| Pages:
1
2 |
DavidJE
Oud Junkie
Posts: 265
Registered: 7-14-2013
Location: Vienna, Austria
Member Is Offline
|
|
Yes, there are differences in the intervalic relationships between Turkish and Arabic makams, among other differences. With Rast for example, the 3rd
and 7th degrees are played a bit flatter in Arabic playing than in Turkish playing, I believe. It SEEMS to me (but I'm not actually very qualified to
say!) that Arabic players tend to play closer to the "quarter tone", that the standardization of the theory has had a little more impact on
performance. Or, maybe the Arabic quarter tone theory happened to match performance more? I don't know.
|
|
|
Brian Prunka
Oud Junkie
Posts: 2951
Registered: 1-30-2004
Location: Brooklyn, NY
Member Is Offline
Mood: Stringish
|
|
The Arabic quarter tone varies geographically, but I don't think it is so much the "quarter tone" idea that is the cause. It's hard to know, but
going by Farabi's calculations you are not far off from the sikah that is used today, so I think that it just happened to match the practice more
rather than it being an influence. I think this is more about a general trend of how people feel about theoretical descriptions; in my experience,
people from the Arabic tradition are not especially concerned with verbal descriptions and names—the music is just what it is. A kind of minimalist
approach to theory, which works well in a mentor-protege type relationship. It is much more challenging for an academic setting or for self-study.
Quote: Originally posted by SV_T_oud  | One more question I'd like to ask my more knowledgeable forum mates is about the difference in intervalic relationships between Turkish and Arabic
makams, maqams. Do they follow the same scheme or are there noticeable differences?
Take for instance Rast (that's one of a few I know by name): is Turkish version the same as Arabic and maybe even Persian versions?
In couple years from now I might be able to tell the difference (if any) by ear but for now please share with me your observations.
I ask because I'm mostly interested in Turkish music but majority of the instructional material is for Arabic music.
|
I find it strange that you say the majority of instructional material is for Arabic music. I know of almost no real instructional materials for
Arabic music (in English, anyway), while there are tons of resources for Turkish music.
What I will say is this: while the concept of seyir does exist in the Arab maqam, it is not taught and verbalized as it is in Turkish music, and it is
not as rigid. In Turkish music, the same scale with a different seyir is considered a different maqam. In Arabic music it's not, usually. It's more
like "here's one kind of Rast, and then there is this other way that goes like this." It would seem from old sources that this used to be somewhat
different in Arabic music, but over time the conception became somewhat simplified.
There are differences in the intonation of the maqamat. Overall they are similar enough that Turkish and Arab musicians can play together, but there
are some noticeable discrepancies and the musicians have to come to agreement . . .
The biggest one that stands out to me is that in Turkish music, the third of hicaz is the same interval as the third of Rast. In Arabic music, that
is not the case. This causes an interesting issue when you modulate to sikah from hijaz . . . in Turkish music it is no problem (hicaz to segah) but
in Arabic music it is not "real sikah". Sami Abu Shumays coined the term "pseudo sikah"—it acts like sikah in every way, but it is higher than the
Arabic version of sikah. Sometimes, a modulation to Rast in a Turkish piece is considered a modulation to ‘Ajam (Acem) by Arab musicians, since
Turkish rast is so close to ‘ajam (if it was learned by ear, rather than from sheet music).
|
|
|
SV_T_oud
Oud Maniac
Posts: 83
Registered: 8-27-2014
Member Is Offline
|
|
Brian, first of all - thanks much again for your detailed and critical answer! I appreciate your help as always.
Regarding your comment above that I quoted. Let me explain what I mean. Imagine I want to get into my ear and ingrain into my mind the sound of some
common makams, let's say 10.
For that the best way to go I think is to get some books with audio examples clearly demonstrating the sound of these makams.
I found these books/DVDs that I think would serve the purpose:
- Cameron Powers, Arabic-Musical-Scales-Basic-Maqam-Teachings available at:
http://www.maqam.com/store/p/19-Arabic-Musical-Scales-Basic-Maqam-T...
