[ValleyBell]

When working on SMPSPlay and DEPlay there was always one thing that bothered me: the DAC frequency
It's something I couldn't just get right, so I decided to do some research.

I startet with Oerg866's SMPS Player, which uses jman2050's DAC driver:
At first I made an SMPS file that plays a few DAC sounds and made each sound play the same sample, but with different DAC rates.
Then I used MESS to make a VGM of it. (MESS because MAME's timing system is extremely accurate.)
After that I wrote a small tool in VB that reads that VGM and calculated the frequency of each sound.
Finally I made another small VB tool that recalculated the formula FreqHz = BaseVal / (RateBase + DACRate) for the DAC rates I logged. With this tool I tested values for RateBase until I found a perfect one. (I made it recalculate BaseVal with the tested frequency of rate 01 and RateBase.)

Then I did the same with plain Sonic 1, Sonic 2, and Barver Battle Saga (because I already had its pointers) for the Data East driver.

And here are my results:
(The columns are DACRate, logged DAC samples, length of the DAC sound in VGM ticks (44100 Hz), DAC samples per VGM sample, DAC frequency.)

jman2050's DAC driver
---------------------
01	8,238	 11,940	  1.45	30,426.784
02	8,238	 13,275	  1.61	27,366.915
03	8,238	 14,600	  1.77	24,883.274
04	8,238	 15,937	  1.93	22,795.746
05	8,238	 17,268	  2.10	21,038.673
06	8,238	 18,602	  2.26	19,529.932
07	8,238	 19,938	  2.42	18,221.276
08	8,238	 21,265	  2.58	17,084.213
0C	8,238	 26,598	  3.23	13,658.764
10	8,238	 31,926	  3.88	11,379.308
20	8,238	 53,243	  6.46	 6,823.353
30	8,238	 74,561	  9.05	 4,872.464
40	8,082	 94,060	 11.64	 3,789.243
80	4,278	 94,054	 21.99	 2,005.867
C0	2,908	 94,020	 32.33	 1,363.995
FF	2,212	 94,038	 42.51	 1,037.338
00	8,238	352,309	 42.77	 1,031.185
-> FreqHz = 272624 / (7.96 + DACRate)
 Calculating with (8 + prim) is still a good approximation. (DACSmpl/VGMSmpl variance is <0.2 for rates <= 80)


Sonic 1 DAC driver
------------------
01	8,236	 15,517	  1.88	23,407.076
02	8,236	 16,846	  2.05	21,560.465
03	8,236	 18,190	  2.21	19,967.433
04	8,236	 19,530	  2.37	18,597.419
05	8,236	 20,866	  2.53	17,406.671
06	8,236	 22,209	  2.70	16,354.073
07	8,236	 23,550	  2.86	15,422.828
08	8,236	 24,882	  3.02	14,597.203
0C	8,236	 30,252	  3.67	12,006.069
10	8,236	 35,602	  4.32	10,201.888
20	8,236	 57,041	  6.93	 6,367.483
40	8,236	 99,906	 12.13	 3,635.493
80	8,236	185,658	 22.54	 1,956.326
C0	5,710	188,159	 32.95	 1,338.288
FF	4,354	188,077	 43.20	 1,020.919
00	8,236	357,141	 43.36	 1,016.987
-> FreqHz = 271053 / (10.58 + DACRate)


Sonic 2 DAC driver
------------------
[without gaps]
01	5,994	 10,634	  1.77	24,857.570
02	5,994	 11,593	  1.93	22,801.294
03	5,994	 12,550	  2.09	21,062.582
04	5,994	 13,505	  2.25	19,573.151
05	5,994	 14,464	  2.41	18,275.401
06	5,994	 15,442	  2.58	17,117.951
07	5,994	 16,400	  2.74	16,118.012
08	5,994	 17,349	  2.89	15,236.348
0C	5,994	 21,209	  3.54	12,463.360
10	5,994	 25,016	  4.17	10,566.653
20	5,994	 40,424	  6.74	 6,539.071
40	5,994	 71,112	 11.86	 3,717.170
80	4,384	 96,853	 22.09	 1,996.163
C0	3,014	 97,425	 32.32	 1,364.305
FF	2,312	 98,042	 42.41	 1,039.954
00	5,994	255,585	 42.64	 1,034.237
-> FreqHz = 275173 / (10.07 + DACRate)

[with gaps]
01	5,994	 11,152	  1.86	23,702.959
02	5,994	 12,146	  2.03	21,763.165
03	5,994	 13,140	  2.19	20,116.849
04	5,994	 14,163	  2.36	18,663.800
05	5,994	 15,155	  2.53	17,442.125
06	5,994	 16,181	  2.70	16,336.160
07	5,994	 17,175	  2.87	15,390.707
08	5,994	 18,178	  3.03	14,541.501
0C	5,994	 22,193	  3.70	11,910.756
10	5,994	 26,203	  4.37	10,087.982
20	5,994	 42,327	  7.06	 6,245.078
40	5,994	 74,027	 12.35	 3,570.797
80	4,384	100,639	 22.96	 1,921.068
C0	3,014	100,638	 33.39	 1,320.748
FF	2,312	 99,898	 43.21	 1,020.633
00	5,994	259,097	 43.23	 1,020.218
-> FreqHz = 265473 / (10.20 + DACRate)


