend you did not make these claims, but that won't get it.
EG: *Werckmeister III*:
Note Exact frequency relation Value in cents
C \frac{1}{1} 0
C♯ \frac{8}{9} \sqrt[4]{2} 96
D \frac{9}{8} 204
D♯ \sqrt[4]{2} 300
E \frac{8}{9} \sqrt{2} 396
F \frac{9}{8} \sqrt[4]{2} 504
F♯ \sqrt{2} 600
G \frac{3}{2} 702
G♯ \frac{128}{81} 792
A \sqrt[4]{8} 900
B♭ \frac{3}{\sqrt[4]{8}} 1002
B \frac{4}{3} \sqrt{2} 1098
Give me a transposition of this, a single one, which prooves the concept.
Enharmonic equivalance is about naming. It means nothing real to point to that concept, as if it's the same as 'all intervals are...' It's not actually even true that all (major) thirds will be as pure in every case in all 'well temperaments'; give me the proof of concept.
It's quite absurd to claim all intervals in all keys are the same, outside of actual equal temperament; in fact, ET would never have been necessary if this were true.
You may wish to pretend you did not make these claims, but that won't get it.