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Letter to Karl Signell on ITU TMDK Congress (11.20.2009)


Dear Karl,

My lengthy and well-studied critique against Nail Yavuzoğlu’s attempt at maqam theory and his utterings in that regard is not just a “complaint” . It is a serious work aimed at bringing down the many fallacies propounded under the guise of factuality. I would have expected you, of all people, not to have underestimated my erudite efforts in that direction, especially toward the unmasking of so much fraud.

Secondly, I believe you are being quite unfair towards the sincere undertaking by Prof. Dr. Lale Berköz, who has concerned herself (even as an outsider) with the quotidian problems of Turkish Classical/Art/Folk musics so much so that she spearheaded this much required event. I don’t see the current administration concerning itself with the outrageously glaring defects of AEU theory which continues to hinder the tradition, do you?

Admittedly, such unfortunate occurences as egotistic sabotages and admission of irrelevant papers that had nothing whatsoever to do with the spirit of the congress undermined Prof. Berköz’s honest intentions, not to mention the fact that her successors have failed to carry the flag after her departure… but isn’t this the way things are with the Turkish bureaucratic academia in general?

The committee responsible for the final report have made the conclusions of the congress (making use of my very own words) quite clear:

“It has been established on the basis of frequency measurements and analyses, that dividing the octave into 24 unequal parts is not a model suitable to express all the intervals of Turkish (Maqam) Music. However, the fact that this tuning has laid the foundation for the discussion of new tunings has been accepted by the scientists partaking in this congress.”

It is not the fault of Lale Berköz or any officer of the late administration in ITU TMDK if this clause is being neglected today.

To continue with the final report:


Solutions as a result of all that followed have been determined to be

1. To reflect the tone-system models that have been developed in conformity with the main theme of this congress to practices in our education and art institutions after evaluating them at broader platforms.

2. Taking Rast as the main maqam instead of Çargah, as defined in “Arel-Ezgi-Uzdilek”, and accepting that scale (as primary).

3. The acceptance of the A4/La = 440 Hz international diapason, which is a requisite of having adopted the Western staff notation,

4. The standardization of scores,

5. Continuity of history and the invigoration of tradition,

6. To regard the solutions to the (discussed) problems with a unified approach that incorporates the common rudiments of Turkish (Maqam) music.

Undoubtedly, there have been other worthwhile suggestions proposed in this congress. It has been understood that wide-ranging research and perspectives are essential and that the need for organizing discussive platforms, panels and briefings on tuning systems has become apparent. This congress held by ITU TMDK is an important step toward the solution of the (discussed) problems. ITU TMDK will continue to pioneer and support detailed and original work on this subject in accordance with contemporary developments.


I don’t need to point out that the (hearsay?) danger of forcing Nail’s theoretical attempt down the throats of every musician studying at ITU TMDK is a gross adulteration of the purposes and conclusions of the congress.

So, do you still deem the conclusions pointless and the congress a sham? That would be a shame. The shame is rather on the current administration, in fact, for failing to uphold the results spelled out by the committee, and failing to maintain its credibility toward the expectations of honest scientists and theorists who partook in the congress.

Dr. Ozan Yarman

“Mujannab zone” ratios


The neutral thirds of Maqam music are more diversified than the ratio 11:9 can encompass. As far as I can tell, middle second intervals roam the “mujannab zone” between 14:13 and 11:10. This is a region 37 cents wide. Then again, it is possible to broaden the zone and take 15:14 and 10:9 as the extremes. In that case, we acquire a zone that is 63 cents wide!

I conjecture that high prime-limit and relatively simple integer ratios such as 14:13, 27:25, 13:12, 12:11, 35:32, 11:10, 54:49 are at play. Then again, instead of middle seconds, we might be looking at neutral thirds, in which case, the performer could be searching for 17:14, 39:32, 11:9, 27:22, 16:13, 21:17 and 31:25.

So many ratios imply the necessity to temper pitches and represent diverse middle seconds as well as neutral thirds with few, yet, variegated microtones that are reasonably spaced apart. That is how I arrived at 79 MOS 159-tET. Check out page 95 of my doctorate dissertation. There, you will see how the 79-tone tuning approximates many such RI intervals.