Data East DAC driver
--------------------
01	5,309	 14,714	  2.77	15,911.846
02	5,309	 15,768	  2.97	14,848.231
03	5,309	 16,817	  3.17	13,922.037
04	5,309	 17,866	  3.37	13,104.607
05	5,309	 18,906	  3.56	12,383.735
06	5,309	 19,951	  3.76	11,735.096
07	5,309	 21,006	  3.96	11,145.716
08	5,309	 22,077	  4.16	10,605.014
0C	5,309	 26,261	  4.95	 8,915.384
10	5,309	 30,457	  5.74	 7,687.129
20	5,309	 47,284	  8.91	 4,951.504
40	5,309	 80,941	 15.25	 2,892.562
80	5,309	148,154	 27.91	 1,580.294
C0	5,309	215,537	 40.60	 1,086.249
FF	5,309	281,543	 53.03	   831.585
00	5,309	282,774	 53.26	   827.965
-> FreqHz = 222766 / (13 + prim)

All formulas are accurate enough that the difference between tested and recauculated (DAC samples per VGM sample) is <= 0.2 for all rates. (except for Sonic 2 with gaps)

The Sonic 2 driver does all work on the Z80, so it needs to interrupt the DAC stream every frame to do update the music engine. [without gaps] means, that I removed the small delays that occur because of this, [with gaps] means that I just took the approximate frequency including these delays.


The next update of SMPSPlay and DEPlay with feature perfect DAC rates. (For SMPSPlay they're adjustable via ini-files.)

[flamewing]

For what is worth, the DAC playback loops in the S3, S&K and S3D sound drivers take the exact same number of cycles as the equivalent code in the S2 driver, so the formula without gaps should be equal for both.

[Andlabs]

Sega was stupid and told developers that they needed to halt the Z80 when reading controllers and performing DMA, so all official games have interrupted DAC. Plus, DAC is played with a CPU loop, and some things (like GEMS) don't even use a loop... this is to say, this is great, but IDK how far this method can be pushed

[ValleyBell]

Something I should note is, that I made sure that the sound driver has absolutely nothing to do but playing a DAC sound every few seconds.

The few drivers I checked had a quite consistent stream of DAC data. They just stopped it for a few samples every frame (both Sonic 2 and Data East). Sonic 1 didn't interrupt the stream at all.
The method worked well with 2 sound engines, and I think it should also work with other ones. (Maybe in some distant future, we can listen to Ristar songs with clean samples.)

btw: I did the Sonic 2 [with gaps] table just to see the differences between the averaged and 'real' frequencies. Don't take it too serious.

[Sik]

What about the differences between NTSC and PAL? There should be timing differences even with a busy loop, since the Z80 speed also changes between both systems.

[ValleyBell]

NTSC vs. PAL is actually a good idea, so I made another test.
Also I recalculated the jman2050 and Sonic 1 DAC frequencies with removed gaps. (The gaps had ~6 samples in Sonic 1, but ~30 in Sonic 2.)

jman2050's DAC driver
---------------------
01	8,238	 11,811	  1.43	30,759.106
02	8,238	 13,151	  1.60	27,624.956
03	8,238	 14,486	  1.76	25,079.097
04	8,238	 15,804	  1.92	22,987.585
05	8,238	 17,108	  2.08	21,235.434
06	8,238	 18,424	  2.24	19,718.617
07	8,238	 19,768	  2.40	18,377.975
08	8,238	 21,082	  2.56	17,232.511
0C	8,238	 26,330	  3.20	13,797.790
10	8,238	 31,618	  3.84	11,490.158
20	8,238	 52,786	  6.41	 6,882.427
30	8,238	 73,877	  8.97	 4,917.577
40	8,082	 93,281	 11.54	 3,820.887
80	4,278	 93,219	 21.79	 2,023.834
C0	2,908	 93,158	 32.04	 1,376.616
FF	2,212	 93,169	 42.12	 1,047.013
00	8,238	349,322	 42.40	 1,040.003
-> FreqHz = 275320 / (7.96 + DACRate)
 Calculating with (8 + DACRate) is still a good approximation, (DACSmpl/VGMSmpl variance is <0.2 for rates <= 80)