- Learn Maqamat on the OUD (VERSION 2) available on eBay:
http://www.ebay.com/itm/251707033501
Both are for Arabic version of makamat. Although the 'Learn Maqamat on the OUD' mentions some Turkish content present I understand the creator comes
from the Arabic tradition anyway.
I didn't find any similar books/DVDs currently readily available for Turkish version. If you pointed me to such books/DVDs etc. I would send you my
big thanks.
|
|
|
SV_T_oud
Oud Maniac
Posts: 83
Registered: 8-27-2014
Member Is Offline
|
|
David, thank you for your input also.
| Quote: Originally posted by DavidJE | | Yes, there are differences in the intervalic relationships between Turkish and Arabic makams, among other differences. With Rast for example, the 3rd
and 7th degrees are played a bit flatter in Arabic playing than in Turkish playing, I believe. It SEEMS to me (but I'm not actually very qualified to
say!) that Arabic players tend to play closer to the "quarter tone", that the standardization of the theory has had a little more impact on
performance. Or, maybe the Arabic quarter tone theory happened to match performance more? I don't know. |
|
|
|
Brian Prunka
Oud Junkie
Posts: 2951
Registered: 1-30-2004
Location: Brooklyn, NY
Member Is Offline
Mood: Stringish
|
|
| Quote: Originally posted by SV_T_oud |
Regarding your comment above that I quoted. Let me explain what I mean. Imagine I want to get into my ear and ingrain into my mind the sound of some
common makams, let's say 10.
For that the best way to go I think is to get some books with audio examples clearly demonstrating the sound of these makams.
I found these books/DVDs that I think would serve the purpose:
- Cameron Powers, Arabic-Musical-Scales-Basic-Maqam-Teachings available at:
http://www.maqam.com/store/p/19-Arabic-Musical-Scales-Basic-Maqam-T...
- Learn Maqamat on the OUD (VERSION 2) available on eBay:
http://www.ebay.com/itm/251707033501
Both are for Arabic version of makamat. Although the 'Learn Maqamat on the OUD' mentions some Turkish content present I understand the creator comes
from the Arabic tradition anyway.
I didn't find any similar books/DVDs currently readily available for Turkish version. If you pointed me to such books/DVDs etc. I would send you my
big thanks.
|
Here is one for Turkish makamlar: http://www.ortav.com/sunshop/index.php?l=product_detail&p=415
Really, the sound of the makam should simply be learned from recordings, there are plenty of examples of taksim in various makams on CDs . . . it's
true that these will have modulations that deviate from the essential notes of the makam, but I don't consider that a big deal. I was really thinking
of actual detailed instructional materials, of which there is an abundance in Turkish music and a relative paucity for Arabic music. The Signell
book is very detailed also, though it doesn't include recordings. There are numerous Turkish oud method books as well.
The main thing is to understand the jins concept in the construction of the maqam . . . if you know rast, bayati/ussak, nahawand/buselik, kurd, hijaz,
nikriz, ajam, saba, sikah, then you know all the sounds, you just need to learn which combinations produce which maqam.
|
|
|
SV_T_oud
Oud Maniac
Posts: 83
Registered: 8-27-2014
Member Is Offline
|
|
Thank you Brian for the link and useful tips!
|
|
|
DavidJE
Oud Junkie
Posts: 265
Registered: 7-14-2013
Location: Vienna, Austria
Member Is Offline
|
|
The book Brian recommended is excellent...highly recommended.
|
|
|
SV_T_oud
Oud Maniac
Posts: 83
Registered: 8-27-2014
Member Is Offline
|
|
However, it's out of stock. Maybe even out of print...
|
|
|
Gocauo
Oud Maniac
Posts: 64
Registered: 8-22-2014
Member Is Offline
|
|
It does not appear to be out of print, but you may have trouble finding it with the CDs.