Yavuzoğlu48’e göre makamlar


Nail Yavuzoğlu’nun 48 eşit perdeli önerisine göre, sent değerinde müteakip aralıklarla makam dizileri:

Uşşak [150+150+200] + [200+100+200+200]
Hüseyni [175+125+200+200] + [150+150+200] yahut [100+200+200]
Rast [200+175+125+200] + [200+175+125] yahut [200+100+200]
Segah [125+200] + [200+175+125] yahut [200+100+200]
Nigar [200+200+100] + [200+200+100+200]
Kürdi [100+200+200] + [200+100+200+200]
Neva [175+125+200] + [200+175+125+200]
Hicaz [125+275+100] + [200+175+125+200]
Hümayun [125+275+100] + [200+100+200+200]
Karcığar [150+150+200] + [150+250+100+200]
Suzinak [200+175+125+200] + [125+275+100] yahut [150+250+100]
Mahur-Acemaşiran [200+200+100+200] + [200+200+100]
Nihavend [200+100+200+200] + [100+200+200] yahut [125+275+100]
Nikriz [200+125+275+100] + [200+175+125] yahut [200+100+200]
Neveser [200+125+275+100] + [125+275+100]
Buselik [200+100+200+200] + [100+200+200] yahut [150+150+200]
Hisar [175+125+200+200] + [125+275+100]
Uzzal [125+275+100+200] + [150+150+200]
Zirguleli Hicaz, Suzidil, Evcara, Şedaraban, Zirguleli Suznak, Hicazkar
[125+275+100+200] + [125+275+100]
                           yahut [100+200+200] (sadece Hicazkar’da…)
Kürdili Hicazkar [100+200+200+200] + [125+275+100] yahut [150+150+200]
Çargah [150+250+100+200] + [125+175+100]
Ferahnak [125+200] + [200+175+125] yahut [200+100+200]
Müstear [200+125] + [200+175+125] yahut [200+100+200]
Nişabur [200+100] + [200+175+125] yahut [200+100+200]
Hüzzam [125+200] + [150+250+100]
Saba [150+150] + [150+250+100+200] + [125+175+100]
Sünbüle [100+200] + [150+250+100+200] + [125+175+100]
Şevkefza [200+200+100+200] + [150+250+100+200] + [125+175+100]
Bestenigar [125+200] + [150+150] + [150+250+100+200] + [125+175+100]
Irak [125+200] + [150+150+200] + [200+100+200+200]
Araban [150+150+200] + [125+275+100] + [200+100+200+200]
Revnaknüma [125+275+100+125+100] + [200+175+125] yahut [200+100+200]
Yegah [200+175+125+200] + [175+125+200] + [200+175+125+200]
                                      yahut [150+150+200] + [200+100+200+200]

Kanunda mandallar


Türkiye’de kanun yapımcıları, Batı’dan ithal edilen akort aygıtlarını referans alarak, mandal tahtasında natürel mandalları yastık mandallardan (eşikten) 100 sent uzağa yerleştirip, arayı 6 eşit parçaya bölerler. Bunun sonucunda, pratikçe 72-ton Eşit Taksimata varılmış olur. Lakin, 72-tET’in bütün perdeleri kullanılmaz, sadece bir alt-kümeye yakınsanır.

72-tET’te en küçük aralık 16.7 senttir. Holder komması 22.6 sentti. Aradaki fark 6 senttir. Şu da var ki, 53-tET perdeleri ile 72-tET perdeleri arasında yer yer yarım Holder kommasını bulan farklar vardır. Diğer bir deyişle, bu iki sistem birbirleriyle tamamen uyumsuzdur.

Bazı hallerde, natürel mandal ile yastık mandal arasının altı yerine 5’e yahut 7’ye bölündüğü olur. Yarım-ses aralığı 100 sent olarak belirlendikten sonra, bölme işlemleri hep 12-tET’in katlarını verir. Nitekim, 5’e bölmekle 60-tET, 7’ye bölmekle 84-tET ortaya çıkar.

72-tET, 13-limite kadar çok iyi sonuç veren, bazı Batılı mikrotonalistlerce standart kabul edilmiş istisnai bir bölünmedir. Sözgelimi, Rast ve Segah makamlarının 3-limitli AEU segahını çok iyi karşılar (rast perdesine göre segah 72-tET’te 383 sent uzaktadır). Bu ses-düzeninin kanun imalatçılarımızca, hangi yolla olursa olsun, keşfedilip tatbik edilmişŸ olması çarpıcıdır.