Sonic 1 DAC driver
------------------
01	8,236	 15,278	  1.86	23,773.243
02	8,236	 16,670	  2.02	21,788.098
03	8,236	 18,116	  2.20	20,048.995
04	8,236	 19,437	  2.36	18,686.402
05	8,236	 20,697	  2.51	17,548.804
06	8,236	 22,112	  2.68	16,425.814
07	8,236	 23,355	  2.84	15,551.599
08	8,236	 24,672	  3.00	14,721.449
0C	8,236	 30,030	  3.65	12,094.825
10	8,236	 35,194	  4.27	10,320.157
20	8,236	 56,326	  6.84	 6,448.312
40	8,236	 98,566	 11.97	 3,684.918
80	8,236	182,903	 22.21	 1,985.794
C0	5,710	185,336	 32.46	 1,358.673
FF	4,354	185,301	 42.56	 1,036.214
00	8,236	351,896	 42.73	 1,032.145
-> FreqHz = 275300 / (10.68 + DACRate)

[PAL version]
01	8,236	 15,550	  1.89	23,357.402
02	8,236	 16,841	  2.04	21,566.867
03	8,236	 18,207	  2.21	19,948.789
04	8,236	 19,561	  2.38	18,567.946
05	8,236	 20,910	  2.54	17,370.043
06	8,236	 22,242	  2.70	16,329.808
07	8,236	 23,513	  2.85	15,447.097
08	8,236	 24,906	  3.02	14,583.137
0C	8,236	 30,227	  3.67	12,015.999
10	8,236	 35,577	  4.32	10,209.056
20	8,236	 56,783	  6.89	 6,396.414
30	8,236	 78,154	  9.49	 4,647.332
40	8,236	 99,445	 12.07	 3,652.347
80	8,236	184,592	 22.41	 1,967.624
C0	6,806	223,024	 32.77	 1,345.795
FF	5,192	222,947	 42.94	 1,027.003
00	8,236	355,891	 43.21	 1,020.559
-> FreqHz = 272853 / (10.68 + DACRate)


Sonic 2 DAC driver
------------------
[without gaps]
01	5,994	 10,634	  1.77	24,857.570
02	5,994	 11,593	  1.93	22,801.294
03	5,994	 12,550	  2.09	21,062.582
04	5,994	 13,505	  2.25	19,573.151
05	5,994	 14,464	  2.41	18,275.401
06	5,994	 15,442	  2.58	17,117.951
07	5,994	 16,400	  2.74	16,118.012
08	5,994	 17,349	  2.89	15,236.348
0C	5,994	 21,209	  3.54	12,463.360
10	5,994	 25,016	  4.17	10,566.653
20	5,994	 40,424	  6.74	 6,539.071
40	5,994	 71,112	 11.86	 3,717.170
80	4,384	 96,853	 22.09	 1,996.163
C0	3,014	 97,425	 32.32	 1,364.305
FF	2,312	 98,042	 42.41	 1,039.954
00	5,994	255,585	 42.64	 1,034.237
-> FreqHz = 275680 / (10.09 + DACRate)

[with gaps]
01	5,994	 11,152	  1.86	23,702.959
02	5,994	 12,146	  2.03	21,763.165
03	5,994	 13,140	  2.19	20,116.849
04	5,994	 14,163	  2.36	18,663.800
05	5,994	 15,155	  2.53	17,442.125
06	5,994	 16,181	  2.70	16,336.160
07	5,994	 17,175	  2.87	15,390.707
08	5,994	 18,178	  3.03	14,541.501
0C	5,994	 22,193	  3.70	11,910.756
10	5,994	 26,203	  4.37	10,087.982
20	5,994	 42,327	  7.06	 6,245.078
40	5,994	 74,027	 12.35	 3,570.797
80	4,384	100,639	 22.96	 1,921.068
C0	3,014	100,638	 33.39	 1,320.748
FF	2,312	 99,898	 43.21	 1,020.633
00	5,994	259,097	 43.23	 1,020.218
-> FreqHz = 270978 / (10.50 + DACRate)


Data East DAC driver
--------------------
01	5,309	 14,714	  2.77	15,911.846
02	5,309	 15,768	  2.97	14,848.231
03	5,309	 16,817	  3.17	13,922.037
04	5,309	 17,866	  3.37	13,104.607
05	5,309	 18,906	  3.56	12,383.735
06	5,309	 19,951	  3.76	11,735.096
07	5,309	 21,006	  3.96	11,145.716
08	5,309	 22,077	  4.16	10,605.014
0C	5,309	 26,261	  4.95	 8,915.384
10	5,309	 30,457	  5.74	 7,687.129
20	5,309	 47,284	  8.91	 4,951.504
40	5,309	 80,941	 15.25	 2,892.562
80	5,309	148,154	 27.91	 1,580.294
C0	5,309	215,537	 40.60	 1,086.249
FF	5,309	281,543	 53.03	   831.585
00	5,309	282,774	 53.26	   827.965
-> FreqHz = 222865 / (13 + DACRate)

The NTSC Z80 clock is 3579545, the PAL clock is 3546895, so PAL should be 99.088% NTSC speed. The ratio of the Sonic 1 frequency base values is 272853 / 275300 = 99.11 %, which is quite close.