This is the ISBN 0974588245
|
|
|
SV_T_oud
Oud Maniac
Posts: 83
Registered: 8-27-2014
Member Is Offline
|
|
... which are the most valuable part of it...
|
|
|
Gocauo
Oud Maniac
Posts: 64
Registered: 8-22-2014
Member Is Offline
|
|
Agreed....
|
|
|
SV_T_oud
Oud Maniac
Posts: 83
Registered: 8-27-2014
Member Is Offline
|
|
I'd like to expand my question a little bit into the theoretical direction if you don't mind. It's still related to Turkish music though.
First, please consider me ignorant in this field for the ease of conversation and do not assume I possess any deep knowledge in the area of Musical
Temperament systems.
There are quite a few temperament systems according to information available on the Internet. Here is a good example of the web site making an attempt
on categorization of the temperaments:
http://leware.net/temper/temper.htm#_nr_308
Apparently there are a few competing systems offering their "best" ratios based on scientific calculations.
Just Intonation, Pythagorean Tuning, a few Regular and Irregular Temperament schools, you name it. All in all I think there are at least 10 commonly
accepted temperament versions including the notorious Equal Temperament which is the basis of the Western Music. And the latter is ironic.
Well, what I'm getting at you may ask? Here we go.
First, how do you know which version of the temperament out of ten is correct? For instance one musician's ear will be fond of the scale coming from
the Pythagorean tuning and other's will favor the Kirnberger's temperament of the Irregular school of thought.
Is it correct to say that there is no "correct" temperament in Nature and there are only preferences?
Second, if the above is not ultimately true which of the temperaments is considered most "natural" by good ear players?
Finally, which of the scientifically justified temperaments most closely matches the temperamet used in the Turkish koma system and why?
By the way, I remember reading somewhere else in the Wikipedia (actually here):
http://en.wikipedia.org/wiki/Pythagorean_tuning
that the shortcoming of the Pythagorean tuning is in accumulating an error when superimposing the stack of pure fiffths with the 3:2 ratio over a
stack of octaves with the 2:1 ratio.
It's said that over the span of 7 octaves the error resulting from superimposing two stacks is approximately a 1/4-th of a semitone.
As far as I can imagine the 1/8-th of a tone Pythagorean comma is somehow related to the 1/9-th of a tone Turkish koma.
Exactly how and why?
|
|
|
Brian Prunka
Oud Junkie
Posts: 2951
Registered: 1-30-2004
Location: Brooklyn, NY
Member Is Offline
Mood: Stringish
|
|
I wrote a detailed answer and lost it. maybe i'll try gain later.
short answer: pythagorean tuning and just intonation are not temperaments, because the intervals are not modified (tempered)
pythagorean comma is in between 1/8 and 1/9 of a tone, and closer to 1/9 than 1/8. Probably the Turkish theorists picked the nearest simple ratio,
which may explain somewhat why the theory doesn't match how people actually play.
1/8 = ~25.49¢
Pythagorean comma = ~23.46¢
1/9 = ~22.66¢
|
|
|
Brian Prunka
Oud Junkie
Posts: 2951
Registered: 1-30-2004
Location: Brooklyn, NY
Member Is Offline
Mood: Stringish
|
|
| Quote: Originally posted by SV_T_oud | I
Just Intonation, Pythagorean Tuning, a few Regular and Irregular Temperament schools, you name it. All in all I think there are at least 10 commonly
accepted temperament versions including the notorious Equal Temperament which is the basis of the Western Music. And the latter is ironic.
|
ET is not the 'basis' for Western music, the basis is 5-limit Just Intonation, but the spread of ET in the 1800s led to musical developments that have
become essentially intertwined with Western musical styles. All of the temperaments are attempts to balance the best sound (Just Intonation) with the
widest musical possibilities in the areas of modulation, modal interchange and chromatic development. JI generally sounds the most "in-tune" if you
stick to one basic key, but sounds the worst if do a lot of modulating (unless you have infinite note possibilities, such as in choral music).
| Quote: |
First, how do you know which version of the temperament out of ten is correct? For instance one musician's ear will be fond of the scale coming from
the Pythagorean tuning and other's will favor the Kirnberger's temperament of the Irregular school of thought.