Logaritmik-aritmetik bölme işlemi

100 sentin bağıl frekansı = 2^(100/1200) = 1.059463094359295 Hz’tir. Yastık mandalda açık telin bağıl frekansı 1 idi. Aradaki fark 0.059463094359295 Hz’tir. Bu değer 6 eşit parçaya taksim edilirse, 0.009910515726549 Hz çıkar. Demek ki, yastık mandaldan natürel mandala doğru aritmetik bölünme sonucu bağıl frekanslar şöyle olacaktır:



1.0 Hz (yastık)
1.009910515726549 Hz
1.019821031453098 Hz
1.029731547179647 Hz
1.039642062906196 Hz
1.049552578632745 Hz
1.059463094359294 Hz (natürel)

Bağıl Frekans x ln(2)/1200 formülü ile sent değerleri aşağıdadır:

0: 0.0 sent
1: 17.073 sent
2: 33.979 sent
3: 50.722 sent
4: 67.304 sent
5: 83.729 sent
6: 100.000 sent

Sonuçlar, oktavın logaritmik olarak 72 eşit parçaya taksim edilmesiyle oluşacak değerler ile pratikçe aynıdır:

aritmetik bölünme logaritmik bölünme farklar
17.073 16.667 0.406
33.979 33.333 0.646
50.722 50 0.722
67.304 66.667 0.637
83.729 83.333 0.396
100.000 100.000 0

Şu da var ki, tize doğru her mandal bir öncekinden az daha yüksek olmak zorundadır. Bundan dolayı, tellerin her mandal kaldırmada ilave gerilmesi sözkonusudur. Yine de, buradan doğacak farkların gözardı edilebilir olduğu varsayılabilir.

Holder komması


Holder komması, oktavın 53. kökü değerinde bir bağıl frekansa sahip aralık birimidir. Diğer bir deyişle, oktavın logaritmik 53 eşit parçaya taksim edilmesiyle ortaya çıkar:

53√2= e (ln(2)/53) = 1.01316 Hz (1200/53 = 22.64151 sent)

“Holder komması sistemi” yahut Alexander J. Ellis’e göre “Mercator devri“, mükemmele yakın saf beşli aralığı ihtiva eden bir temperamandır:

(1200/53) * 31 = 701.88679 sent

Mercator devrinin beşlisi ile saf beşli arasında Mercator kommasının 53’te biri kadar, yani insan kulağının algılayamayacağı bir fark vardır:

(log 1.5) * 1200/(log 2) = 701.955 sent

701.95500 – 701.88679 =

53√((353)/(284)) 53√1.00209 = 0.06821 sent

Mercator kommasının 53’te biri kadar tempere edilmiş beşliler üst üste bindirildiğinde 54. adımda başlanılan noktaya (oktav farkıyla) dönülür (Bkz. 53-ton_Eşit_Taksimat).

Marin Mersenne’in 1637 yılında bildirdiğine göre, bu aralık birimini Batı’da ilk keşfeden Belçikalı Mühendis Jean Galle’dir. 1660’ta Nicolaus Mercator tarafından yukarıdaki kesin matematiksel tarifi verilmiştir. Daha sonra, Isaac Newton’un yayınlanmamış 1664-65 tarihli risalelerinde ve William Holder’in 1684 tarihli müzik teorisi kitabında, “Mercator devrinden” sözedildiği görülür.

Holder kommasına Yılmaz Uysal “uygun”, Mildan Niyazi Ayomak ise “minik” der. Mercator kommasıyla karıştırılmaması ve 17. Yüzyıl Avrupası ile irtibatlandırılması bakımından, Holder komması tabiri daha doğrudur.

Avant-garde tuning terms


Gene Ward Smith had had commented on the tuning list on 10 April 2006:

<Any otonal chord can be considered a utonal chord, and vice-versa … 1-11/9-3/2-11/6. The inverted chord is 1-9/11-2/3-6/11, and upon multiplication by 11/6 that becomes 1-11/9-3/2-11/6 again. Have these been considered and given a name? If not, “outonal chord” is my proposal.>

I had had then suggested the term “ubi-tonal” for suchlike chords.