Is it correct to say that there is no "correct" temperament in Nature and there are only preferences?
|
No, it is not correct, because the first 12 notes of the harmonic series, representing simple vibrational ratios (1/1, 2/1, 3/1, 4/1, etc) are
unassailably in tune.
| Quote: | which of the temperaments is considered most "natural" by good ear players?
|
This gets more complicated because while there is a simple physical basis for tuning, culturally we learn to identify with various sounds in a
somewhat arbitrary way. The more complex the reference ratio, the more we seem to accept culturally determined or ambiguous tuning cues. For
example, nearly everyone agrees that 1/1 unisons and 2/1 octaves are ideal. 3/2 fifths and 4/3 fourths likewise are nearly universally desirable.
5/4 major thirds are a little less universal, but still very widely considered ideal (largely dependent on the degree to which third-based harmony is
stylistically desirable, or possibly the other way around). Beyond that, there is a lot of variability.
However, when you consider music that employs the sounding of many simultaneous tones, it becomes quite clear that (5-limit) Just Intonation
is most widely considered ideal. That is, we can accept a lot of tuning variation melodically that we cannot accept harmonically.
Vocalists and instruments with flexible pitch most naturally gravitate toward this tuning most of the time.
| Quote: |
Finally, which of the scientifically justified temperaments most closely matches the temperamet used in the Turkish koma system and why?
|
53 tone equal temperament is essentially identical to the Turkish 1/9 tone system.
53tET results in a division being ~22.64¢, compared to the theoretical turkish comma of 22.66¢ if we are going by 1/9 of the pythagorean whole tone.
Considering that the size of a "whole tone" is arguably not fixed, I think this difference is a computational error of some kind. Or put another
way, they figured out that if you divide the octave equally into 53 parts, 9 of those parts is a rounding error with respect to a pure pythagorean
whole tone (remember that we can't really hear discrepancies less than 2¢ in most cases, this is 203.91¢ for a Pythagorean WT, compared to 203.77¢
for 53tET WT).
None of the traditional Western temperaments approximate the Koma system because they were devised for different purposes entirely.
| Quote: |
the shortcoming of the Pythagorean tuning is in accumulating an error when superimposing the stack of pure fiffths with the 3:2 ratio over a stack of
octaves with the 2:1 ratio.
|
This is not really the problem, Pythagorean tuning is perfectly usable, as long as you stay in one key. The real problem with it is that major thirds
and minor sixths, and to a lesser extent minor thirds and major sixths sound pretty terrible, which is why music composed in this system (medieval
Western music like Gregorian chant, etc.) shies away from using those intervals harmonically (or treats them as dissonances rather than
consonances).
| Quote: | | It's said that over the span of 7 octaves the error resulting from superimposing two stacks is approximately a 1/4-th of a semitone.
|
this is the basis of the Pythagorean comma, however, note that the situation where you would notice this doesn't necessarily come up in non-modulating
music.
|
|
|
Brian Prunka
Oud Junkie
Posts: 2951
Registered: 1-30-2004
Location: Brooklyn, NY
Member Is Offline
Mood: Stringish
|
|
Warning Okay, this is going to get nerdy:
The degree to which Pythagorean tuning is usable (leaving aside the issue of harmony using thirds and 6ths), depends largely on the extent to which
the scale is devised of contiguous 5ths.
Consider, any mode of the major scale can be constructed entirely of contiguous 5ths:
Dorian:
F C G D A E B
Originating tone is in the center.
Mixolydian:
F C G D A E B
Originating tone is slightly off center
Ionian:
F C G D A E B
originating tone fairly off center
Lydian:
F C G D A E B
originating tone is the lowest.
Aeolian:
F C G D A E B
originating tone slightly off-center above
Phrygian:
F C G D A E B
originating tone fairly off-center above
Locrian:
F C G D A E B
originating tone extremely off-center above
All else being equal, notes below the tonic and intervals farther away from the tonic (based on the harmonic relationships) are harder to hear. This
would point to Dorian or Mixolydian being the easiest to hear in Pythagorean tuning, and Locrian and Phrygian as the hardest to hear.
Dorian's hardest note to hear is F; it's 3 fifths away in the downward direction.
Mixolydian's hardest note to hear is B, it's 4 fifths away in the upward direction.
Lydian's hardest note is B, it's 6 fifths away in the upward direction. Compare to Phrygian, which has the F located 5 fifths away in the downward
direction.
As a rough estimate, lets assign difficulty this way: 3 points for each fifth in the upward direction, 4 points for each fifth in the downward
direction.
As presumably the most difficult, let's consider Locrian as the reference for 100% difficulty:
Locrian: F(24) C(20) G(16) D(12) A(8) E(4) B(0), total difficulty: 84 (100%)
Dorian: F(12) C(8) G(4) D(0) A(3) E(6) B(9), total difficulty: 42 (50%)
Mixolydian: F(8) C(4) G(0) D(3) A(6) E(9) B(12), total difficulty: 42 (50%)
Ionian: F(4) C(0) G(3) D(6) A(9) E(12) B(15), total difficulty: 49 (58%)
Aeolian: F(16) C(12) G(8) D(4) A(0) E(3) B(6), total difficulty: 49 (58%)
Lydian: F(0) C(3) G(6) D(9) A(12) E(15) B(18), total difficulty: 63 (75%)
Phrygian: F(20) C(16) G(12) D(8) A(4) E(0) B(3), total difficulty: 63 (75%)
Now let's look at what happens if you use five limit JI and assume that 3rds above are 4 points and 3rds below are 5 points:
Dorian: F(8) C(8) G(4) D(0) A(3) E(6) B(8), total difficulty: 37 (44%)
Mixolydian: F(8) C(4) G(0) D(3) A(6) E(8) B(4), total difficulty: 33 (39%)
Major: F(4) C(0) G(3) D(6) A(8) E(4) B(7), total difficulty: 32 (38%)
Minor: F(5) C(8) G(8) D(4) A(0) E(3) B(6), total difficulty: 34 (40%)
Lydian: F(0) C(3) G(6) D(9) A(4) E(7) B(10), total difficulty: 39 (46%)
Phrygian: F(9) C(5) G(8) D(8) A(4) E(0) B(3), total difficulty: 37 (44%)
So we can see that in the case of Dorian and Mixolydian, it is not a big difference to use Pythagorean tuning, but in the case of Major, Minor, Lydian
and Phrygian, it makes it much easier to use Just Intonation.
Of course, my point assignments are a bit arbitrary, just to get a rough sense of how far removed each scale note is from the tonic. In reality, I
would say that fourths (that is, 5ths below) are slightly harder than 3rds above, and 3rds below slightly harder still, so maybe better values would
be 6, 8, 7 and 9.
This yields: Dorian (difficulty 50% Pythagorean, 43% JI), Mixolydian (50% Pythagorean, 38% JI), Lydian (75% Pythagorean, 45% JI), Phrygian (75%
Pythagorean, 42% JI) etc.
Again this is just to get a rough sense of how our intuition works regarding how easy intervals are to hear (note this works both ways: it is harder
to hear it to get it in tune, but as a listener we are also more forgiving of variation)
TL;DR: I am a nerd
|
|
|
SamirCanada
Moderator
Posts: 3405
Registered: 6-4-2004
Member Is Offline
|
|
 = Brian
Just missing a peice of white tape in the middle of the glasses
@samiroud Instagram
samiroudmaker@gmail.com
|
|
|
SV_T_oud
Oud Maniac
Posts: 83
Registered: 8-27-2014
Member Is Offline
|
|
Brian, great job!
A few of my miserable comments follow.
| Quote: Originally posted by Brian Prunka | No, it is not correct, because the first 12 notes of the harmonic series, representing simple vibrational ratios (1/1, 2/1, 3/1, 4/1, etc) are
unassailably in tune.
|
This is interesting. What is essentially "in tune"?
Is this something you have to believe or is there a scientific definition of this term that makes it unassailably true?
Or is it the thing-in-itself, the self-defining substance? That is: because this is the natural harmonic series the notes coming out of the series are
considered "unassailably in tune"?
It's interesting also because before I used to believe that natural harmonics of the brass instruments (I play some trumpet) are "out of tune" and
only today I noticed the following in the commonly used description: "Some harmonics are 'out of tune', meaning that they don't lie close to notes
on the familiar, equal-tempered scale."
| Quote: Originally posted by Brian Prunka | ET is not the 'basis' for Western music, the basis is 5-limit Just Intonation, but the spread of ET in the 1800s led to musical developments that have
become essentially intertwined with Western musical styles.
|
Did I have to say "...basis of the modern Western music" to be accepted?
Otherwsie, Brian, thanks again and I need couple weeks to digest your dissertation.
|
|
|
SV_T_oud
Oud Maniac
Posts: 83
Registered: 8-27-2014
Member Is Offline
|
|
That sounds "cool", but what's the humor? I want to understand...
Sort of "stays cool all the time" with the tape between the glasses or anything else? I'm not that good at English idiomatic thinking.
|
|
|
SV_T_oud
Oud Maniac
Posts: 83
Registered: 8-27-2014
Member Is Offline
|
|
Brian, please help me with one more thing.
When cents are used to estimate the intervals, how are they calculated?
For instance when you say: "...53tET results in a division being ~22.64¢, compared to the theoretical turkish comma of 22.66¢..." how do we get
22.64 cents per a fraction when deviding the whole (octave) into 53 bits? In other words 22.64 cents of what?
I actually never understood "cents" widely used in English texts compared to what we use in our country which are "percents". If I interpreted the
22.64 cents as our "percents" applied to an octave I would get something between 1/4 and 1/5 of an octave.
And one more thing. You say about "equal division" of an octave into 53 tones. Is it literally equal, that is if we take the Western tone A and
calculate the frequency increment per a microtone over an octave we get: 880-440 Hz = 440 Hz / 53 = 8.3 Hz per microtone over an octave? Of course
that value will vary when applied to different octaves, ie. 440-220 Hz = 220 / 53 = 4.1 Hz per microtone in that octave. It would make no sense.
Or does "equal" means "equal logarithmic" increment? I vote for the latter but who knows?
Brian knows!
I would appreciate if you gave me a small example in cents and Hz applied to an octave of A = 440 Hz. Just a few first values, please like:
A + 0/53 = 440 Hz, x cents
A + 1/53 = ...
A + 2/53 = ...
the rest will be self explanatory if the formula is known.
Thanks in advance!
|
|
|
Brian Prunka
Oud Junkie
Posts: 2951
Registered: 1-30-2004
Location: Brooklyn, NY
Member Is Offline
Mood: Stringish
|
|
Calculating cents from Hz is difficult and calculating Equal Temperaments in Hz is based on logarithmic increments.
"Cents" is a system of measurements that takes into account the logarithmic scale of frequency and is based on Equal Temperament. It is convenient
because it simply expresses the way we hear (ratios do this as well, but only for integer relationships, while cents can express irrational numbers).
100 cents = an ET half step. "Cent" just means hundred or hundredth. "Percent" is actually a compound word from "per cent", meaning "out of a
hundred", so 100% just means "100 out of 100". In this case "100" is a half step, not an octave. An octave is 1200 cents (12 half steps).
Here is a calculator for converting between Hz and ¢
(It also shows the formula)
|
|
|
Brian Prunka
Oud Junkie
Posts: 2951
Registered: 1-30-2004
Location: Brooklyn, NY
Member Is Offline
Mood: Stringish
|
|
| Quote: Originally posted by SV_T_oud |
That sounds "cool", but what's the humor? I want to understand...
Sort of "stays cool all the time" with the tape between the glasses or anything else? I'm not that good at English idiomatic thinking.
|
It's just a stereotypical image of a nerd or geeky scientist type.
|
|
|
Brian Prunka
Oud Junkie
Posts: 2951
Registered: 1-30-2004
Location: Brooklyn, NY
Member Is Offline
Mood: Stringish
|
|
| Quote: Originally posted by SV_T_oud |
For instance when you say: "...53tET results in a division being ~22.64¢, compared to the theoretical turkish comma of 22.66¢..." how do we get
22.64 cents per a fraction when deviding the whole (octave) into 53 bits? In other words 22.64 cents of what?
|
Hopefully it is clear from my post above, but an octave is 1200 cents, so 1200/53 = ~22.64
| Quote: |
And one more thing. You say about "equal division" of an octave into 53 tones. Is it literally equal, that is if we take the Western tone A and
calculate the frequency increment per a microtone over an octave we get: 880-440 Hz = 440 Hz / 53 = 8.3 Hz per microtone over an octave? Of course
that value will vary when applied to different octaves, ie. 440-220 Hz = 220 / 53 = 4.1 Hz per microtone in that octave. It would make no sense.
|
"Equal" in this case would mean perceptually equal, which is based on a logarithmic scale; it would be equal ratios, not equal frequency increments.
Think of it this way:
A is 110 Hz. To get to the next A we double it, adding another 110 Hz, so the next A is 220 Hz. If we added another 110 Hz, we would have 330 Hz.
This isn't A, it is E. Since every octave requires a proportional doubling, going by equal frequency increments isn't equal at all.
| Quote: |
I would appreciate if you gave me a small example in cents and Hz applied to an octave of A = 440 Hz. Just a few first values, please like:
A + 0/53 = 440 Hz, x cents
A + 1/53 = ...
A + 2/53 = ...
the rest will be self explanatory if the formula is known.
Thanks in advance! |
You can't express Hz in cents because cents measures a ratio, not a frequency. If you said Just Intonation, A(440Hz) to E (660Hz), you could
calculate the 3/2 ratio as being 701.955 cents. An ET fifth, we can simply count up 7 half steps and get 700 cents.
The calculator from the site below gives me ~1.013164/1 as the ratio of succeeding intervals in 53tET. So in your example:
440Hz +1/53 = 440Hz*1.013164 = 445.79216Hz
440Hz +2/53 = 440Hz(1.013164^2) = 451.66057Hz
440Hz +3/53 = 440Hz(1.013164^3) = 457.60623Hz
etc.
From this site (which has more calculation options than the one above):
| Quote: |
Formula for converting the interval frequency ratio f2 / f1 to cents (c or ¢).
¢ or c = 1200 × log2 (f2 / f1)
log 2 = 0.301029995
This formula employs a log 2, or logarithm base 2 function. This formula can also be
written using a log 10 function, available on most scientific calculators via the log button:
c = 1200 × 3.322038403 log10 (f2 / f1)
1/log 2 = 1/0.301029995 = 3.322038403
The formula expressed using log10 rather than log 2.
3.322038403 is a conversion factor that converts base 2 logarithms to base 10 logarithms.
1 Cent = 2(1/1200) = 1.0005777895065548592967925757932
One cent is thus the number that multiplied by itself 1200 times results in the number 2.
The cent is an interval which is calculated from the interval frequency ratio as follows:
(In of the interval frequency ratio / ln 2)×1200 = cents value of the interval.
An interval of a halftone is equivalent to: 2(1/12) = 1,0594630943592952645618252949463.
That is: [ln (2(1/12)) / ln (2)]×1200 cent = 100 cent.
The Pythagorean comma is the frequency ratio (3 / 2)12 / 27 =
312 / 219 = 531441 / 524288 = 1.0136432647705078125.
The resulting is converted to 23.460010384649013 cent.
Twelve perfect fifths (3 / 2) reveals 8423.46 cents and
seven octaves, however, reveals only 8400 cents. |
I am not about to do that math, so I use the handy calculator provided (and also some rule-of-thumb estimates based on simpler math).
|
|
|
SV_T_oud
Oud Maniac
Posts: 83
Registered: 8-27-2014
Member Is Offline
|
|
Thanks Brian, I of course could be more persistent in finding all that info myself, especially the 100 cents for 1/2 step but sometimes the amount of
information given on the realted web sites is overwhelming and I get lost before finding the answers.
Big thanks for all your teaching!
|
|
|
| Pages:
1
2 |
|
![[*]](images/xmbprobrown/default_icon.gif){kind=icon}
